×

Computational algorithms for calculating least absolute value and Chebyshev estimates for multiple regression. (English) Zbl 0547.65098

Summary: Least absolute value (LAV) and Chebyshev estimation are two possible alternatives to least squares estimation in multiple regression models. This paper reviews the historical development of these two alternatives with special attention placed on the development of algorithms for producing LAV and Chebyshev estimates of the regression parameters. Recent algorithmic improvements are reviewed, sources for obtaining computer programs are identified and recommendations are made for the most efficient algorithm for specific cases. In the final section suggestions for future research are made for the development and refinement of computational algorithms and inference procedures.

MSC:

65C99 Probabilistic methods, stochastic differential equations
65D10 Numerical smoothing, curve fitting
41A45 Approximation by arbitrary linear expressions
62J05 Linear regression; mixed models
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abdelmalek N. N., Mathematics of Computation 29 pp 844– (1975)
[2] Abdelmalek N. N., BIT 15 pp 117– (1975) · Zbl 0303.65032 · doi:10.1007/BF01932684
[3] Abdelmalek N. N., International Journal for Numerical Methods in Engineering 10 pp 1197– (1976) · Zbl 0333.65018 · doi:10.1002/nme.1620100521
[4] Abdelmalek N. N., ACM Transactions on Mathematical Software 6 pp 220– (1980) · Zbl 0457.65023 · doi:10.1145/355887.355894
[5] Abdelmalek N. N., ACM Transactions on Mathematical Software 6 pp 228– (1980) · doi:10.1145/355887.355895
[6] Armstrong R. D., Communications in Statistics - Simulation and Computation 6 pp 383– (1977) · doi:10.1080/03610917708812052
[7] Armstrong R. D., Naval Research Logistics Quarterly 26 pp 79– (1979) · Zbl 0406.62052 · doi:10.1002/nav.3800260109
[8] Armstrong R. D., Communications in Statistics - Simulation and Computation 8 pp 175– (1979) · Zbl 0439.65049 · doi:10.1080/03610917908812113
[9] Armstrong R. D., SIAM Journal on Numerical Analysis 14 pp 555– (1977) · Zbl 0367.65032 · doi:10.1137/0714034
[10] Armstrong R. D., Applied Statistics 27 pp 363– (1978) · Zbl 0437.62065 · doi:10.2307/2347181
[11] Armstrong R. D., Applied Statistics 28 pp 93– (1979) · Zbl 0454.65090 · doi:10.2307/2346829
[12] Armstrong R. D., Mathematical Programming 19 pp 186– (1980) · Zbl 0442.90052 · doi:10.1007/BF01581640
[13] Armstrong R. D., Applied Mathematics and Computation 7 pp 267– (1980) · Zbl 0445.65061 · doi:10.1016/0096-3003(80)90049-1
[14] Armstrong R. D., TIMS Studies in the Management Sciences 19 pp 67– (1982)
[15] ArmstrongR. D. and SklarM. G. (1979). A dual method for discrete Chebychev curve fitting with linear restrictions on the parameters.American Statistical Association Statistical Computing Section Proceedings, Washington, B.C. 293–296.
[16] Barrodale I., ACM Transactions on Mathematical Software 1 pp 264– (1975) · Zbl 0309.65015 · doi:10.1145/355644.355651
[17] Barrodale I., SIAM Journal on Numerical Analysis 10 pp 839– (1973) · Zbl 0266.65016 · doi:10.1137/0710069
[18] Barrodale I., Communications of the ACM 17 pp 319– (1974) · doi:10.1145/355616.361024
[19] Barrodale I., SIAM Journal on Numerical Analysis 15 pp 603– (1978) · Zbl 0387.65027 · doi:10.1137/0715040
[20] Barrodale I., Numerische Mathematlk 8 pp 295– (1966) · Zbl 0173.18801 · doi:10.1007/BF02162565
[21] Bartels R. H., ACM Transactions on Mathematical Software 6 pp 594– (1980) · Zbl 0448.49017 · doi:10.1145/355921.355930
[22] Bartels R. H., ACM Transactions on Mathematical Software 6 pp 609– (1980) · Zbl 0448.49018 · doi:10.1145/355921.355931
[23] Bartels R. H., SIAM Journal on Numerical Analysis 15 pp 255– (1978) · Zbl 0376.65019 · doi:10.1137/0715017
[24] Bartels R. H., A FORTRAN program for solving overdetermined systems of linear equations in the Li sense (1976)
[25] Bartels R. H., SIAM Journal on Numerical Analysis 15 pp 224– (1978) · Zbl 0376.65018 · doi:10.1137/0715015
[26] Bartels R. H., Communications of the ACM 11 pp 401– (1968) · Zbl 0162.20701 · doi:10.1145/363347.363364
[27] Bartels R. H., Communications of the ACM 11 pp 428– (1968) · doi:10.1145/363347.363389
[28] Bassett G., Journal of the American Statistical Association 73 pp 618– (1978) · doi:10.1080/01621459.1978.10480065
[29] Bassett G., Journal of the American Statistical Association 77 pp 407– (1982)
[30] Bloomfield P., SIAM Journal on Scientific and Statistical Computing 1 pp 290– (1980) · Zbl 0471.65007 · doi:10.1137/0901019
