L1‎‎ Norm Based Data Analysis and Related Methods

(1632-1989)

  • Bijan Bidabad Professor, Economics and Chief Islamic Banking Advisor, Bank Melli, Iran
Keywords: L1 norm, Regression, Algorithm, Computer

Abstract

This paper gives a rather general view on the L1‎ norm criterion on the area of data analysis and related topics. We tried to cover all aspects of mathematical properties, historical development, computational algorithms, simultaneous equations estimation, statistical modeling, and application of the L1‎ norm in different fields of sciences.

References

N.N. Abdelmalek (1971) Linear approximation for a discrete point set and L1‎‎ solutions of overdetermined linear equations. J. ACM, 18, 41-47.

N.N. Abdelmalek (1974) On the discrete linear L1‎‎ approximation and L1‎‎ solutions of overdetermined linear equations. J. of Approx. Theory, 11, 38-53.

N.N. Abdelmalek (1975a) An efficient method for the discrete L1‎‎ approximation problem. Math. Comput., 29, 844-850.

N.N. Abdelmalek (1975b) Chebyshev solution of overdetermined system of linear equations. BIT, 15, 117-129.

N.N. Abdelmalek (1976) A computer program for the Chebyshev solution of overdetermined system of linear equations, Inter.J. Numer. Meth. in Engin. 10, 1197-1202.

N.N. Abdelmalek (1977a) Computing the strict Chebyshev solution of overdetermined linear equations, Math. Comp., 31, 974-983.

N.N. Abdelmalek (1977b) The discrete linear restricted Chebyshev approximation. BIT, 17, 249-261.

N.N. Abdelmalek (1980a) L1‎‎ solution of overdetermined systems of linear equations. ACM Trans. Math. Soft., 6, 220-227.

N.N. Abdelmalek (1980b) A Fortran subroutine for the L1‎‎ solution of overdetermined systems of linear equations. ACM Trans. Math. Soft., 6, 228-30.

S. Abou-Jaoude (1976a) Sur une condition necessaire et suffisante de L1‎‎-convergence pres que complete de l'estimateur de la partition fixe pour une densite. C. R. Acad. Sci. Paris, Ser. A 283, 1107-1110.

S. Abou-Jaoude (1976b) Sur la convergence L1‎‎ at Ll de l'estimateur de la partition aleatoire pour une densite. Ann. Inst. Henri Poincare, 12, 299-317.

S. Abou-Jaoude (1976c) Conditions necessaires et suffisantes de convergence L1‎‎ en probabilite de l'histogramme pour une densite. Ann. Inst. Henri Poincare, 12, 213-231.

J.F. Affleck-Graves, A.H. Money, K. Carter ( ) An evaluation of an alternative methods of estimating the beta coefficient in the market model. Univ. of Capetown, South Africa.

T. Amemiya (1982) Two stage least absolute deviations estimators. Econometrica, 50,3, 689-711.

T.W. Anderson (1962) Least squares and best unbiased estimates. Ann. of Math. Stat., 33, 266-272.

D.W. Anderson (1965) Linear programming time estimating equations. J. of Indus. Engin. 16, 136-138

D.H. Anderson (1975) Linear programming and the calculation of maximum norm approximations. Ph.D. thesis, Australian National University..

D.H. Anderson, M.R. Osborne (1976) Discrete linear approximation problems in polyhedral norms. Numer. Math. 26, 179-189.

D.H. Anderson, M.R. Osborne (1977a) Discrete nonlinear approximation problems in polyhedral norms, a Levenberg-like algorithm. Numer. Math. 28, 157-170.

D.H. Anderson, M.R. Osborne (1977b) Discrete nonlinear approximation in polyhedral norms. Numer. Math., 28, 143-156.

D. Anderson, W.L. Steiger (1982) A comparison of methods for discrete L1‎‎ curve-fitting. Tech. Rep. DCS-TR-96. Dept. of Comp. Sci., Hill center for the Math. Sci. Busch Campus, New Brunswick, N.J.

F.J. Anscombe (1976) Topics in the investigation of linear relations fitted by the method of least square (with discussion), J. of Royal Stat. Soc. B series, 1-52.

J. Antoch, A. Bartkowiak, J. Pekalska (1986) Computing L1‎‎ norm, a-trimmed and a-winsorized regression on the ODRA 1305 computer. Rep. N-159, Institute of Computer Science, Wroclaw Univ., Poland.

J. Antoch (1987) Variable selection in linear model based on trimmed least squares estimator. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 231-246.

H. Anton, C.S. Duris (1973) On an algorithm for best approximate solutions to Av=b in normed linear spaces. J. Approx. Theory 8, 133-141.

M. Aoki (1965) Successive generation of Chebyshev approximate solution. J. Basic Engin., 87, 17-22.

G. Appa, C. Smith (1973) On L1‎‎ and Chebyshev estimation.Math. Prog. 5, 73-87.

R.D. Armstrong, J.J. Elam, J.W. Hultz (1977) Obtaining least absolute value estimates for a two-way classification model.Comm. Stat., B6, 365-81.

R.D. Armstrong, E.L. Frome (1976a) A comparison of two algorithms for absolute deviation curve fitting. JASA, 71, 328-330.

R.D. Armstrong, E.L. Frome (1976b) The calculation of least absolute value estimates for two-way tables. Proc. of the statistical computing section, ASA, Washington D.C., 101-106.

R.D. Armstrong, E.L. Frome (1977) A special purpose linear programming algorithm for obtaining least absolute value estimates in a linear model with dummy variables. Comm. Stat., B6, 383-98.

R.D. Armstrong, E.L. Frome (1979) Least-absolute-value estimators for one-way and two-way tables. Naval Res. Log. Quart., 26, 79-96.

R.D. Armstrong, E.L. Frome, D.S. Kung (1979) A revised simplex algorithm for the absolute deviation curve fitting problem. Comm. Stat. B8, 175-190.

R.D. Armstrong, E.L. Frome, M.G. Sklar (1980) Linear programming in exploratory data analysis. J. of Educ. Stat., 5, 293-307.

R.D. Armstrong, J. Godfrey (1979) Two linear programming algorithms for the discrete L1‎‎ problem. Math. Comput., 33, 289-300.

R.D. Armstrong, J.W. Hultz (1977) An algorithm for a restricted discrete approximation problem in L1‎‎ norm. SIAM J. Numer. Anal., 14, 555-565.

R.D. Armstrong, D.S. Kung (1978) AS132: Least absolute value estimates for a simple linear regression problem. Appl. Stat., 27, 363-366.

R.D. Armstrong, D.S. Kung (1979) AS135: Mini-max estimates for a linear multiple regression problem. Appl. Stat., 93-100.

R.D. Armstrong, D.S. Kung (1980) An algorithm for a least absolute value regression problem with bounds on the parameters. Appl. Math. Comput. 7, 267-279.

R.D. Armstrong, D.S. Kung (1982a) An algorithm to select the best subset for a least absolute value regression problem. TIMS studies in the management sciences, 19, 67-80.

R.D. Armstrong, D.S. Kung (1982b) A dual algorithm to solve linear least absolute value problems. J. Oper. Res. Soc., 33, 931-936.

K.J. Arrow, M. Hoffenberg (1959) A time series analysis of interindustry demands. NHPC, Amsterdam. T.S. Arthanari, Y. Dodge (1981) Mathematical programming in statistics. John Wiley, Interscience division, New York.

W.C. Ashar, T.D. Wallace (1963) A sampling of minimum absolute deviations estimators. Oper. Res., 11, 747-752.

P. Assouad (1977) Un espace hypermetric non plongeable dans un espace L1‎‎. C. R. Acad. Sc. Paris, T. 285, Serie A, 361-363.

S. Baboolal, G.A. Watson (1981) Computational experience with an algorithm for discrete L1‎‎ approximation. Computing, 27, 245-252.

S.C. Banks, H.L. Taylor (1980) A modification to the discrete L1‎‎ linear approximation algorithm of Barrodale and Roberts.

SIAM J. on Scientific and Stat. Comput. 1, 187-190. G.D.I. Barr, J.F. Affleck-Graves, A.H. Money, M.L. Hart (1980a) Performance of a generalized algorithm for Lp-norm regression estimates. Tech. Rep. no. ALS-4, Aug., Univ. of Capetown, Dept. of Math. Stat. South Africa.

G.D.I. Barr, J.F. Affleck-Graves, A.H. Money, M.L. Hart (1980b) Lp norm estimation and the choice of p, a practical approach. Univ. of Capetown, Dept. of Math. Stat., Tech. Rep. no. ALS-3, July.

G.D.I. Barr, A.H. Money, J.F. Affleck-Graves, M.L. Hart (1980) Lp-norm estimation of a symmetric distribution. Univ. of Capetown, Dept. of Math. Stat., Tech. Rep. no. ALS-5, Oct..

G.D.I. Barr, A.H. Money, J.F. Affleck-Graves, M.L. Hart (1981a) Estimation of location for skewed data sets: a comparative study utilizing the data set published by Stigler. Univ. of Capetown, Dept. of Math. Stat., Tech. Rep. no. ALS-7, may.

G.D.I. Barr, A.H. Money, J.F. Affleck-Graves, M.L. Hart (1981b) Estimation of location for skewed data sets. Univ. of Capetown, Dept. of Math. Stat., Tech. Rep. no. ALS-6, April.

Barrodale (1967) Approximation in the L1‎‎ and Ll norms by linear programming. Ph.D. thesis, Univ. of Liverpool, Liverpool, England.

Barrodale (1968) L1‎‎ approximation and analysis of data. Appl. Stat. 17,51-57.

Barrodale (1970) On computing best L1‎‎ approximations. In A. Talbot, Approximation theory Academic Press, New York, 205-215.

Barrodale, C. Phillips (1975) Solution of an over-determined system of linear equations in Chebyshev norm. ACM Trans. on Math. Soft. 264-70.

Barrodale, M.J.D. Powell, F.D.K. Roberts (1972) The differential correction algorithm for rational Ll approximation. SIAM J. Num. Anal., 9, 493-503.

Barrodale, F.D.K. Roberts (1970) Application of mathematical programming to Lp approximation. In J.B. Rosen,O.L. Mangasarian, K. Ritter, Nonlinear programming Academic Press, New York, 447-64.

Barrodale, F.D.K. Roberts (1973) An improved algorithm for discrete L1‎‎ linear approximation. SIAM J. Numer. Anal., 10,839-848.

Barrodale, F.D.K. Roberts (1974) Algorithm 478: Solution of an overdetermined system of equations in the L1‎‎ norm. Comm. ACM, 17, 319-320.

Barrodale and F.D.K. Roberts (1977) Algorithms for restricted least absolute value estimation. Comm. Stat. B6(4), 353-363.

Barrodale and F.D.K. Roberts (1978) An efficient algorithm for discrete L1‎‎ linear approximation with linear constraints. SIAM J. Numer. Anal., 15, 603-611.

Barrodale and F.D.K. Roberts, C.R. Hunt (1970) Computing best Lp approximations by functions nonlinear in one parameter. Comp. J., 13, 382-386.

