×

Inaccuracy rates and Hodges-Lehmann large deviation rates for parametric inferences with nuisance parameters. (English) Zbl 0845.62021

Summary: In the context of parametric inference for a scalar parameter \(\beta\) in the presence of a finite-dimensional nuisance parameter \(\lambda\) based on a large random sample \(X_1, \dots, X_n\), this paper calculates an exact one-sided inaccuracy rate for maximum-likelihood and \(M\)-estimators, as well as the Hodges-Lehmann large deviation rate [J. L. Hodges and E. L. Lehmann, Ann. Math. Stat. 27, 324-335 (1956; Zbl 0075.29206)] for type-II error probabilities under fixed alternatives. The method is to couple the large-deviation theorems of P. Groeneboom et al. [Ann. Probab. 7, 553-586 (1979; Zbl 0425.60021)] for empirical measures with a characterization via the Implicit Function Theorem of ‘least favorable measures’ extremizing the Kullback-Leibler information functional over statistically interesting sets of measures.

MSC:

62F12 Asymptotic properties of parametric estimators
62F05 Asymptotic properties of parametric tests
60F10 Large deviations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andersen, P. K.; Gill, R. D., Cox’s regression model for counting processes: a large approach, Ann. Statist., 10, 1100-1120 (1982) · Zbl 0526.62026
[2] Bahadur, R. R., On the asymptotic efficiency of tests and estimators, Sankhya, 22, 229-252 (1960) · Zbl 0109.12503
[3] Bahadur, R. R., Rates of convergence of estimates and tests, Ann. Math. Statist., 38, 303-324 (1967) · Zbl 0201.52106
[4] Bahadur, R. R., Some Limit Theorems in Statistics (1971), SIAM: SIAM Philadelphia · Zbl 0257.62015
[5] Bahadur, R. R.; Gupta, J. C.; Zabell, S. L., Large deviations, tests and estimates, (Chakravarti, I. M., Asymptotic Theory of Statistical Tests and Estimation (1980), Academic Press: Academic Press New York), 33-64 · Zbl 0601.62037
[6] Beckenbach, E. F.; Bellman, R., Inequalities (1961), Springer: Springer Berlin · Zbl 0206.06802
[7] Cramer, H., Mathematical Methods of Statistics (1946), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0063.01014
[8] Cziszar, I., \(I\)-divergence geometry of probability distributions and minimization problems, Ann. Probab., 3, 146-158 (1975) · Zbl 0318.60013
[9] Fu, J. C., The rate of convergence of consistent point estimators, Ann. Statist., 3, 234-240 (1975) · Zbl 0303.62020
[10] Fu, J. C., Large sample point estimation: a large deviation theory approach, Ann. Statist., 10, 762-771 (1982) · Zbl 0489.62031
[11] Groeneboom, P.; Oosterhoff, J.; Ruymgaart, F. H., Large deviation theorems for empirical probability measures, Ann. Probab., 7, 553-586 (1979) · Zbl 0425.60021
[12] Hodges, J. L.; Lehmann, E. L., The efficiency of some nonparametric competitors of the \(t\) test, Ann. Math. Statist., 27, 324-335 (1956) · Zbl 0075.29206
[13] Kester, A. D.M.; Kallenberg, W. C.M., Large deviations of estimators, Ann. Statist., 14, 648-664 (1986) · Zbl 0603.62028
[14] Kallenberg, W. C.M.; Ledwina, T., On local and nonlocal measures of efficiency, Ann. Statist., 15, 1401-1420 (1987) · Zbl 0651.62040
[15] Kelley, J. L., General Topology (1955), Van Nostrand Reinhold Company: Van Nostrand Reinhold Company New York · Zbl 0066.16604
[16] Koutsoukos, A., Probabilities of moderate and large deviations of test Statistics and estimators in the presence of nuisance parameters, (Dissertation (1990), University of Maryland at College Park)
[17] Mogulskii, A. A., Large deviations for the maximum likelihood estimators, Lecture Notes in Mathematics, 1299, 526-531 (1988)
[18] Serfling, R. J., Approximation Theorems of Mathematical Statistics (1980), Wiley: Wiley New York · Zbl 0423.60030
[19] Sievers, G. L., Estimates of location: a large deviation comparison, Ann. Statist., 6, 610-618 (1978) · Zbl 0395.62034
[20] Yosida, K., Functional Analysis (1965), Springer: Springer Berlin · Zbl 0126.11504
[21] Wald, A., Note on the consistency of maximum likelihood estimate, Ann. Math. Statist., 20, 595-601 (1949) · Zbl 0034.22902
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.