| 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:
|
http://hdl.handle.net/10338.dmlcz/134412 |
| . |
| 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 |
| . |