Barrodale, C. Phillips (1974) An improved algorithm for discrete Chebyshev linear approximation. Proc. 4th Manitoba conf. on numer. math. Univ. of Manitoba, Winnipeg, Manitoba, 177-190.

Barrodale, A. Young (1966) Algorithms for best L1‎‎ and Ll linear approximations on a discrete set. Numer. Math., 8, 295-306.

R.H. Bartels, A.R. Conn (1977) LAV regression: A special case of piecewise linear minimization. Comm. Stat., B6, 329-340.

R.H. Bartels, A.R. Conn (1980a) Linearly constrained discrete L1‎‎ problems. ACM Trans. Math. Soft. 6, 594-608.

R.H. Bartels, A.R. Conn (1980b) Algorithm 563, A program for linearly constrained discrete L1‎‎ problems. ACM trans. Math. Soft., 6, 609-614.

R.H. Bartels, A.R. Conn (1982) An approach to nonlinear L1‎‎ data fitting. In J.P. Hennart (ed.), Numerical analysis, Cocoyoc, Springer Verlag, 45-58.

R.H. Bartels, A.R. Conn, C. Charalambous (1978) On Cline's direct method for solving overdetermined linear system in the Ll sense, SIAM J. Numer. Anal., 15, 255-270.

R.H. Bartels, A.R. Conn, Y. Li (1987) Primal methods are better than dual methods for solving overdetermined linear systems in the Ll sense? Res. Rep. CS-87-12, Univ. of Waterloo, Comp. Sci. Dept.

R.H. Bartels, A.R. Conn, J. Sinclair (1976) The L1‎‎ solution to an overdetermined linear system. Proc. 9th Ann. Symp. Interface Stat. In C.D.C. Hoaglin (ed.) Boston, Prindle, Weber and Schmidt Inc., 120-7.

R.H. Bartels, A.R. Conn, J. Sinclair (1978) Minimization technique for piecewise differentiable functions: The L1‎‎ solution to an overdetermined linear system. SIAM J. Numer. Anal. 15, 224-241.

R.H. Bartels, G.H. Golub (1968a) Algorithm 328: Chebyshev solution to an overdetermined linear system. Comm. ACM 11, 428-430. R.H. Bartels, G.H. Golub (1968b) Stable numerical methods for obtaining the Chebyshev solution to an overdetermined system of equations, Comm. ACM, 11, 401-406.

R.H. Bartels, G.H. Golub (1969) The simplex method of linear programming using LU decomposition. Comm. ACM, 12, 266-268.

G.W. Bassett (1973) Some properties of the least absolute error estimator. Ph.D. thesis. Dept. of Econ., Univ. of Michigan.

G.W. Bassett, Jr. (1987) A p-subset property of L1‎‎ and regression quantile estimates. Work. Pap., Dept. of Economics, Univ. of Illinois-Chicago.

G.W. Bassett, Jr. (1988a) A p-subset property of L1‎‎ and regression quantile estimates. CSDA, 6(3), 297-304.

G.W. Bassett, Jr. (1988b) A property of the observations fit by the extreme regression quantiles. CSDA, 6(4), 353-360.

G.W. Bassett, Jr., R. Koenker (1978) Asymptotic theory of least absolute error regression. JASA, Sep., 73, no. 363, 618-22.

G.W. Bassett, Jr., R. Koenker (1982) An empirical quantile function for linear models with iid errors. JASA, 77, no. 378, 407-415.

J. Bejar (1956) Regression en mediana y la programacion lineal, Trabajos de Estadistica 7, 141-58.

J. Bejar (1957) Calculo practico de la regression en mediana Trabajos de Estadistica, 8, 157-173.

Bijan, Bidabad (1984a) Goal programming, optimal decision process with different priorities and its computer program. Plan and Budget Ministry, Tehran, Iran.

Bijan, Bidabad (1984b) Determining industrial location for 1992 by using goal programming method. Plan and Budget Ministry, Tehran, Iran.

Bijan, Bidabad (1984c) Zero-one programming, optimal decision making with bivalent variables and its computer program. Plan and Budget ministry, Tehran, Iran.

Bijan, Bidabad (1987a) Least absolute error estimation. The First International Conference on Statistical Data Analysis Based on the L1‎‎ norm and Related Methods, Neuchatel, Switzerland. http://www.bidabad.com/doc/lae-I.pdf

Bijan, Bidabad (1987b) Least absolute error estimation, part II. Submitted to the First International Conference on Statistical Data Analysis Based on the L1‎‎ norm and Related Methods, Neuchatel, Switzerland. http://www.bidabad.com/doc/lae-II.pdf

Bijan, Bidabad (1988a) A proposed algorithm for least absolute error estimation. Proc. of the Third Seminar of Mathematical Analysis. Shiraz Univ., 24-34, Shiraz, Iran.

Bijan, Bidabad (1988b) A proposed algorithm for least absolute error estimation, part II. Proc. of the Third Seminar of Mathematical Analysis, Shiraz Univ., 35-50, Shiraz, Iran.

Bijan, Bidabad (1989a) Discrete and continuous L1‎‎ norm regressions, proposition of discrete approximation algorithms and continuous smoothing of concentration surface, Ph.D. thesis, Islamic Azad Univ., Tehran, Iran. http://www.bidabad.com/doc/L1-norm-thesis-en.pdf

Bijan, Bidabad (1989b) Discrete and continuous L1‎‎ norm regressions, proposition of discrete approximation algorithms and continuous smoothing of concentration surface, Ph.D. thesis, Islamic Azad Univ., Tehran, Iran. Farsi translation. http://www.bidabad.com/doc/L1-norm-thesis-fa.pdf

Bijan Bidabad (2005). L1 norm based computational algorithms. http://www.bidabad.com/doc/l1-article6.pdf

Bijan Bidabad (2005). L1 norm solution of overdetermined system of linear equations. http://www.bidabad.com/doc/l1-article5.pdf

Bijan Bidabad (2005). L1 norm based data analysis and related methods. http://www.bidabad.com/doc/l1-articl1.pdf

Bijan Bidabad (2005). New algorithms for the L1 norm regression. http://www.bidabad.com/doc/l1-article2.pdf

Bijan Bidabad (2005). Comparative study of the L1 norm regression algorithms. http://www.bidabad.com/doc/l1-articl3.pdf

Bijan Bidabad (2005). Continuous L1 norm estimation of Lorenz curve. http://www.bidabad.com/doc/l1-articl4.pdf

Bijan Bidabad (1993). Estimating Lorenz curve for Iran by using continuous L1 norm estimation, Economics and Management Journal, Islamic Azad University, No. 19, winter 1993, pp. 83-101. http://www.bidabad.com/doc/iraninc-l1.pdf

Bijan Bidabad (2005). Continuous L1 norm estimation of Lorenz curve when probability density function is known.

Bijan Bidabad (2005). USA Income distribution counter-business-cyclical trend (Estimating Lorenz curve using continuous L1 norm estimation). First meeting of the Society for the Study of Economic Inequality (ECINEQ), Palma de Mallorca, Spain, July 20-22, 2005.

http://www.uib.es/congres/ecopub/ecineq/general.html

http://www.uib.es/congres/ecopub/ecineq/papers/039Bidabab.pdf

http://www.bidabad.com/doc/estimating-lorenz-us.pdf

Bijan Bidabad, Hamid Shahrestani. (2008) An implied inequality index using L1 norm estimation of Lorenz curve. Global Conference on Business and Finance Proceedings. Mercedes Jalbert, managing editor, ISSN 1931-0285 CD, ISSN 1941-9589 Online, Volume 3, Number 2, 2008, The Institute for Business and Finance Research, Ramada Plaza Herradura, San Jose, Costa Rica, May 28-31, 2008, pp. 148-163.

http://www.bidabad.com/doc/L1-Implied-inequality-index-4.pdf

http://www.theibfr.com/archive/ISSN-1941-9589-V3-N2-2008.pdf

Global Journal of Business Research, Vol. 4, No. 1, 2010, pp.29-45.

http://www.bidabad.com/doc/SSRN-id1631861.pdf

R. Blattberg, T. Sargent (1971) Regression with non-Gaussian stable disturbances: some sampling results, Econometrica, May, 501-510.

P. Bloomfield, W. Steiger (1980) Least absolute deviations curve fitting. SIAM J. Sci. Stat. Comput. 1, 290-301.

P. Bloomfield, W. Steiger (1983) Least absolute deviations: theory, applications and algorithms. Birkhauser, Boston.

P.T. Boggs (1974) A new algorithm for Chebyshev solution of overdetermined linear system. Math. Comp., 28, 203-218.

M.S. Borowsky (1976) Algorithms for solving the dual problem for Av=b with varying norms. J. Approx. Theory, 17, 107-114.

R.J. Boscovich (1757) De litteraria expeditione per pontificiam ditionem, et synopsis amplioris operis..., 'Bononiensi' scientiarum et artum instituto atque academia commetarii, vol.4, 353-396. Reprinted with a Serbo-Croatian translation by N. Cubranic, Institute of higher geodesy, Univ. of Zagreb 1961.

R.J. Boscovich (1760) De recentissimis graduum dimensionibus et figura, ac magnitudine terrae inde derivanda. Philosophiae recentioris, a benedicto stay in Romano archigynasis publico eloquentare professore, vesibus traditae, libri X, cum adnotianibus et supplementas P. Rugerii Joseph Boscovich, S.J., 2, 406-426.

A.L. Bowley (1902) Methods of representing the statistics of wages and other groups not fulfilling the normal law of error, II: applications to wage statistics and other groups. J. of the Roy. Stat. Soc., 65, 331-54.

A.L. Bowley (1928) F.Y. Edgeworth's contributions to mathematical statistics. London, Roy. Stat. Soc..

G.E.P. Box, G.C. Tiao (1962) A further look at robustness vi Bayes' theorem. Biometrika, 419-432.

D. Bradu (1987a) L1‎‎ fit, median polish and conjugate gradients. CSIR Tech. Rep. TWISK 509, National Res. Inst. for Math Sci. CSIR, Pretoria.

D. Bradu (1987b) An n-median polish algorithm. CSDA, 5, 327-336.

M. Brannigan, S.A. Gustafson (1987) H-sets in convex programming and constrained approximation. Work. Pap., Rogaland Univ., Stavanger.

H.D. Brecht (1976) Regression methodology with observation errors in the explanatory variables. Decis. Sci. 7, 57-65.

J.J. Brennan, L.M. Seiford (1987) Linear programming and L1‎‎ approximation using the method of vanishing Jacobians. CSDA, 5, 263-276.

B.M. Brown (1980) Median estimates in a simple linear regression. Australian J. of Stat., 22, 154-165.

B.M. Brown, T.P. Hettmansperger (1987) Invariant tests in bivariate models and the L1‎‎-criterion. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods, NHPC, 333-344.

C. Bruen (1938) Methods for the combination of observations modal points or most lesser-deviations, mean loci or least squares, and mid point of least range or least greatest-deviation. Metron 13, 61-140.

