Previous |  Up |  Next


Title: $M$-estimators of structural parameters in pseudolinear models (English)
Author: Liese, Friedrich
Author: Vajda, Igor
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 44
Issue: 4
Year: 1999
Pages: 245-270
Summary lang: English
Category: math
Summary: Real valued $M$-estimators $\hat{\theta }_n:=\min \sum _1^n\rho (Y_i-\tau (\theta ))$ in a statistical model with observations $Y_i\sim F_{\theta _0}$ are replaced by $\mathbb{R}^p$-valued $M$-estimators $\hat{\beta }_n:=\min \sum _1^n\rho (Y_i-\tau (u(z_i^T\,\beta )))$ in a new model with observations $Y_i\sim F_{u(z_i^t\beta _0)}$, where $z_i\in \mathbb{R}^p$ are regressors, $\beta _0\in \mathbb{R}^p$ is a structural parameter and $u:\mathbb{R}\rightarrow \mathbb{R}$ a structural function of the new model. Sufficient conditions for the consistency of $\hat{\beta }_n$ are derived, motivated by the sufficiency conditions for the simpler “parent estimator” $\hat{\theta }_n$. The result is a general method of consistent estimation in a class of nonlinear (pseudolinear) statistical problems. If $F_\theta $ has a natural exponential density $\mathrm{e}^{\theta x-b(x)}$ then our pseudolinear model with $u=(g\circ \mu )^{-1}$ reduces to the well known generalized linear model, provided $\mu (\theta )= {\mathrm d}b(\theta )/{\mathrm d}\theta $ and $g$ is the so-called link function of the generalized linear model. General results are illustrated for special pairs $\rho $ and $\tau $ leading to some classical $M$-estimators of mathematical statistics, as well as to a new class of generalized $\alpha $-quantile estimators. (English)
Keyword: $M$-estimator
Keyword: generalized linear models
Keyword: pseudolinear models
MSC: 62F10
MSC: 62F12
MSC: 62F35
idZBL: Zbl 1060.62029
idMR: MR1698768
DOI: 10.1023/A:1023027929079
Date available: 2009-09-22T18:00:45Z
Last updated: 2020-07-02
Stable URL:
Reference: [1] L. D. Brown: Fundamentals of Statistical Exponential Families. Lecture Notes No. 9.Institute of Mathematical Statistics, Hayward, California, 1986. MR 0882001
Reference: [2] L. Fahrmeir, H. Kaufman: Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models.Annals of Statistics 13 (1985), 342–368. MR 0773172, 10.1214/aos/1176346597
Reference: [3] F. R. Hampel, P. J. Rousseeuw, E. M. Ronchetti, W. A. Stahel: Robust Statistics: The Approach Based on Influence Functions.Wiley, New York, 1986. MR 0829458
Reference: [4] J. Jurečková, B. Procházka: Regression quantiles and trimmed least squares estimator in nonlinear regression model.Nonparametric Statistics 3 (1994), 201–222. MR 1291545
Reference: [16] J. Jurečková, P. K. Sen: Robust Statistical Procedures.Wiley, New York, 1996. MR 1387346
Reference: [5] R. Koenker, G. Basset: Regression quantiles.Econometrica 46 (1978), 33–50. MR 0474644, 10.2307/1913643
Reference: [6] F. Liese, I. Vajda: Consistency of $M$-estimates in general regression models.J. Multivar. Analysis 50 (1994), 93–114. MR 1292610, 10.1006/jmva.1994.1036
Reference: [7] F. Liese, I. Vajda: Necessary and sufficient conditions for consistency of generalized $M$-estimates.Metrika 42 (1995), 291–324. MR 1380211, 10.1007/BF01894328
Reference: [8] E. L. Lehman: Theory of Point Estimation.Wiley, New York, 1983. MR 0702834
Reference: [9] S. Morgenthaler: Least-absolute-deviations fits for generalized linear models.Biometrika 79 (1992), 747–754. Zbl 0850.62562, 10.1093/biomet/79.4.747
Reference: [10] J. Pfanzagl: Parametric Statistical Theory.De Gruyter, Berlin, 1994. Zbl 0807.62016, MR 1291393
Reference: [11] D. Pollard: Convergence of Stochastic Processes. Springer, New York.1984. MR 0762984
Reference: [15] D. Pollard: Empirical Processes: Theory and Applications.IMS, Hayward, 1990. Zbl 0741.60001, MR 1089429
Reference: [12] D. Pollard: Asymptotics for least absolute deviation regression estimators.Econometric Theory 7 (1991), 186–199. MR 1128411, 10.1017/S0266466600004394
Reference: [13] A. van der Vaart, J. A. Wellner: Weak Convergence and Empirical Processes.Springer, New York, 1996. MR 1385671
Reference: [14] S. Zwanzig: On $L_1$-norm estimators in nonlinear regression and in nonlinear error-in-variables models.IMS Lecture Notes 31, 101–118, Hayward, 1997. Zbl 0935.62074, MR 1833587


Files Size Format View
AplMat_44-1999-4_1.pdf 460.6Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo