Asymptotically Optimal Tests under a General Dependence Set-Up

G.G. Roussas and D. Bhattacharya (2010). Asymptotically Optimal Tests under a General Dependence Set-Up. Journal of Statistical Research, Vol. 44, No. 1, pp. 57-83.

Abstract

Let the random variables $X_0, X_1, \ldots, X_n$ be $(n+1)$ observations from a general discrete parameter stochastic process $\{X_n\}$ , $n \geq 0$ , whose probability laws are of a known functional form, but dependent on a finite dimensional parameter $\theta \in \Theta \subset \mathbb{R}^k$ , $k \geq 1$ . Asymptotically optimal tests for testing a null hypothesis $H_0:~\theta = \theta_0$ against a composite alternative for Locally Asymptotically Normal (LAN) and Locally Asymptotically Mixture of Normal (LAMN) models are derived, using the results on asymptotic expansion of the log-likelihood ratio statistic (in the probability sense), its asymptotic distribution, asymptotic distribution of certain random quantities which are closely related to the log-likelihood ratios, and an exponential approximation result on the log-likelihood ratio statistic. The concepts of contiguity, differentially equivalent probability measures and differentially sufficient statistics play a key role in deriving the results. The testing hypothesis problem is restricted to the case that $k=1$ , although all other underlying results hold for $k \geq 1$ . The general case ( $k \geq 1$ ) will be discussed elsewhere.