Let the random variables be observations from a general
discrete parameter stochastic process , , whose probability laws are
of a known functional form, but dependent on a finite dimensional parameter
. Asymptotically optimal tests for testing a null hypoth-
esis 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 , although all other
underlying results hold for . The general case () will be discussed
elsewhere.