[31] Boscovich R. J., Bononiensi Scientiarum et Artum Institute Atque Academia Commentarii 4 pp 353– (1757)
[32] Boscovich R. J., Philosophiae Recentioris, a Benedicto Stay in Romano Archigynasis Publico Eloquentare Professore, resibustradltae, Libri X, cum adnotianibus et Supplementis P. Rogerii Joseph Boscovich, S. J. 2 pp 406– (1760)
[33] Brownlee K. A., Statistical Theory and Methodology in Science and Engineering (1965) · Zbl 0136.39203
[34] Charnes A., Management Science 1 pp 138– (1955) · Zbl 0995.90590 · doi:10.1287/mnsc.1.2.138
[35] Claerbout J. F., Geophysics 38 pp 826– (1973) · doi:10.1190/1.1440378
[36] Cline A. K., SIAM Journal on Numerical Analysis 13 pp 293– (1976) · Zbl 0337.65027 · doi:10.1137/0713027
[37] Cogger K. O., Statistical Inference in Minimum Absolute Deviation Regression (1979)
[38] Daniel C., Fitting Equations to Data (1971)
[39] Davis K. R., Quantitative Methods for Management (1981)
[40] Dielman T. E., TIMS Studies in the Management Sciences 19 pp 31– (1982)
[41] Draper N. R., Applied Regression Analysis (1966) · Zbl 0158.17101
[42] Edgeworth F. Y., Hermathena 6 pp 279– (1887)
[43] Edgeworth F. Y., Philosophical Magazine 25 pp 184– (1888) · doi:10.1080/14786448808628170
[44] Eisenhart C., Roger Joseph Boscovich pp 200– (1961)
[45] El-Attar R., SIAM Journal on Numerical Analysis 16 pp 70– (1979) · Zbl 0401.90089 · doi:10.1137/0716006
[46] Forsythe A. B., Technometrics 14 pp 159– (1972) · doi:10.1080/00401706.1972.10488892
[47] Fourier J. B. J., Histoire de l’Academie des Sciences pour pp 47– (1824)
[48] Frome E. L., Communications in Statistics - Simulation and Computation 9 pp 201– (1980) · Zbl 0451.62050 · doi:10.1080/03610918008812150
[49] Gauss C. F., Theoria Motus Corporum Coelestium in Sectionibus Conlcis Solum Ambientium (1809)
[50] GaussC. F. (1823). Theoria Combinationis Observatlonum Erroribus Mlnlmlm Obnoxlae. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores, 5; German summary, Gottingische Gelehrte Anzeigen (1821), 321–327 and (1823), 313–318.
[51] GaussC. F. (1828). Supplementum Theoriae Combinationis Observationum Erroribus Minimis Obnoxiae. Commentationes Societatis Regine Scientiamm Gottingensis Recentiores, 6; German summary, Gottingische Gelehrte Anzeigen (1826), 1521–1527.
[52] Gentle J. E., Communications in Statistics - Theory and Methods 6 pp 839– (1977) · Zbl 0366.62090 · doi:10.1080/03610927708827535
[53] Goldstein A. A., On the method of descent in complex domains and its application to the minimal approximation of overdetermined systems of linear equations (1956)
[54] Goldstein A. A., Journal of the Association for Computing Machinery 4 pp 341– (1957) · doi:10.1145/320881.320890
[55] Harter H. L., International Statistical Review 42 pp 147– (1974) · Zbl 0291.62002 · doi:10.2307/1403077
[56] Harter H. L., International Statistical Review 42 pp 245– (1974)
[57] Harter H. L., International Statistical Review 43 pp 1– (1975) · Zbl 0311.62033 · doi:10.2307/1402658
[58] Harter H. L., International Statistical Review 43 pp 125– (1975) · Zbl 0311.62034 · doi:10.2307/1402897
[59] Harter H. L., International Statistical Review 43 pp 269– (1975) · Zbl 0331.62054 · doi:10.2307/1403110
[60] Harter H. L., The Chronological Annotated Bibliography of Order Statistics (1983) · Zbl 0515.62045
[61] Hoffman K. L., ACM Transactions on Mathematical Software 6 pp 587– (1980) · Zbl 0445.41006 · doi:10.1145/355921.355929
[62] Hogg R. V., Journal of the American Statistical Association 69 pp 909– (1974) · doi:10.1080/01621459.1974.10480225
[63] Hogg R. V., The American Statistician 33 pp 108– (1979)
[64] International Mathematical and Statistical Libraries, Inc., IMSL Library 3, Reference Manual (1980)
[65] Josvanger L. A., Communications in Statistics - Simulation and Computation 12 pp 215– (1983) · doi:10.1080/03610918308812313
[66] Karst O. J., Journal of the American Statistical Association 53 pp 118– (1958) · doi:10.1080/01621459.1958.10501430
[67] Kelley J. E., Journal of the Society for Industrial and Applied Mathematics 6 pp 15– (1958) · Zbl 0084.15804 · doi:10.1137/0106002
[68] Kennedy W. J., Communications in Statistics -Simulation and Computation 6 pp 21– (1977) · doi:10.1080/03610917708812024
[69] Klingman D., TIMS Studies in the Management Sciences 19 pp 53– (1982)
[70] Koenker R., Econometrica 50 pp 1577– (1982) · Zbl 0497.62057 · doi:10.2307/1913398
[71] LaplaceP. S. (1786). Memoire sur la figure de la terre. Memoires de 1’Academic royale des Sciences de Paris, Annee 1783, 17–46.