J.J. Buckley, A.H. Kvanli (1981) The MAD estimator: when is it non-linear? Applications to two-way design models. Comm. Stat. A10, 2581-2590.

Burgoyne (1965) Polynomial approximation by Edgeworth's method. Univ. of London.

S. Busovaca (1985) Handling degeneracy in a nonlinear L1‎‎ algorithm. Ph.D. thesis, Univ. of Waterloo, Waterloo, Ontario.

V.A. Cabot, R.L. Francis, M.A. Stary (1970) A network flow solution to a rectilinear distance facility location problem. AIIE trans., vol. 2.

P.H. Calamai, A.R. Conn (1987) A projected Newton method for Lp norm location problems. Math. Prog. 38, 75-109.

G. Le Calve (1987) L1‎‎-embeddings of data structure (I,D). In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. 195-202., NHPC.

J.M. Chambers (1977) Computational methods for data analysis Wiley, New York.

D.G. Champernowne (1953) A model of income distribution. Economic J., 63, 318-351.

Charnes, W.W. Cooper, R.O. Ferguson (1955) Optimal estimation of executive compensation by linear programming. Manag. Sci. 1, 138-151.

Charalambous, A.R. Conn (1978) An efficient method to solve the minimax problem directly. SIAM J. Numer. Anal., 15, no.1, 162-187.

Charalambous (1979) On conditions for optimality of nonlinear L1‎‎ problem. Math. Prog. 17, 123-135.

E.W. Cheney (1966) Introduction to approximation theory McGraw-Hill, New York.

W. Cheney, A.A. Goldstein (1958) Note on a paper by Zuhovickii concerning the Chebyshev problem for linear equations. SIAM J. Numer. Anal. 6, 233-239.

J.A. Chisman (1966) Using linear programming to determine time standards, J. Ind. Engin. 17, 189-191.

Su Chun (1987) Some results on the Lp-convergence (pr1) of U-statistics. CSDA, 5, 321-326.

D.I. Clarke (1981) Finite algorithms for linear optimization problems. Ph.D. thesis, Australian National Univ., Canberra.

F.H. Clarke (1983) Optimization and nonsmooth analysis. Wiley, New York.

J.F. Claerbout, F. Muir (1973) Robust modeling with erratic data. Geophysics 38, 826-844.

A.K. Cline (1970) Uniform approximation as a limit of approximations. Ph.D. thesis, Univ. of Michigan, Ann, Arbor, Michigan.

A.K. Cline (1972) Rate of convergence of Lawson's algorithm, Math. of Comp., 26, 167-176.

A.K. Cline (1976) A descent method for the uniform solution of overdetermined system of linear equations. SIAM J. Numer. Anal., 13, 293-309.

K.O. Cogger (1979) Time-series analysis and forecasting with an absolute error criterion. TIMS Studies in Manag. Sci., S. Markidakis, S.C. Wheelwright (eds.) NHPC, 189-201.

T.F. Coleman (1978) A note on 'New algorithms for constrained minimax optimization', Math. Prog., 15, 239-242.

A.R. Conn (1975) Optimization of microwave networks. IEEE Tran. on Microwave Theory and Techniques, Oct., 834-838.

A.R. Conn (1976) linear programming via a nondifferentiable penalty function. SIAM J. Numer. Anal., 13, 145-154.

A.R. Conn (1984) Nonlinear programming, exact penalty functions and projection techniques for non-smooth functions.

Tech. Rep. no. CS-84-26. Dept. of Comp. Sci., Univ. of Waterloo, Ontario, Canada.

A.R. Conn, N.I.M. Gould (1987) An exact penalty function for semi-infinite programming. Math. Prog., 37, 19-40.

B. Cornell, J.K. Dietrich (1978) Mean-absolute-deviation versus least-squares regression estimation of beta coefficients. J. of Financial and Quantitative analysis 13, 123-131.

F. Critchley (1980) Optimal norm characterizations of multidimensional scaling methods and some related data analysis problem. In E. Diday et al (eds.) Data analysis and informatics, NHPC, 209-229.

D.C. Crocker (1969) Linear programming technique in regression analysis, the hidden danger. A.I.E.E. Trans., 1, 112-126.

M. Csorgo, L. Horvath (1987) Asymptotics for Lp-norms of naive estimators of densities. Tech. Rep. series of Lab. Res. Stat. Prob. no. 96, Carleton Univ., Ottawa, Canada.

M. Csorgo, L. Horvath (1988) Asymptotics for Lp-norms of kernel estimators of densities. CSDA, 6(3), 241-250.

M. Csorgo, E. Gombay, L. Horvath (1987) Asymptotics for Lp-norms of kernel estimators of density under random censorship. Tech. Rep. series of Lab. Res. Stat. Prob., Carleton Univ., Ottawa, Canada.

M. Davies (1976) Linear approximation using the criterion of least total deviations. J. Roy. Stat. Soc. B29, 101-109.

J. Descloux (1963) Approximation in Lp and Chebyshev approximations. SIAM J. 11, 1017-1026.

L. Devroye (1983) The equivalence of weak, strong and complete convergence in L1‎‎ for kernel density estimates. Ann. of Stat., 11, 896-904.

L. Devroye (1985) A note on the L1‎‎ consistency of variable kernel estimates. Ann. Stat., 13, 1041-1049.

L. Devroye, L. Gyorfi (1985) Non parametric density estimation, the L1‎‎ view, Wiley, New York.

L. Devroye, T.J. Wagner (1979) The L1‎‎ convergence of kernel density estimates. Ann. Stat., 7, 1136-1139.

L. Devroye, T.J. Wagner (1980) On the L1‎‎ convergence of kernel estimators of regression functions with applications in discrimination. Z. Wahrsch, verw. Gebiete, 51, 15-25.

T.E. Dielman (1984) Least absolute value estimation in regression models: An annotated bibliography. Comm. Stat. 13, 513-41.

T. Dielman, R. Pfaffenberger (1982) LAV (Least Absolute Value) estimation in linear regression: A review, TIMS studies in the Manag. Sci., 19, 31-52.

T. Dielman, R. Pfaffenberger (1984) Computational algorithms for calculating least absolute value and Chebyshev estimates for multiple regression. Amer. J. Math. Manag. Sci., 4, 169-197.

J.J. Dinkel, R.C. Pfaffenberger (1981) Constrained L1‎‎ estimation via geometric programming. European J. Oper. Res. 7, 299-305.

P.J. Dhrymes (1978) Mathematics for econometrics. Springer-Verlag, New York.

Y. Dodge (1984) Robust estimation of regression coefficients by minimizing a convex combination of least squares and least absolute deviations. Comp. Stat. Quart., 1, 139-153.

Y. Dodge (1987) An introduction to statistical data analysis L1‎‎-norm based. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. Reprinted in CSDA, 5, 239-254.

Y. Dodge, J. Jureckova (1987) Adaptive combination of least squares and least absolute deviations estimators. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎-norm and related methods. 275-287, NHPC.

Y. Dodge, J. Jureckova (1988) Adaptive combination of M-estimator and L1‎‎-estimator. In Y. Dodge, V.V. Fedorov, H.P. Wynn (eds.) Optimal design and analysis of experiments, 167-176, NHPC.

W. Domschke, A. Drext (1984) An international bibliography for location-and layout-planning. Instut fuer Unternehmensforschung und Informatik, Hochschule der Eundeswehr, postach 700822 D-2000 Hamburg 70.

E.L. Dougherty, S.T. Smith (1966) The use of linear programming to filter digitized map data. Geophysics, 31, 253-259.

Z. Drezner, G.O. Wesolowsky (1978) A trajectory method for the optimization of multi-facility location problem with Lp distances. Manag. Sci., 24, no. 14, 1507-1514.

A.F. Dufton (1928) Correlation. Nature, 121, 866.

J. Dupacova (1987a) Asymptotic properties of restricted L1‎‎-estimates of regression. Feb., WP-87-18, Work. Pap. Inter. Inst. for App. Sys. Analysis, A-2361 Laxenburg, Austria.

J. Dupacova (1987b) Asymptotic properties of restricted L1‎‎-estimates of regression. In Y. Dodge (ed.) statistical data analysis based on the L1‎‎-norm and related methods. 263-274, NHPC.

C.S. Duris, V.P. Sreedharan (1968) Chebyshev and L1‎‎ solutions of linear equations using least squares solutions. SIAM J. Num. Anal., 5, 491-505.

C.S. Duris, M.G. Temple (1973) A finite step algorithm for determining the "strict" Chebyshev solution to Ax=b. SIAM J. Num. Anal. 10, 690-699.

R. Dutter (1977) Numerical solution of robust regression problems, computational aspects, a comparison. J. of Stat. Computation and Simulation, 5, 207-238.

R. Dutter (1987) A Fortran program for robust and bounded influence regression. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. 139-144. NHPC.

F.Y. Edgeworth (1883) The method of least squares. Philosophical Magazine, 16, 360-375.

F.Y. Edgeworth (1887a) On observations relating to several quantities. Hermathena, 6, 279-285.

F.Y. Edgeworth (1887b) A new method of reducing observations relating to several quantities. Philosophical Magazine, 24, 222-223.

F.Y. Edgeworth (1888) On a new method of reducing observation relating to several quantities. Philosophical Magazine, 25, 184-191.

F.Y. Edgeworth (1902) Method of representing statistics of wage and other groups not fulfilling the normal law of error, I: mathematical considerations. J. Roy. Stat. Soc., 65, 325-331.

F.Y. Edgeworth (1923) On the use of medians for reducing observations relating to several quantities. Philosophical Magazine, 6th series, 46, 1074-1088.

Eisenhart (1961) Boscovich and the combination of observations. Ch. 9 of Whyte (1961, 200-212) reprinted in Kendall and Plackett (1977) studies in the history of statistics and probability, vol.II, Charles Griffin and Co. Ltd., High Wycombe 88-100.

H. Ekblom (1973a) A note on nonlinear median estimators. JASA, 68, 431-2.

H. Ekblom (1973b) Calculation of linear best Lp-approximations. BIT 13, 292-300.

H. Ekblom (1974) Lp methods for robust regression. BIT 14, 22-32.

H. Ekblom (1987) The L1‎‎ estimate as limiting case of an Lp-or Huber estimate. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 109-116.

H. Ekblom, S. Henriksson (1969) Lp-criteria for the estimation of location parameters. SIAM J. Appl. Math., 17, 1130-1141.

R.A. El-Attar, M. Vidyasagar, S.R.K. Dutta (1976) Optimality conditions for L1‎‎-norm minimization Proc. 19th midwest symp. on circuits and systems, 272-275.

R.A. El-Attar, M. Vidyasagar, S.R.K. Dutta (1979) An algorithm for L1‎‎-norm minimization with application to nonlinear L1‎‎-approximation. SIAM J. Numer. Anal., 16, 70-86.

