Asymptotically optimal tests under a general dependence set-up

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
$\theta \in \Theta\subset \mathbb{R}$ . Asymptotically optimal tests for testing a null hypoth-
esis $H_0 :\theta =\theta_0$ 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 distri-
bution of certain random quantities which are closely related to the log-likelihood
ratios, and an exponential approximation result on the log-likelihood ratio statis-
tic. The concepts of contiguity, differentially equivalent probability measures and
differentially suffcient statistics play a key role in deriving the results. The test-
ing 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.

44n1_4

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