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 fi nite 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.

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