J.E. Estienne (1926-28) Introduction a une theorie rationnelle des erreurs d'observation. Revue d'artillerie 97(1926), 421-441; 98(1928), 542-562; 100(1927), 471-487.

R.C. Fair (1974) On the robust estimation of econometric models. Ann. Econ. Soc. Measurement, 3, 667-77.

Fama (1965) The behavior of stock market prices. J. Bus. 38, 34-105.

E.F. Fama, R. Roll (1971) Parameter estimates for symmetric stable distributions. JASA 66, 331-338.

R.W. Farebrother (1985) Unbiased L1‎‎ and Ll estimation. Comm. Stat., A, 14,1941-1962.

R.W. Farebrother (1987a) The theory of committee decisions and the double median method. CSDA, 5, 437-442.

R.W. Farebrother (1987b) The historical development of the L1‎‎ and Ll estimation procedures. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 37-64.

R.W. Farebrother (1987c) A simple recursive procedure for the L1‎‎ norm fitting of a straight line. Work. Pap., Dept. of Econometrics and Social Stat. Univ. of Manchester, Manchester, M13 9PL, UK.

R.W. Farebrother (1987d) Mechanical representation of the L1‎‎ and L2 estimation problems. In Y. Dodge (1987) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 455-464.

R.W. Farebrother (1987e) A remark on AS108: multiple linear regression with minimum sum of absolute errors. Appl. Stat., 36, no. 1, 118.

D.W. Fausett, J.H. Weber (1978) Mass spectral pattern recognition via techniques of mathematical programming. Analytical Chemistry, 50, 722-731.

V.V. Fedorov (1987) Various discrepancy measures in model testing (two competing regression models). In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC, 357-366.

B. Fichet (1987a) The role played by L1‎‎ in data analysis. In Y. Dodge (ed.), Statistical data analysis based on the L1‎‎ norm and related methods. NHPC, 185-194.

B. Fichet (1987b) Lp-space in data analysis. First Conference of the International Federation of Classification Societies. Aachen.

W.D. Fisher (1961) A note on curve fitting with minimum deviations by linear programming. JASA, 11, 359-362.

R. Fletcher (1981) Numerical experiments with an exact L1‎‎ penalty function method. In O.L. Mangasarian, R.R. Meyer, S.M. Robinson (eds.) Nonlinear programming 4. Academic Press, New York, 99-129.

R. Fletcher (1984) An L1‎‎ penalty method for nonlinear constraints. Rep. NA/81, Dept. of Math. Sci., Univ. of Dundee, Scotland.

R. Fletcher, J.A. Grant, M.D. Hebden (1971) The calculation of best Lp approximations. Comp. J., 14, 276-279.

R. Fletcher, J.A. Grant, M.D. Hebden (1974a) Linear minimax approximation as the limit of best Lp approximation. SIAM J. Numer. Anal., 11, 123-136.

R. Fletcher, J.A. Grant, M.D. Hebden (1974b) The continuity and differentiability of parameters of best linear Lp approximations. J. Approx. Theory, 10, 69-73.

A.B. Forsythe (1972) Robust estimation of straight line regression coefficients by minimizing pth power deviations. Technometrics, 14, 159-166.

C.R. Forth (1974) Robust estimation techniques for population parameters and regression coefficients. M.S. thesis. Air Force Institute of Technology, Wright-Patterson,AFB, Ohio.

R. Fourer (1985a) A simplex algorithm for piecewise-linear programming I: derivation and proof. Math. Prog., 33, 204-233.

R. Fourer (1985b) A simplex algorithm for piecewise-linear programming II: Finiteness, feasibility and degeneracy. Tech. Rep., 85-03 (revised), Dept. of Ind. Engin. and Manag. Sci., The Tech. Inst., Northwestern Univ., Evanston, Illinois.

R. Fourer (1986) A simplex algorithm for piecewise-linear programming III: Computational analysis and applications. Tech. Rep., 86-03, Dept. of Ind. Engin. and Manag. Sci., The Tech. Inst., Northwestern Univ., Evanston, Illinois.

J.B.I. Fourier (1824) Solution d'une question particuliere au calcul des inegalites, second extrait. Histoire de l'academie des sciences pour 1824, 47-55. Reprinted in oeuvres de Fourier, 2. Paris, 1980, Gauthier-Villars, 325-328.

E.L. Frome, R.D. Armstrong (1977) A robust procedure for estimating the trend-cycle component of an economic time series. In D. Hogben (ed.) Proc. of the tenth symposium on the interface. Gaithersburg: National Bureau of Standards.

E.L. Frome, G.J. Yakatan (1980) Statistical estimation of the pharmokinetic parameters in the one compartment open model. Comm. Stat. B9, 201-222.

Gaivoronski (1987) Numerical techniques for finding estimates which minimize the upper bound of absolute deviations. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 247-262.

Galilei (1632) Dialogo dei massimi sistemi. J.S. GaLpin (1986) Robust and bounded influence regression. National Res. Inst. for Math. Sci. WNNR, CSIR, TWISK, Pretoria, South Africa.

J.S. GaLpin, D.M. Hawkins (1987) Methods of L1‎‎ estimation of a covariance matrix. CSDA, 5, 305-319.

C.W. Ganger, D. Orr (1972) Infinite variance and research strategy in time series analysis. JASA 67, 275-285.

C.B. Garcia, F.G. Gould (1983) An application of homotopy to solving linear programs. Math. Prog. 27, 263-282.

C.F. Gauss (1809) Theoria motus corporum coelestium. In F. Perthes, I.H. Besser, Sectionbus conicis solem ambientium, Hamburg. Reprinted in his werke, vol. 7, F. Pethes, Gotha 1871. English translation by C.H. Davis, Little, Brown and Co., Boston, 1857. Reprinted by Dover Pub. New York, 1963.

J.E. Gentle (1977) Least absolute value estimation: an introduction. Comm. Stat., B6, 313-28.

J.E. Gentle, T.A. Hansen (1977) Variable selection under L1‎‎, Proceedings of the statistical computing section A.S.A., 228-230.

J.E. Gentle, W.J. Kennedy, V.A. Sposito (1976) Properties of the L1‎‎ estimate space. Proc. Stat. Comp. section A.S.A. 163-164.

J.E. Gentle, W.J. Kennedy, V.A. Sposito (1977) On least absolute values estimation. Comm. Stat., A6, 839-845.

J.E. Gentle, S.C. Narula, V.A. Sposito (1987) Algorithms for unconstrained L1‎‎ linear regression. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 83-94.

J.E. Gentle, V.A. Sposito (1976) On the invariance of certain estimators. Bull. Austral. Math. Soc., vol.14, 405-408.

J.E. Gentle, V.A. Sposito, W.J. Kennedy (1977) On some properties of L1‎‎ estimators. Math. Prog., 12, 139-140.

J.E. Gentle, V.A. Sposito, S.C. Narula (1988) Algorithms for unconstrained L1‎‎ simple linear regression. CSDA, 6(4), 335-340.

W.M. Gentleman (1965) Robust estimation of multivariate location by minimizing p-th power deviations. Ph.D. thesis, Princeton Univ., New Jersey.

J. Gilsinn, K. Hoffman, R.H.F. Jackson, E. Leyendecker, P. Saunder, D. Shier (1977) Methodology and analysis for comparing discrete L1‎‎ approximation codes., Comm. Stat., B6, 399-413.

F.R. Glahe, J.G. Hunt (1970) The small sample properties of simultaneous equation least absolute estimators vis-a-vis least squares estimators. Econometrica, 38, 742-753. K. Glashoff, R. Schultz (1979) Uber die genaue Berechnung von besten L1‎‎-approximierenden. J. Approx. Theory 25, 280-293.

S.M. Goldfeld, R.E. Quandt (1981) Econometric modelling with non-normal disturbances. J. of Econometrics, Nov., 17(2), 141-55.

A.A. Goldstein, W. Cheney (1958) A finite algorithm for the solution of consistent linear equations and inequalities and for Tchebycheff approximation of inconsistent linear equations. Pacific J. Math., 8, 415-427.

R. Gonin (1983) A contribution to the solving of nonlinear estimation problems. Ph.D. thesis, Univ. of Capetown.

R. Gonin (1986) Numerical algorithms for solving nonlinear Lp-norm estimation problems: part I; a first-order gradient algorithm for well-conditioned small residual problems. Comm. Stat. B, 15(3), 801-813.

R. Gonin, A.H. Money (1985a) Nonlinear Lp-norm estimation: part I, on the choice of the exponent, p, where the errors are additive Comm. Stat. A14, 827-840.

R. Gonin, A.H. Money (1985b) Nonlinear Lp-norm estimation: part II, asymptotic distribution of the exponent, p, as a function of the sample kurtosis. Stat. A14, 841-849.

R. Gonin, A.H. Money (1987a) Outliers in physical processes: L1‎‎-or adaptive Lp norm estimation. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 477-454.

R. Gonin, A.H. Money (1987b) A review of computational methods for solving the nonlinear L1‎‎ norm estimation problem In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 117-130.

R. Gonin, A.H. Money (1987c) Nonlinear Lp-norm parameter estimation. Draft manuscript, Marcel Dekker, New York.

R. Gonin, S.H.C. du Toit (1987) Numerical algorithms for solving nonlinear Lp-norm estimation problem, part II-a mixture method for large residual and ill-conditioned problems. Comm. Stat. A16, no. 4.

S. Gross, W.L. Steiger (1979) Least absolute deviation estimates in autoregression with infinite variance. J. Appl. Prob., 16, 104-116.

Groucher (1971) Best L1‎‎ and Ll approximations. M.Sc. thesis, Birkbeck College, London Univ., London, England.

S.A. Gustafson, K.O. Kortanek, W. Rom (1970) Non-Chebyshevian moment problems. SIAM J. Numer. Anal., vol. 7, no. 3, 335-342.

L. Gyorfi (1987) Density estimation from dependent sample. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 393-404.

L. Gyorfi, E.C. Van der Meulen (1987) Density-free convergence properties of various estimators of entropy. CSDA, 5(4), 425-436.

E.E. Hagen (1975) The economics of development. Irwin inc, Illinois.

J. Hald (1981a) A 2-stage algorithm for nonlinear L1‎‎ optimization. Rep. no. 81-03, Numerisk Instut. Danmark Tekniske Hojskole, 2800 Lyngby, Denmark.

J. Hald (1981b) A two stage algorithm for linearly constrained nonlinear L1‎‎ optimization. Methods of Oper. Res., 43, 87-103.

J. Hald, K. Madsen (1985) Combined Lp and quasi-Newton methods for nonlinear L1‎‎ optimization. SIAM J. Numer. Anal., 22, no.1, 68-80.

M.L. Hand, V.A. Sposito (1980) Using the least squares estimator in the Chebyshev estimation. Comm. Stat., B9(1), 43-49.

W. Hardle (1987) XploRe, a computing environment for exploratory regression. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. North-Holland. 163-174.

T.E. Harris (1950) Regression using minimum absolute deviations. Am. Stat., 4, 14-15.

