## Efficient estimation of Banach parameters in semiparametric models.(English)Zbl 1065.62053

Summary: Consider a semiparametric model with a Euclidean parameter and an infinite-dimensional parameter, to be called a Banach parameter. Assume:
(a) There exists an efficient estimator of the Euclidean parameter. (b) When the value of the Euclidean parameter is known, there exists an estimator of the Banach parameter, which depends on this value and is efficient within this restricted model.
Substituting the efficient estimator of the Euclidean parameter for the value of this parameter in the estimator of the Banach parameter, one obtains an efficient estimator of the Banach parameter for the full semiparametric model with the Euclidean parameter unknown. This hereditary property of efficiency completes estimation in semiparametric models in which the Euclidean parameter has been estimated efficiently. Typically, estimation of both the Euclidean and the Banach parameter is necessary in order to describe the random phenomenon under study to a sufficient extent. Since efficient estimators are asymptotically linear, the above substitution method is a particular case of substituting asymptotically linear estimators of a Euclidean parameter into estimators that are asymptotically linear themselves and that depend on this Euclidean parameter. This more general substitution case is studied for its own sake as well, and a hereditary property for asymptotic linearity is proved.

### MSC:

 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 46N30 Applications of functional analysis in probability theory and statistics 62J05 Linear regression; mixed models 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text:

