Tests of composite hypotheses for random variables and random processes

Let a sequence of random variables $X(t)$, $t=\dots,-1,0,1,\dots$, or of a random process $X(t)$, $-\infty<t<\infty$, be observed at $t=1,\dots,T$ or $0\le t\le T$. The general problem is considered of testing composite hypotheses for probability distributions of $X(t)$ on the basis of observations. Two different tests are proposed, and it is shown that the asymptotic properties of both these tests coincide quite often with the asymptotic properties of the likelihood-ratio test. It is also shown that the proposed tests generalize many previously known tests. The special case of Gaussian and stationary $X(t)$ is studied.