H.L. Harter (1974a) The method of least squares and some alternative, I. Int. Stat. Rev., 42, 147-174.

H.L. Harter (1974b) The method of least squares and some alternative, II. Int. Stat. Rev., 42, 235-264.

H.L. Harter (1975a) The method of least squares and some alternative, III. Int. Stat. Rev., 43, 1-44.

H.L. Harter (1975b) The method of least squares and some alternative, IV Int. Stat. Rev., 43, 125-190, 273-278.

H.L. Harter (1975c) The method of least squares and some alternative, V. Int. Stat. Rev., 43, 269-272.

H.L. Harter (1976) The method of least squares and some alternative, VI. Int. Stat. Rev., 44, 113-159.

H.L. Harter (1977) Nonuniqueness of absolute value regression. Comm. Stat. B6, 829-838.

H.L. Harter (1981), Method of least p-th powers. In Encyclopedia of statistical science, 5, 464-467.

A.C. Harvey (1977) A comparison of preliminary estimators for robust regression. JASA, 72, 910-13.

A.C. Harvey (1978) On the unbiasedness of robust regression estimators. Comm. Stat., A7, 779-783.

P. Hattenschwiler (1988) Goal programming becomes most useful using L1‎‎-smoothing functions CSDA, 6(4), 369-384.

W.M. Haussler (1984) Computational experience with an eigen vector algorithm for robust Lp-discrimination. Com. Stat. Q. 1, 233-244.

W.J. Heiser (1987) Correspondence analysis with least absolute residuals CSDA, 5, 337-356.

W.J. Heiser (1988) Multidimensional scaling with least absolute residuals. To appear in H.H. Boc (ed.), Classification and related methods of data analysis, (IFCS'87). NHPC.

S. Henriksson (1972) On a generalization of Lp-approximation and estimation. Thesis, Dept. of Computer Sci., Lund Univ., Sweden.

R.W. Hill, P.W. Holland (1977) Two robust alternatives to least squares regression. JASA, 72, 828-833.

C.R. Hobby, J.R. Rice (1965) A moment problem in L1‎‎ approximation. Proc. Amer. Math. Soc., 16, 665-670.

K.L. Hoffman, D.R. Shier (1980a) A test problem generator for discrete linear L1‎‎ approximation problems. ACM Trans. Math. Soft., 6, 587-593.

K.L. Hoffman, D.R. Shier (1980b) A test problem generator for discrete linear L1‎‎ approximation problems. ACM Trans. Math. Soft., 6, 615-617.

W.W. Hogan (1976) Norm minimizing estimation and unbiasedness. Econometrica, vol. 44, no.3, May.

P.W. Holland, R.E. Welsch (1977) Robust regression using iteratively reweighted least-squares. Comm. Stat., A6, 813-827.

L. Horvath (1987) Asymptotic normality of Lp-norms of density estimators. Tech. Rep. series of Lab Res. Stat. Prob., no.3, Carleton Univ., Ottawa, Canada.

Th.V. Hromadka II, Ch.Ch. Yen, G.F. Pinder (1987) The best approximation method. An introduction. Springer-Verlag, Berlin.

P.J. Huber (1987) The place of the L1‎‎-norm in robust estimation. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. Reprinted in CSDA, 5, 255-262.

C.R. Hunt (1970) Best Lp approximation by certain nonlinear functions. M.Sc. thesis, Univ. of Victoria, Univ. of Victoria, B.C. Canada.

J.G. Hunt, J.M. Dowling, F.R. Glahe (1974) L1‎‎ estimation in small samples with Laplace error distributions Decision Sci., 5, 22-29.

Imai, K. Kato, P. Yamamoto (1987) A linear-time algorithm for linear L1‎‎ approximation of points. Tech. Rep. CSCE-87-C30. Dept. of Comp. Sci. and Comm. Engin., Kyushu Univ. 36, Fukuoka 812, Japan.

K. Jajuga (1987) A clustering method based on the L1‎‎-norm. CSDA, 5, 357-371.

K. Jittorntrum, M.R. Osborne (1980) Strong uniqueness and second order convergence in nonlinear discrete approximation. Numer. Math., 34, 439-455.

Joe, R. Bartels (1983) An exact penalty method for constrained, discrete linear Ll data fitting. SIAM J. Sci. Stat. Comput., vol.4, no.1, 69-84.

L.A. Josvanger, V.A. Sposito (1983) L1‎‎-norm estimates for the simple regression problem. Comm. Stat. B12, 215-21.

Jureckova (1983) Trimmed polynomial regression. Commentationes Mathematicae Universitatis Carolinae, 24, 4, 597-607.

Jureckova (1984) Regression quantiles and trimmed least squares estimator under a general design. Kybernetika, vol.20, no.5, 345-357.

Jureckova, P.K. Sen (1984) On adaptive scale-equivalent M-estimators in linear models. Stat. and Decision supplement issue, no.1, 31-46.

K.R. Kadiyala (1972) Regression with non-Gaussian stable disturbances: some sampling results. Econometrica, July.

N. Kaergard (1987) Estimation criterion, residuals and prediction evaluation. CSDA, 5, 443-450.

S.W. Kahng (1972) Best Lp approximation. Math. of Comp., 26, 505-508.

L.V. Kantorovich, G.P. Akilov (1964) Functional analysis in normed spaces. Pergamon Press, Oxford England.

L.A. Karlovitz (1970a) An algorithm for the best Lp approximation. J. Approx. Theory. 3, 123-127.

L.A. Karlovitz (1970b) Construction of nearest points in the Lp, even and Ll norms. J. Approx. Theory, 3, 123-127. O.J.

Karst (1958) Linear curve fitting using least deviations. JASA, 53, 118-132.

Kaufman, P.J. Rousseeuw (1987) Clustering by means of medoids. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 405-416.

Y. Kawara (1979) Straight line fitting by minimizing the sum of absolute deviations. J. of the Japan Stat. Soc., 9, 47-64.

J.E. Kelley (1958) An application of linear programming to curve fitting. SIAM J. Appl. Math., 6, 15-22.

J.H.B. Kemperman (1984) Least absolute value and median polish. In Y.L. Tong (ed.), Inequalities in statistics and probability (IMS Lecture notes monograph series, vol.5), Inst. of Math. Stat., Hayward, CA, 84-113.

J.H.B. Kemperman (1987) The median of a finite measure on a Banach space. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 217-230.

W.J. Kennedy, J.E. Gentle (1977) Examining rounding error in least absolute values regression computations. Comm. Stat., B6, 415-420.

W.J. Kennedy, J.E. Gentle (1978) Comparisons of algorithms for minimum Lp norm linear regression. Proc. of Computer Sci. and Stat., Tenth Annual Symposium on Interface. D. Hogben (ed.), U.S. government printing office. Washington D.C., 373-378.

W.J. Kennedy, J.E. Gentle (1980) Statistical computing. New York, Marcel Dekker.

W.J Kennedy, J.E. Gentle, V.A. Sposito (1977) Comparisons of algorithms for L1‎‎ estimation in the linear model. Paper presented at Midwestern Regional Meeting of IMS, Madison, WI. (Available from the second author).

W.J Kennedy, J.E. Gentle, V.A. Sposito (1977) A computer oriented method for generating test problems for L1‎‎ regression. Comm. Stat., B6, 21-27.

B. Kim ( ) Lp norm estimation procedures and an L1‎‎ norm algorithm for unconstrained and constrained estimation for linear models. VPI & SU, Blacksburg, VA 24061 USA.

E.A. Kiountouzis (1971) Optimal Lp approximation techniques and data analysis. Bull. Soc. Math. Greece, 12, 191-206.

E.A. Kiountouzis (1972) Mathematical programming and best linear Lp approximation. Extrait du Bull. de la Soc. Math. de Grece Novelle Serie, Tom 13, Fassc. 1, 46-57.

E.A. Kiountouzis (1973) Linear programming techniques in regression analysis. Appl. Stat., 22, 69-73.

Klingman, J. Mote (1982) Generalize network approaches for solving least absolute value and Tchebycheff regression problems. TIMS studies in the Mana. Sci., 19.

J. Kmenta (1986) Elements of econometrics. MacMillan, New York.

R. Koenker (1987) A comparison of asymptotic testing methods for L1‎‎-regression. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. North-Holland. 287-295.

R. Koenker, G. Bassett, Jr. (1978) Regression quantile. Econometrica, vol. 46, no. 1 (Jan.).

R. Koenker, G. Bassett, Jr. (1982a) Test of linear hypothesis and L1‎‎ estimation. Econometrica, vol.50, no. 6, 1577-1583.

R. Koenker, G. Bassett, Jr. (1982b) Robust tests for heteroskedasticity based on the regression quantiles. Econometrica, vol. 50, no. 1, Jan., 43-61.

R. Koenker, G. Bassett, Jr. (1984) Four (pathological) examples in asymptotic statistics. The Amer. Statistician, Aug., vol.38, no.3, 209-212.

R. Koenker, G. Bassett, Jr. (1985) On Boscovich's estimator. Ann. of Stat., 13, 1625-1628.

W.W. Kotiuga (1982) Power system state estimation using least absolute value techniques. Ph.D. thesis, Univ. of Waterloo.

W.W. Kotiuga, M. Vidyasagar (1982) Bad data rejection properties of weighted least absolute value techniques applied to static state estimation. IEEE Trans. on Power Apparatus and Systems. PAS-101, 844-853.

B.R. Kripke, T.J. Rivlin (1965) Approximation in the metric of L1‎‎(X,u). Trans. Amer. Math. Soc., 119, 101-22.

K. Kveton (1987) Method of averages as an alternative to L1‎‎-and L2-norm methods in special linear regression problems. CSDA, 5, 407-414.

P.S. Laplace (1793) Sur quelques points du system du monde. Memoires de l'Academie Royale des Science de Paris. Annee 1789, 1-87. Reprinted in Oeuvres completes de Laplace II. Paris, Gauthier-Villars, 1985, 477-558.

P.S. Laplace (1799) Traite des mecanique celeste, 2. Paris; J.B.M. Depart. Reprinted as oeuvres completes de Laplace, 2. Paris; Gauthier-Villars 1878, 116-165.

P.S. Laplace (1812) Theorie analytique des probabilites, Mme courcier Paris 1820 Reprinted in his oeuvres, vol.7, Imprimerie Royale, Paris, 1847, and Gauthier-Villars et fils, Paris 1886.

P.S. Laplace (1818) Duexieme supplement to Laplace (1812).

J.L. Larson, A.H. Sameh (1980) Algorithms for round of error analysis a relative error approach. Computing 24, 275-297.

K.D. Lawrence, D.R. Shier (1981) A comparison of least square and least absolute deviation regression models for estimation Weibull parameters. Comm. Stat., B10, 315-326.

C.L. Lawson (1961) Contribution to the theory of linear least maximum approximations. Ph.D. thesis, Univ. of California, Los Angeles, California.

Lazarski (1975a) Approximation of continuous functions in the space L1‎‎. Automatika, 487, 85-93. E. Lazarski (1975b) The approximation of the continuous function by the polynomials of power functions in L1‎‎ space. Automatika, 487, 95-106.

