Sequential Probability Ratio Test for the Mean Value Function of a Gaussian Process

Abstract

Sequential probability ratio tests are defined for testing a simple hypothesis against a simple alternative for the mean value function of a real Gaussian process with known covariance kernel. Exact formulas are obtained for the error probabilities and the OC function using the fact that the $\log$ likelihood ratio process is a Gaussian process with independent increments and have continuous sample paths. An identity of a familiar nature holds for the expected value of the $\log$ likelihood ratio process at a random stopping time. In certain situations this identity yields an exact formula for the ASN function. Two examples are given. The analysis employs the theory of reproducing kernel Hilbert spaces.