Bounds for the Power of Likelihood Ratio Tests and Their Asymptotic Properties

Abstract

Let $P_0$ and $P_1$ be two probability distributions on a measurable space $(\mathscr{H}, \mathscr{B})$. We consider the problem of testing the simple hypothesis $H: P = P_0$ against the simple alternative $K: P = P_1$ on the basis of $n$ independent random variables $X_1, X_2, \cdots, X_n$ with common distribution $P$. The method that is widely used in the literature for investigating large sample properties of optimal tests is the following: according to the Neyman-Pearson lemma an optimal test can be described by means of sums of independent random variables. Theorems from the theory of probabilities of large deviations or ergodic theorems then allow one to obtain results on the asymtotic behavior of the error probabilities of optimal tests. In this paper we use a representation of the power of an optimal test which has its origin in the duality theory of infinite linear programming, in order to derive upper and lower bounds for the power. The bounds holds for any sample size $n$. This is done in Section 1 for two different types of tests, namely for most powerful tests at leve $\alpha_n$ and for tests which minimize a weighted sum of the error probabilities. It turns out that those bounds allow one to derive the well-known asymptotic properties under slightly more general conditions, i.e., $\alpha_n$ is permitted to tend to zero faster than any negative power of $n$ (but not exponentially fast) and the weight $\lambda_n$ to tend exponentially fast to zero. This and another application are discussed in Section 2.