Lazarski (1975c) On the necessary conditions of the uniqueness of approximation by the polynomials of power functions in L1‎‎ space. Automatika, 487, 107-117.

Lazarski (1977) Approximation of continuous functions by exponential polynomials in the L1‎‎ space. Automatika, 598, 82-87.

M.G. Lejeune, P. Sarda (1988) Quantile regression: a non parametric approach. CSDA, 6(3) 229-240.

J.T. Lewis (1969) Approximation with convex constraints. Doctoral thesis, Brown Univ., Providence, R.I.

J.T. Lewis (1970) Computation of best one-sided L1‎‎ approximation. Math. Comp., 24, 529-536.

R.F. Love (1974) The dual of a hyperbolic approximation to the generalized constrained multi-facility location problem with Lp distances Manag. Sci., vol. 21, 22-23.

R.F. Love, J.G. Morris (1975) The use of nonlinear programming for solving facility location problems involving Lp distances using convex programming. Oper. Res., vol. 23, no.3, 581-588.

G.S. Maddala (1977) Econometrics. McGraw-Hill.

K. Madsen (1975) An algorithm for minimax solution of overdetermined systems of nonlinear equations. J. Inst. Math. and Appl., 321-328.

K. Madsen (1985) Minimization of non-linear approximation functions. Copenhagen.

B. Mandelbrot (1960) The Pareto-Levy law and the distribution of income. Inter. Econ. Rev., 1, 79-106.

B. Mandelbrot (1961) Stable Paretian random functions and multiplicative variation of income. Econometrica, 29, 517-543.

B. Mandelbrot (1963) New methods in statistical economics. J. of Political Economy, Oct.,421-440.

Marazzi (1987) Solving bounded influence regression with ROBSYS. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 145-163.

Marazzi (1988) Algorithms for the computation of weights in bounded influence regression. CSDA 6(3), 251-276.

Marazzi, A. Randriamiharisoa (1985) ROBETH-ROBSYS: a software for robust statistical computing. Doc. no. 0,1,2,3,4,6. Institut Universitaire de Medecin Sociale et Preventive Lausanne, Switzerland.

J.S. Marron (1987) What does optimal bandwidth selection mean for non parametric regression estimation. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC, 379-392.

C.L. Mathieu (1816) Sur les experiences du pendule, faites par les navigateurs espagnol, en differens points du globe. Connaissance des tems, 314-332.

J.W. McKean, R.M. Schrader (1984) A comparison of the methods for studentizing the sample median. Comm. Stat., B 13(16), 751-773.

J.W. McKean, R.M. Schrader (1987) Least absolute errors analysis of variance. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. North-Holland. 297-306.

J.W. McKean, G.L. Sievers (1987) Coefficients of determination for least absolute deviation analysis. Stat. and Prob. Letters, 5, 49-54.

R.A. McLean, G.A. Watson (1980) Numerical methods for nonlinear discrete L1‎‎ approximation problems. In L. Collatz,

Meinardus, H. Werner (eds.) Numerical methods of approximation theory. ISNM 52, Birkhauser Verlag, Basel.

C.R. McConnell (1987) On computing a best discrete L1‎‎ approximation using the method of vanishing Jacobians. CSDA, 5, 277-288.

G.F. McCormick, V.A. Sposito (1975) A note on L1‎‎ estimation based on the median positive quotient. Appl. Stat., 24, 347-350.

G.F. McCormick, V.A. Sposito (1976) Using the L2-estimator in L1‎‎-estimation. SIAM J. Numer. Anal., 13, 337-343.

Megiddo, A. Tamir (1983) Finding least-distances line. SIAM J. Alg. Disc. Meth., 4, no. 2, 207-211.

J. Meier (1987) A fast algorithm for clusterwise linear absolute deviations regression. OR Spektrum, 9, 187-189.

M.S. Meketon (1986) Least absolute value regression. Work. Pap., AT&T Bell Laboratories, Holmdel, N.J.

Melaku, G. Sadasivan (1987) L1‎‎-norm and other methods for sample allocation in multivariate stratified surveys. CSDA, 5, 415-424.

J.A. Menendez, B. Salvador (1987) An algorithm for isotonic median regression. CSDA, 5, 399-406.

Merle, H. Spath (1974) Computational experiences with discrete Lp-approximation. Computing, 12, 315-321.

J.R. Meyer, R.R. Glauber (1964) Investment decisions, Economic forecasting and public policy. Harvard Business School Press, Cambridge, Massachusetts.

Militky, J. Cap (1987) Application of Bayes approach to adaptive Lp nonlinear regression. CSDA, 5, 381-390.

J.S.B. Mitchell (1987) Shortest rectilinear paths among obstacles. School of Oper. Res. and Ind. Engin. College of Engin. Cornell Univ.. Ithaca, New York, 14853.

M.J. Mojarrad (1977) The application of comparative Monte Carlo methods to econometrics: an efficient Monte Carlo study of finite sample properties of iterative instrumental variables estimation. Ph.D. Diss., Univ. of Pennsylvania.

Mond, M. Schechter (1976) A programming problem with an Lp norm in the objective function. J. Austral. Math. Soc., Ser., B, 19, 333-342.

A.H. Money, J.F. Affleck-Graves, M.L. Hart (1978a) A review of some alternatives to least squares regression. Tech Rep. no. ALS-1, Sep., Univ. of CapeTown, South Africa.

A.H. Money, J.F. Affleck-Graves, M.L. Hart (1978b) Least squares and some alternative: a simulation study. Tech. Rep. ALS-2. Univ. of CapeTown, South Africa.

A.H. Money, J.F. Affleck-Graves, M.L. Hart, G.D.I. Barr (1982) The linear regression model: Lp norm estimation and the choice of p. Comm. Stat., 11, 89-109.

R.M. Moroney (1961) The Haar problem in L1‎‎. Proc. Amer. Math. Soc., 12, 793-795.

J.G. Morris, W.a Verdini (1979) Minisum Lp distance location problems solved via a perturbed problem and Weiszfeld's algorithm. Oper. Res., 27, 1180-1188.

Munoz Perez, A. Fernandez Palacin (1987) Estimating the quantile function by Bernstein polynomials. CSDA, 5, 391-398.

V.I. Mudrov, V.L. Kushko, V.I. Mikhailov, E. Osvitskii (1968) Some experiments on the use of the least-modul: method in processing orbital data. Cosmic Res., 6, 421-431.

W. Murray, M. Overton (1981) A projected Lagrangian algorithm for nonlinear L1‎‎ optimization. SIAM J. Sci. Stat. Comput., 207-224.

S.C. Narula (1987) The minimum sum of absolute errors regression. J. Quality Tech. 19, 37-45.

S.C. Narula, J.F. Wellington (1977a) An algorithm for the minimum sum of weighted absolute errors regression. Comm. Stat., B(6), 341-352.

S.C. Narula, J.F. Wellington (1977b) Prediction, linear regression and minimum sum of relative errors. Technometrics, 19, 185-190.

S.C. Narula, J.F. Wellington (1977c) AS108, multiple linear regression with minimum sum of absolute error. Appl. Stat., 26, 106-111.

S.C. Narula, J.F. Wellington (1979) Selection of variables in linear regression using the minimum sum of weighted absolute errors criterion. Technometrics, 21, no.3 Aug.

S.C. Narula, J.F. Wellington (1982) The minimum sum of absolute errors regression, a state of the art survey. Inter. Stat. Rev., 50, 317-326.

S.C. Narula, J.F. Wellington (1983) Selection of variables in linear regression, a pragmatic approach. J. of Stat. Comput. and Simul., 17, 159-172.

S.C. Narula, J.F. Wellington (1985) Interior analysis for the minimum sum of absolute errors regression. Technometrics, 27, 181-188.

S.C. Narula, J.F. Wellington (1987) An efficient algorithm for the MSAE and MMAE regression problems. Work. Pap., Virginia CommonWealth Univ., Richmond, VA 23284.

H. Nikaido (1970) Introduction to sets and mappings in modern economics. North Holland, Amsterdam.

H. Nyquist (1980) Recent studies on Lp-norm estimation. Ph.D. thesis, Univ. of Umea, Sweden.

H. Nyquist (1983) The optimal Lp-norm estimator in linear regression models. Comm. Stat. A12, 2511-2524.

H. Nyquist (1988) Least orthogonal absolute deviations. CSDA, 6(4), 361-368.

H. Nyquist, A. Westlund (1977) L1‎‎-versus L2-norm estimation in interdependent systems when residual distributions are stable. Dept. of Stat., Univ. of Umea Presented at European Meeting of the Econometric Soc., Vienna, 5-9 Sep.

W. Oberhofer (1982) The consistency of nonlinear regression minimizing the L1‎‎ norm. Ann. of Stat., 10, 316-319.

W. Oettli (1975) Symmetric duality, and a convergent subgradient method for discrete linear, constrained approximation problems with arbitrary norms appearing in the objective function and in the constraints. J. Approx. Theory, 14, 43-50.

M.R. Osborne (1980) An algorithmic approach to nonlinear approximation problems. Approx. Theory III, 705-710.

M.R. Osborne (1985) Finite algorithms in optimization and data analysis. Wiley, Chichester.

M.R. Osborne (1987) The reduced gradient algorithm. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 95-108.

M.R. Osborne, S.A. Pruess, R.S. Womersley (1986) Concise representation of generalized gradients. J. of Austra. Math. Soc., Ser. B, 28, 57-74.

M.R. Osborne, G.A. Watson (1967) On the best linear Chebyshev approximation. Computer J., 10, 172-177.

M.R. Osborne, G.A. Watson (1971) On an algorithm for discrete nonlinear L1‎‎ approximation. Computer J., 14, 184-188.

M.R. Osborne, G.A. Watson (1978) Nonlinear approximation problems in vector norms. In Dundee, G.A. Watson (eds.) Numerical analysis. Springer Verlag.

M.R. Osborne, G.A. Watson (1985) An analysis of the total approximation problem in separable norms, and an algorithm for the total L1‎‎ problem. SIAM J. Sci. Stat. Comp., 6, 410-424.

M. Overton (1982) Algorithms for nonlinear L1‎‎ and Ll fitting. In M.J.D. Powell (ed.) Nonlinear optimization. Academic Press, London, 91-101

R.M. Oveson (1968) Regression parameter estimation by minimizing the sum of absolute errors. Doctoral dissertation Harvard Univ., Cambridge, Massachusetts.

H.J. Paarsch (1984) A Monte Carlo comparison of estimates for censored regression models. J. of Econometrics, 24, 197-213.

M.J. Panik (1976) Classical optimization: foundation and extensions. NHPC.

U. Peters, C. Willms (1983) Up-and down-dating procedures for linear L1‎‎ regression. OR Spektrum 5, 229-239.

R.C. Pfaffenberger, J.J. Dinkel (1978) Absolute deviations curve fitting: an alternative to least squares. In H.A. David (ed.) Contributions to survey sampling and applied statistics. Academic Press, New York, 279-294.

