## Brazilian Journal of Probability and Statistics

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

### Contiguity and irreconcilable nonstandard asymptotics of statistical tests

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

#### 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 *χ̅*^{2}- and *E̅*-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