A CLASS OF EPSILON BAYES TESTS OF A SIMPLE NULL HYPOTHESIS ON MANY PARAMETERS IN EXPONENTIAL FAMILIES.

Xl,...,Xk are observed as independent one-dimensional random variables, Xi being the sufficient statistic for a parameter theta i in an exponential family. The problem of testing the hypothesis Ho theta i theta oi is considered when k is large. The usual hypothesis testing loss structure and near alternatives for moderate size and power are assumed. If epsilon 0 is given and k is large enough, it is shown that the Bayes test for a prior distribution pi making the theta i independent under Hl has Bayes risk at most epsilon less than a test based on a non-negative definite quadratic form which rejects Ho when summation aiXi biXi squared or c. The constants ai and bi are determined by the first two moments of theta i under pi, reducing the class of decision procedures under consideration. The approximate Bayes risk and location of the prior distribution under pi is determined. The quadratic forms are shown to be proper Bayes for fixed k. For large k, epsilon-minimax tests are determined. Applications of the above results are made to the binomial, negative binomial, Poisson, normal and gamma distributions by introducing the concept of quadratic moment structure. Some large deviation theorems are proved for exponential families. Author