Pilibossian (1987) A direct solving algorithm for a linear regression according to L1‎‎-norm criteria. Work. Pap., L.S.T.A. Universite, Paris VI

M.A. Porter, D.J. Winstanley (1979) Remark ASR29. Remarks on AS110: Lp norm fit of a straight line. Appl. Stat., 28, 112-113.

S. Portnoy (1987) Using regression fractiles to identify outliers. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC, 345-356.

J.L. Powell (1983) The asymptotic normality of two-stage least absolute deviation estimators. Econometrica, 51, 1569-1575.

J.L. Powell (1984) Least absolute deviations estimation for the censored regression model. J. of Econometrics, 25, 303-325.

J.L. Powell (1986) Censored regression quantiles. J. of Econometrics, 32, 143-155.

M.J.D. Powell, Y. Yuan (1984) Conditions for super linear convergence in L1‎‎ and Ll solutions of overdetermined nonlinear equations. IMAJ Num. Anal., 4, 241-251.

Prochazka (1988) Regression quantiles and trimmed least squares estimator in nonlinear regression model. CSDA, 6(4), 385-392.

R. Prony (1804) Recherches physico-mathematiques sur la theorie des eaus courantes. Paris, l'imprimerie imperiale.

V. Ptak (1958) On approximation of continuous functions in the metric uatb3x(t)3dt Czechoslovak Math. J. 8(83), 267-273.

Rabinowitz (1968) Application of linear programming to numerical analysis. SIAM Rev., 10, 121-159.

Rabinowitz (1970) Mathematical programming and approximation. In A. Talbot (ed.) Approximation Theory. Academic Press, 217-231.

Ralston, P. Rabinowitz (1985) A first course in numerical analysis. Wiley, New York.

J.O. Ramsay (1977) A comparative study of several robust estimates of slopes, intercept, and scale in linear regression. JASA, 72, 608-615

M.R. Rao, V. Srinivasan (1972) A note on Sharpe's algorithm for minimum sum of absolute deviations in a simple regression problem. Manag. Sci., 19, 222-225.

R.H. Rasche, J. Gaffney, A.Y.C. Koo, N. Obst (1980) Functional forms for estimating the Lorenz curve. Econometrica, 48, 1061-1062.

H.D. Ratliff, J.C. Picard (1978) A cut approach to rectilinear distance facility location problem. Oper. Res., 26, 422-433.

W. Rey (1975) On the least pth power methods in multiple regressions and location estimations. BIT, 15, 174-185.

E.C. Rhodes (1930) Reducing observations by the method of minimum deviations. Philo. Mag., 7th series, 9, 974-92.

J.R. Rice (1964a) On computation of L1‎‎ approximations by exponentials, rationals, and other functions. Math. Comp., 18, 390-396.

J.R. Rice (1964b) On nonlinear L1‎‎ approximation. Arch. Rational Mech. Anal., 17 61-66.

J.R. Rice (1964c) The approximation of functions, vol. I, linear theory. Reading Mass:, Addison-Wesley.

J.R. Rice (1969) The approximation of functions, vol. II, linear theory. Reading Mass:, Addison-Wesley.

J.R. Rice (1985) Numerical methods, software, and analysis. McGraw-Hill, ch. 11.

J.R. Rice, J.S. White (1964) Norms for smoothing and estimation. SIAM Rev., 6, 243-256.

P.D. Robers, A. Ben-Israel (1969) An interval programming algorithm for discrete linear L1‎‎ approximation problem. J. Approx. Theory, 2, 323-336.

P.D. Robers, S.S. Robers (1973) Algorithm 458: discrete linear L1‎‎ approximation by interval linear programming. Comm. ACM, 16, 629-633.

Ronchetti (1987) Bounded influence in regression: a review. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC, 65-80.

A.E. Ronner (1977) P-norm estimators in a linear regression model. Doctoral thesis, Rijkuniversiteit te Groningen.

A.E. Ronner (1984) Asymptotic normality of p-norm estimators

in multiple regression. Z. Wahrschein lickeitstheorie verw. Gebiete 66, 613-620.

G. Roodman (1974) A procedure for optimal stepwise MSAE regression analysis. Oper. Res., 22, 393-399.

Rosenberg, D. Carlson (1977) A simple approximation of the sampling distribution of least absolute residuals regression estimates. Comm. Stat., B6, 421-437.

Rossi, H.D. Brunk (1987) L1‎‎ and L2 cross-validation for density estimation with special reference to orthogonal expansions. Tech. Rep. 120, Dept. of Stat., Oregon State Univ..

Rossi, H.D. Brunk (1988) L1‎‎ and L2 cross-validation for density estimation with special reference to orthogonal expansions. CSDA, 6(3), 203-228.

P.J. Rousseeuw (1984) Least median of squares regression. J. Amer. Stat. Asso., 79, 871-80.

P.J. Rousseeuw (1987) An application of L1‎‎ to astronomy. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC, 437-446.

P.J. Rousseeuw, A. Leroy (1987) Robust regression and outlier detection. Wiley-Interscience, New York.

A.N. Sadovski (1974) AS74: L1‎‎-norm fit of a straight line. Appl. Stat. 23, 244-248.

A.K.Md.E. Saleh, P.K. Sen (1987) On the asymptotic distributional risk properties of pre-test and shrinkage L1‎‎ estimators. CSDA, 5, 289-300.

J.P. Schellhorn (1987) Fitting data through homotopy methods In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 131-138.

E.J. Schlossmacher (1973) An iterative technique for absolute deviations curve fitting. JASA 68, 857-865.

R.M. Schrader, J.W. McKean (1987) Small sample properties of least absolute errors analysis of variance. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 307-322.

Seneta (1983) The weighted median and multiple regression. Austral. J. Stat., 25(2), 370-377.

Seneta, W.L. Steiger (1984) A new LAD curve-fitting algorithm: slightly overdetermined equation system in L1‎‎. Discrete App. Math., 7, 79-91.

Shanno, R.L. Weil (1970) Linear programming with absolute value functionals. Oper. Res., 19, 120-124.

W.F. Sharpe (1971) Mean-absolute deviation characteristic lines for securities and portfolios. Manag. Sci., 18, B1-B13.

S.J. Sheather (1986) A finite sample estimate of the variance of the sample median. Stat. and Prob. Letter., 4, 337-342.

S.J. Sheather (1987) Assessing the accuracy of the sample median: estimated standard errors versus interpolated confidence intervals. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. North-Holland. 203-216.

S.J. Sheather, J.W. McKean (1987) A comparison of testing and confidence interval methods for the median. Stat. and Prob. Letter, 6, 31-36.

H.D. Sherali, B.O. Skarpness, B. Kim (1987) An assumption-free convergence analysis for a perturbation of the scaling algorithm for linear programs, with application to the L1‎‎ estimation problem. Dept. of Ind. Engin. and OR, Virginia Polytechnic Inst. and State Univ., Blacksburg, Virginia.

O.B. Sheynin (1973) R.J. Boscovich's work on probability. Archive for history of exact sciences, vol. 9, 306-324, and vol. 28, 173.

D.R. Shier, C.J. Witzgall (1978) Norm approximation problems and norm statistics., J. Res. Nat. Bur. Standards, 83, 71-74.

O. Shisha (1974) A remark on an algorithm for best Lp approximation. J. Approx. Theory, 11, 283-284.

R.I. Shrager, E. Hill (1980) Nonlinear curve-fitting in the L1‎‎ and Ll norms. Math. Comput., 34, 529-541.

A.F. Siegel (1983) Low median and least absolute residual analysis of two-way tables. JASA, 78, 371-374.

R.L. Sielken, H.O. Hartley (1973) Two linear programming algorithms for unbiased estimation of linear models. JASA, 68, 639-.

H.A. Simon (1955) On a class of skew distribution functions. Biometrika, 42, 425-440. Reprinted in H.A. Simon (1957) Models of man. New York, Wiley.

R.R. Singleton (1940) A method for minimizing the sum of absolute values of deviations. Ann. of math. Stat., 11, 301-310.

M.G. Sklar, R.D. Armstrong (1982) Least absolute value and Chebyshev estimation utilizing least squares results. Math. Prog., 24, 346-352.

V.K. Smith, T.W. Hall (1972) A comparison of maximum likelihood versus BLUE estimators. Rev. Econ. Stat., 54, 186-190.

S.A. Soliman, G.S. Christensen, A. Rouhi (1988) A new technique for curve fitting based on minimum absolute deviations. CSDA, 6(4), 341-352.

D.L. Souvaine, J.M. Steele (1987) Time-and space-efficient algorithms for least median of squares regression. JASA, 82, no. 399, 794-801.

Spath (1976) L1‎‎ cluster analysis. Computing, 16, 379-387.

Spath (1982) On discrete linear orthogonal Lp-approximation. Numerische Analysis, ZAMM 62, T354-T355.

Spath (1985) Cluster dissection and analysis. Horwod, Chichester.

H. Spath (1986a) Clusterwise linear least squares versus least absolute deviations regression, a numerical comparison for a case study. In W. Gaul, M. Schader (eds.) Classification as a tool of research. Elsevier, Amsterdam.

H. Spath (1986b) Orthogonal least squares fitting with linear manifolds. Numer. Math., 48, 441-445.

H. Spath (1986c) Algorithm, Clusterwise linear least absolute deviations regression. Computing, 37, 371-378.

H. Spath (1987) Using the L1‎‎ norm within cluster analysis. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 427-434.

H. Spath, G.A. Watson (1987) On orthogonal linear L1‎‎ approximation. Numer. Math., 51, 531-543.

V.A. Sposito (1976) A remark on algorithm AS74, L1‎‎ norm fit of a straight line. Appl. Stat., 25, 96-97.

V.A. Sposito (1982) On the unbiasedness of the Lp-norm estimators. JASA, 77, 652-653.

V.A. Sposito (1987a) On median polish and L1‎‎ estimators. CSDA, 5, 155-162.

V.A. Sposito (1987b) Some properties of Lp-estimators in robust procedures. Marcel Dekker. In print.

V.A. Sposito, M.L. Hand (1980) Optimal Lp estimators for symmetric distributions. Proc. of ASA, Stat. Comp. Sec.

V.A. Sposito, M. Hand, G. McCormick (1977) Using an approximate L1‎‎ estimator. Comm. Stat., B6, 263-268.

V.A. Sposito, M.L. Hand, B. Skarpness (1983) On the efficiency of using the sample kurtosis in selecting optimal Lp estimators. Comm. Stat., B12, 265-272.

V.A. Sposito, W.J. Kennedy, J.E. Gentle (1977) AS110: Lp norm fit of a straight line. Appl. Stat., 26, 114-118.

V.A. Sposito, W.J. Kennedy, J.E. Gentle (1980) Useful generalized properties of L1‎‎ estimators. Comm. Stat., A9, 1309-1315.

