Brazilian Journal of Probability and Statistics

Contiguity and irreconcilable nonstandard asymptotics of statistical tests

Pranab K. Sen and Antonio C. Pedroso-de-Lima

Full-text: Open access

Abstract

Wald-type test statistics based on asymptotically normally distributed estimators (not necessarily maximum likelihood estimation or best asymptotically normal) provides an easy access to have tests for statistical hypotheses, far beyond the parametric paradigms. The methodological perspectives rest on a basic consistent asymptotic normal (CAN) condition which is interrelated to the well-known local asymptotic normality (LAN) condition. Contiguity of probability measures facilitates the C(L)AN condition in a relatively easier way. For many regular families of distributions, when statistical hypotheses do not involve nonstandard constraints, verification of contiguity of probability measures is facilitated by the well-known LeCam’s First Lemma [see Hájek, Šidák and Sen Theory of Rank Tests (1999), Chapter 7]. For nonregular families, though contiguity may hold under different setups, CAN estimators are not fully exploitable in the Wald type testing theory. This simple feature is illustrated by a two-parameter exponential model. Guided by this simple example, mixture of distributions are appraised in the context of Wald-type tests and the so-called χ̅2- and -test theory is thoroughly appraised. A general result on counter examples is presented in detail.

Article information

Source
Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 444-470.

Dates
First available in Project Euclid: 22 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1313973404

Digital Object Identifier
doi:10.1214/11-BJPS156

Mathematical Reviews number (MathSciNet)
MR2832896

Zentralblatt MATH identifier
1270.62082

Keywords
CAN statistics exponential family of densities Hellinger-distance LA(M)N likelihood ratio maximum likelihood estimator nonparametrics Rao’s score statistics regular family of estimators resampling plans

Citation

Sen, Pranab K.; Pedroso-de-Lima, Antonio C. Contiguity and irreconcilable nonstandard asymptotics of statistical tests. Braz. J. Probab. Stat. 25 (2011), no. 3, 444--470. doi:10.1214/11-BJPS156. https://projecteuclid.org/euclid.bjps/1313973404


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