### References:

 [1] Begun, J. M., Hall, W. J., Huang, W.-M. and Wellner, J. A. (1983). Information and asymptotic efficiency in parametric-nonparametric models. Ann. Statist. 11 432–452. JSTOR: · Zbl 0526.62045 · doi:10.1214/aos/1176346151 [2] Beran, R. (1974). Asymptotically efficient adaptive rank estimates in location models. Ann. Statist. 2 63–74. JSTOR: · Zbl 0284.62016 · doi:10.1214/aos/1176342613 [3] Bickel, P. J. (1982). On adaptive estimation. Ann. Statist. 10 647–671. JSTOR: · Zbl 0489.62033 · doi:10.1214/aos/1176345863 [4] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models . Johns Hopkins Univ. Press, Baltimore. · Zbl 0786.62001 [5] Bolthausen, E., Perkins, E. and van der Vaart, A. W. (2002). Lectures on Probability Theory and Statistics . Lecture Notes in Math. 1781 . Springer, New York. · Zbl 0996.00039 · doi:10.1007/b93152 [6] Breslow, N. E. (1974). Covariance analysis of censored survival data. Biometrics 30 89–99. [7] Butler, S. and Louis, T. (1992). Random effects models with non-parametric priors. Statistics in Medicine 11 1981–2000. [8] Cosslett, S. (1981). Maximum likelihood estimator for choice-based samples. Econometrica 49 1289–1316. · Zbl 0494.62097 · doi:10.2307/1912755 [9] Cox, D. R. (1972). Regression models and life tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187–220. · Zbl 0243.62041 [10] Dionne, L. (1981). Efficient nonparametric estimators of parameters in the general linear hypothesis. Ann. Statist. 9 457–460. JSTOR: · Zbl 0478.62025 · doi:10.1214/aos/1176345414 [11] Gilbert, P. B., Lele, S. R. and Vardi, Y. (1999). Maximum likelihood estimation in semiparametric selection bias models with application to AIDS vaccine trials. Biometrika 86 27–43. · Zbl 0917.62061 · doi:10.1093/biomet/86.1.27 [12] Gong, G. and Samaniego, F. J. (1981). Pseudo maximum likelihood estimation: Theory and applications. Ann. Statist. 9 861–869. JSTOR: · Zbl 0471.62032 · doi:10.1214/aos/1176345526 [13] Huang, J. (1996). Efficient estimation for the proportional hazards model with interval censoring. Ann. Statist. 24 540–568. · Zbl 0859.62032 · doi:10.1214/aos/1032894452 [14] Johansen, S. (1983). An extension of Cox’s regression model. Internat. Statist. Rev. 51 165–174. JSTOR: · Zbl 0526.62081 · doi:10.2307/1402746 [15] Klaassen, C. A. J. (1987). Consistent estimation of the influence function of locally asymptotically linear estimates. Ann. Statist. 15 1548–1562. JSTOR: · Zbl 0629.62041 · doi:10.1214/aos/1176350609 [16] Klaassen, C. A. J. (1989). Efficient estimation in the Cox model for survival data. In Proc. Fourth Prague Symposium on Asymptotic Statistics (P. Mandl and M. Hušková, eds.) 313–319. Charles Univ., Prague. · Zbl 0698.62030 [17] Klaassen, C. A. J. and Putter, H. (1997). Efficient estimation of the error distribution in a semiparametric linear model. In International Symposium on Contemporary Multivariate Analysis and its Applications (K. Fang and F. J. Hickernell, eds.) B.1–B.8. Hong Kong. [18] Klaassen, C. A. J. and Putter, H. (2000). Efficient estimation of Banach parameters in semiparametric models. Technical report, Korteweg-de Vries Institute for Mathematics. · Zbl 1065.62053 [19] Koul, H. L. (1992). Weighted Empiricals and Linear Models . IMS, Hayward, CA. · Zbl 0998.62501 [20] Koul, H. L. (2002). Weighted Empirical Processes in Dynamic Linear Models , 2nd ed. Springer, New York. · Zbl 1007.62047 · doi:10.1007/978-1-4613-0055-7 [21] Koul, H. L. and Susarla, V. (1983). Adaptive estimation in linear regression. Statist. Decisions 1 379–400. · Zbl 0574.62056 [22] Le Cam, L. (1956). On the asymptotic theory of estimation and testing hypotheses. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 129–156. Univ. California Press, Berkeley. · Zbl 0074.13504 [23] Lehmann, E. L. (1999). Elements of Large-Sample Theory . Springer, New York. · Zbl 0914.62001 [24] Loynes, R. M. (1980). The empirical distribution function of residuals from generalised regression. Ann. Statist. 8 285–298. JSTOR: · Zbl 0451.62040 · doi:10.1214/aos/1176344954 [25] Müller, U. U., Schick, A. and Wefelmeyer, W. (2001). Plug-in estimators in semiparametric stochastic process models. In Selected Proc. Symposium on Inference for Stochastic Processes (I. V. Basawa, C. C. Heyde and R. L. Taylor, eds.) 213–234. IMS, Beachwood, OH. [26] Murphy, S. A., Rossini, A. J. and van der Vaart, A. W. (1997). Maximum likelihood estimation in the proportional odds model. J. Amer. Statist. Assoc. 92 968–976. · Zbl 0887.62038 · doi:10.2307/2965560 [27] Murphy, S. A. and van der Vaart, A. (2000). On profile likelihood (with discussion). J. Amer. Statist. Assoc. 95 449–485. · Zbl 0995.62033 · doi:10.2307/2669386 [28] Nielsen, G. G., Gill, R. D., Andersen, P. K. and Sørensen, T. I. A. (1992). A counting process approach to maximum likelihood estimation in frailty models. Scand. J. Statist. 19 25–43. · Zbl 0747.62093 [29] Owen, A. (1991). Empirical likelihood for linear models. Ann. Statist. 19 1725–1747. JSTOR: · Zbl 0799.62048 · doi:10.1214/aos/1176348368 [30] Pfanzagl, J. and Wefelmeyer, W. (1982). Contributions to a General Asymptotic Statistical Theory . Springer, New York. · Zbl 0512.62001 [31] Pierce, D. (1982). The asymptotic effect of substituting estimators for parameters in certain types of statistics. Ann. Statist. 10 475–478. JSTOR: · Zbl 0488.62012 · doi:10.1214/aos/1176345788 [32] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300–325. JSTOR: · Zbl 0799.62049 · doi:10.1214/aos/1176325370 [33] Randles, R. (1982). On the asymptotic normality of statistics with estimated parameters. Ann. Statist. 10 462–474. JSTOR: · Zbl 0493.62022 · doi:10.1214/aos/1176345787 [34] Schick, A. (2001). On asymptotic differentiability of averages. Statist. Probab. Lett. 51 15–23. · Zbl 1059.62512 · doi:10.1016/S0167-7152(00)00132-2 [35] Scholz, F. W. (1971). Comparison of optimal location estimators. Ph.D. dissertation, Univ. California, Berkeley. [36] Stone, C. (1975). Adaptive maximum likelihood estimators of a location parameter. Ann. Statist. 3 267–284. JSTOR: · Zbl 0303.62026 · doi:10.1214/aos/1176343056 [37] Tsiatis, A. A. (1981). A large sample study of Cox’s regression model. Ann. Statist. 9 93–108. JSTOR: · Zbl 0455.62019 · doi:10.1214/aos/1176345335 [38] van der Vaart, A. W. (1991). On differentiable functionals. Ann. Statist. 19 178–204. JSTOR: · Zbl 0732.62035 [39] van Eeden, C. (1970). Efficiency-robust estimation of location. Ann. Math. Statist. 41 172–181. · Zbl 0218.62043 · doi:10.1214/aoms/1177697197
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.