V.A. Sposito, G.F. McCormick, W.J. Kennedy (1975) L1‎‎ estimation strategies based on the simplex algorithm. In Proc. of the eighth symposium on the interface, J.W. France (ed.) Health science computing facility. UCLA, Los Angeles.

V.A. Sposito, W.C. Smith (1976) On a sufficient and necessary condition for L1‎‎ estimation. Appl. Stat., 25, 154-157.

V.A. Sposito, W.C. Smith, G.F. McCormick (1978) Minimizing the sum of absolute deviations. J. of Appl. Stat. and Econometrics, Vandenhoeck and Ruprecht in Gottingen and Zurich series 12.

V.A. Sposito, M. Tvejte (1984) The estimation of certain parameters used in L1‎‎ interface. Proc. of Stat. Comp. Sec. of ASA, 267-270.

Spyropoulos, E. Kiountouzis, A. Young (1973) Discrete approximation in the L1‎‎ norm. Comp. J., 16, 180-186.

V.P. Sreedharan (1969) Solution of overdetermined linear equation which minimize error in an abstract norm. Numer. Math., 13, 146-151.

V.P. Sreedharan (1971) Least squares algorithms for finding solutions which minimize error in an abstract norm. Numer. Math., 17, 387-401.

G. Stangenhaus (1987) Bootstrap and interface procedures for L1‎‎ regression. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 323-332.

G. Stangenhaus, S.C. Narula (1987) Inference procedures for the L1‎‎ regression. Work. Pap., Universidade Estadual de Compainas, Brasil.

J.M. Steele, W.L. Steiger (1986) Algorithms and complexity for least median of squares regression. Discrete Appl. Math., 14, 39-100.

Stiefel (1960) Note on Jordan elimination, linear programming and Tchebycheff approximation. SIAM J. Numer. Math., 2, 1-17.

Steindl (1965) Random processes and growth of the firms. London, Griffin.

W.L. Steiger (1980) Linear programming via L1‎‎ curve fitting beats simplex. Abstracts, AMS, 80T-C26, 385-386.

W. Steiger, P. Bloomfield (1980) Remark on a paper of Narula and Wellington. Technometrics, 22, 450.

S.M. Stigler (1981) Gauss and invention of least squares. Ann. of Stat., 9, 465-474.

S.M. Stigler (1984) Studies in the history of probability and statistics XL, Boscovich, Simpson and a 1760 manuscript note on fitting a linear relation. Biometrica, 71, 3, 615-620.

Svanberg (1805) Exposition des operations faites en lappnie pour la determination d'un arc du meridien en 1801, 1802 et 1803,... Stockholm.

Taguchi (1972a) On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-I. Ann. of the Inst. of Stat. Math., vol. 24, no.2, 355-381.

Taguchi (1972b) On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-II. Ann. of the Inst. of Stat. Math., vol. 24, no.3, 599-619.

T. Taguchi (1972c) Concentration polyhedron, two dimensional concentration coefficient for discrete type distribution and some new correlation coefficients etc. The Inst. of Stat. Math., 77-115.

T. Taguchi (1973) On the two-dimensional concentration surface and extensions of concentration coefficient and Pareto distribution to the two dimensional case-III. Ann. of the Inst. of Stat. Math., vol. 25, no.1, 215-237.

T. Taguchi (1974) On Fechner's thesis and statistics with norm p. Ann. of the Inst. of Stat. Math., vol. 26, no.2, 175-193.

T. Taguchi (1978) On a generalization of Gaussian distribution. Ann. of the Inst. of Stat. Math., vol. 30, no.2, A, 211-242.

T. Taguchi (1981) On a multiple Gini's coefficient and some concentrative regressions. Metron, vol. XXXIX -N.1-2, 5-98.

T. Taguchi (1983) Concentration analysis of bivariate Paretoan distribution. Proc. of the Inst. of Stat. Math., vol. 31, no.1, 1-32.

T. Taguchi (1987) On the structure of multivariate concentration. Submitted to the First International Conference on Statistical Data Analysis Based on the L1‎‎ Norm and Related Methods, Neuchatel, Switzerland.

T. Taguchi (1988) On the structure of multivariate concentration -some relationships among the concentration surface and two variate mean difference and regressions. CSDA, 6, 307-334.

Takayama (1974) Mathematical economics. The Dryden Press, Illinois.

L.D. Taylor (1974) Estimating by minimizing the sum of absolute errors. In P. Zarembka (ed.) Frontiers in econometrics. Academic Press.

H.H. Taylor, S.C. Banks, J.F. McCoy (1979) Deconvolution with the L1‎‎ norm. Geophysics, 44, 39-52.

H. Theil (1965) The analysis of disturbances in regression analysis. JASA, 67, 1067-1079.

H. Theil (1971) Principles of econometrics. Wiley.

A Tishler, L. Zang (1982) An absolute deviations curve fitting algorithm for nonlinear models. TIMS studies in Manag. Sci., 19.

D.S. Tracy, K.A. Khan (1987) MRPP tests in L1‎‎ norm. CSDA, 5, 373-380.

Trauwaert (1987) L1‎‎ in fuzzy clustering. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 417-426.

J.W. Tukey (1977) Exploratory data analysis. Reading, Mass. Addison-Wesley.

H.H. Turner (1887) On Mr. Edgeworth's method of reducing observations relating to several quantities. Phil. Mag. (5th series), 24, 466-470.

K.H. Usow (1967a) On L1‎‎ approximation: computation for continuous functions and continuous dependence. SIAM J. of Numer. Anal., 4, 70-88.

K.H. Usow (1967b) On L1‎‎ approximation: computation for discrete functions and discretization effect. SIAM J. Numer. Anal., 4, 233-244.

Vajda (1987) L1‎‎-distances in statistical inference: comparison of topological, functional and statistical properties. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 177-184.

C.W. Valentine, C.P. Van Dine (1963) An algorithm for minimax polynomial curve fitting for discrete data. J. ACM, 10, 283-290.

J.F. Van Beeck-Calkoen (1816) Ver de theoric der Gemiddelde Waardij. Verhandlingen der K. Nederlandandsch Instituut Can Wetenschappen, 2, 1-19.

Veidinger (1960) On the numerical determination of the best approximation in the Chebychev sense. Numer. Math., 2, 1-17.

B.A. Von Lindenau (1806) Uber den Gebrauch der Gradmessungen zur bestimmung der gestalt der erde. Monatliche correspondenz zur befar derung der Erd-und Himmels-kunde, 14, 113-158.

B.Z. Vulikh (1976) A brief course in the theory of functions of a real variable. Mir Publishers, Moscow.

H.M. Wagner (1959) Linear programming technique for regression analysis. JASA, 54, 202-212.

H.M. Wagner (1962) Nonlinear regression with minimal assumption. JASA, 57, 572-578.

G.A. Watson (1973) On the best linear Chebyshev approximation. J. Approx. Theory, 7, 48-58.

G.A. Watson (1973) The calculation of best linear one-side Lp approximations. Math. Comp. 27, 607-620.

G.A. Watson (1977) On two methods for discrete Lp-approximation. Computing, 18, 263-266.

G.A. Watson (1978) A class of programming problems whose objective function contains a norm. J. approx. Theory, 23, 401-411.

G.A. Watson (1980) Approximation theory and numerical methods. Wiley, New York.

G.A. Watson (1981) An algorithm for linear L1‎‎ approximation of continuous functions. IMA J. Num. Anal., 1, 157-167.

G.A. Watson (1982a) A globally convergent method for (constrained) nonlinear continuous L1‎‎ approximation problems. In Numerical methods of approximation theory. ISNM59, Birkhauser Verlag.

G.A. Watson (1982b) Numerical methods for linear orthogonal Lp approximation. IMA J. of Numer. Anal., 2, 275-287.

G.A. Watson (1984a) Discrete L1‎‎ approximation by rational functions. IMA J. Num. Anal., 4, 275-288.

G.A. Watson (1984b) The numerical solution of total Lp approximation problems. In D.F. Griffiths (ed.) Numerical analysis. Dundee 1983, Lecture notes in mathematics, 1066, Springer Verlag, 221-238.

G.A. Watson (1985a) On the convergence of eigenvector algorithms for robust Lp-discrimination. Comp. Stat. Quart., 4, 307-314.

G.A. Watson (1985b) On a class of algorithms for total approximation. J. Approx. Theory, 45, no.3, 219-231.

G.A. Watson (1986) Methods for best approximation and regression problems. Rep. NA/96, Dept. of Math. Sci., Univ. of Dundee, DD1 4hn, Scotland, UK.

G.A. Watson (1987) Data fitting by sums of exponentials using the L1‎‎ norm. Inter. Series of Numer. Math., 81, 246-261.

J.F. Wellington, S.C. Narula (1981) Variable selection in multiple linear regression using the minimum sum of weighted absolute errors criterion. Comm. Stat., B10, 641-648.

J.F. Wellington, S.C. Narula (1984) An algorithm for regression quantiles. Comm. Stat., B13(5), 683-704.

A.H. Welsh (1987) Kernel estimates of the sparsity function. In Y. Dodge (ed.) Statistical data analysis based on the L1‎‎ norm and related methods. NHPC. 369-378.

G.O Wesolowsky (1981) A new descent algorithm for least absolute value regression problem. Comm. Stat., B10, 479-491.

G.O. Wesolowsky, R.F. Love (1971) The optimal location of new facilities using rectangular distances. Oper. Res., Jan-Feb.

G.O. Wesolowsky, R.F. Love (1972) A nonlinear approximation method for solving a generalized rectangular distance Weber problem. Manag. Sci., 18, 56-63.

H.C. Wilson (1978) Least squares versus minimum absolute deviations estimation in linear models. Dec. Sci., 322-335.

H.G. Wilson (1979) Upgrading transport costing methodology. Transportation J., 18, 49-55.

C.S. Withers (1986) The bias and skewness of L1‎‎-estimates in regression. CSDA, 5, 301-303.

J.M. Wolfe (1979) On the convergence of an algorithm for a discrete Lp-approximation. Numer. Math., 32, 439-459.

R.S. Womersley (1986) Censored discrete linear L1‎‎ approximation. SIAM J. Sci. Comput., 7, no.1, 105-122.

Y. Wu (1988) Strong consistency and exponential rate of the minimum L1‎‎ norm estimates in linear regression models. CSDA 6(3), 285-296.

Wulff (1983) Numerische verfahren zur linearen orthogonalen Lp-regression. Diplomarbeit, Universitat Oldenburg.

J.D. Young (1971) Smoothing data with tolerances by use of linear programming. J. Inst. Math. Appl., 8, 69-79.

R. Zeckhauser, M. Thompson (1970) Linear regression with non-normal error terms. Rev. Econ. Stat., 52, 280-286.

Published
2019-06-15
How to Cite
Bidabad, B. (2019). L1‎‎ Norm Based Data Analysis and Related Methods. Australian Finance & Banking Review, 3(1), 43-81. https://doi.org/10.46281/afbr.v3i1.317
Section
Research Paper/Theoretical Paper/Review Paper/Short Communication Paper