[72] Laplace P. S., Histoire de 1’Academic Royale des Inscriptions et Belles Lettres, aver les Memoires de litterature Tirez des Registres de cette Academie (1792)
[73] LaplaceP. S. (1793). Sur quelques points du systeme du monde. Memoires de 1 Academie royal des Sciences de Paris, Annee 1789, 1–87.
[74] LaplaceP. S. (1799). Traite’ des Mecanique Celeste, 2, Paris: J. B. M. Deprat.
[75] Laplace P. S., Theorie Analytique des Probabllltes (1812)
[76] Legendre A. M., Nouvelles Methodes pour la Determination des Orbites des Cometes (1805)
[77] Mathieu C. L., Connaissance des Tems, Annee 1816 pp 314– (1813)
[78] Meyer J. R., Investment Decisions, Economic Forecasting and Public Policy (1964)
[79] Mudrov V. I., Cosmic Research 6 pp 421– (1968)
[80] Murray W., SIAM Journal on Scientific and Statistical Computing 2 pp 207– (1981) · Zbl 0468.65036 · doi:10.1137/0902018
[81] Narula S. C., TIMS Studies in the Management Sciences 19 pp 11– (1982)
[82] Narula S. C., Communications in Statistics - Simulation and Computation 6 pp 341– (1977) · doi:10.1080/03610917708812049
[83] Narula S. C., Subset selection using minimization of the maximum weighted absolute error (1977)
[84] Narula S. C., Technometrics 21 pp 299– (1979) · doi:10.1080/00401706.1979.10489773
[85] Osborne M. R., The Computer Journal 12 pp 63– (1969) · Zbl 0164.45802 · doi:10.1093/comjnl/12.1.63
[86] Osborne M. R., The Computer Journal 14 pp 184– (1971) · Zbl 0222.65015 · doi:10.1093/comjnl/14.2.184
[87] Prony R., Recherches Physico-Mathematlques sur la Theorie des Eauz (1804)
[88] Rhodes E. C., Philosophical Magazine 9 pp 974– (1930)
[89] Rice J. R., SIAM Review 6 pp 243– (1964) · Zbl 0121.34102 · doi:10.1137/1006061
[90] Sadovski A. N., Applied Statistics 23 pp 244– (1974) · doi:10.2307/2347013
[91] Schryer N. L., Communications of the ACM 12 pp 326– (1969) · doi:10.1145/363011.363150
[92] Shrager R. I., Mathematics of Compulation 34 pp 529– (1980)
[93] Sielken R. L., Journal of the American Statistical Association 68 pp 639– (1973) · doi:10.1080/01621459.1973.10481398
[94] Singleton J., Annals of Mathematical Statistics 11 pp 301– (1940) · Zbl 0023.34403 · doi:10.1214/aoms/1177731869
[95] Stlefel E., Numerlsche Mathematlk 1 pp 1– (1959) · Zbl 0083.11501 · doi:10.1007/BF01386369
[96] Stlefel E., Numerlsche Mathematlc 2 pp 1– (1960) · Zbl 0097.32306 · doi:10.1007/BF01386203
[97] Tishler A., TIMS Studies in the Management Sciences 19 pp 81– (1982)
[98] Usow K. H., SIAM Journal on Numerical Analysis 4 pp 233– (1967) · Zbl 0166.41902 · doi:10.1137/0704022
[99] Wagner H. M., Journal of the American Statistical Association 54 pp 206– (1959) · doi:10.1080/01621459.1959.10501506
[100] Watson G. A., Journal of the Institute of Mathematical Applications 18 pp 351– (1976) · Zbl 0352.65008 · doi:10.1093/imamat/18.3.351
[101] Wellington J. F., Coimnunlcatlons In Statistics - Simulation and Computation 10 pp 641– (1981) · doi:10.1080/03610918108812240
[102] Wesolowsky G. O., Communications in Statistics - Simulation and Computation 10 pp 479– (1981) · Zbl 0463.65096 · doi:10.1080/03610918108812224
[103] Wilson H. G., Transportation Journal 18 pp 49– (1979)
[104] Zukhovitskll S. I., Doklady Akademll Nauk SSSR 79 pp 561– (1951)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.