Asymptotic Behavior of Expected Sample Size in Certain One Sided Tests

Let R be the set of real numbers, B₁ the set of Borel sets of R, and a -finite nonnegative measure on B₁. Let be an open real number interval (which may be infinite). Throughout we consider a Koopman-Darmois family equation*1\h () (x), \equation* of generalized probability density functions on the measure space (R, B₁, ). We consider one sided tests T of the hypothesis 0. In general, in this paper, T will be a sequential procedure. Associated with T is a stopping variable N (mention of the dependence of N on T is usually omitted). N 0. N = n means that sampling stopped after n observations and a decision was made. In this context we consider to be an integer, and N = means that sampling does not stop. In the discussion of Section 1 we will assume that if and 0 then P_ (N 0. The main result of this paper may be stated as follows. Theorem 1. Suppose (R, B₁, ), , and \h () (x), \ are as described above. Define align*_ = ^- h () x (x) (dx), \\ 2 \\ ² = ^- h (0) x² (dx), align* and assume ₀ = 0. Suppose 0 0} P_ (decide 0). equation* Then align* ₀+ ²_||_|\|^-1E_ N 2² P₀ (N \\ 4 ₀- ²_||_|\|^-1E_ N 2² P₀ (N &=). align* If + 0 then ₀ ²E_ N =. In Section 1, (7) and (8), it is shown that P₀ (N =) 1 - -. Consequently the relations (4) and (5) of Theorem 1 are not vacuous. We were led to formulate Theorem 1 by a problem of constructing bounded length confidence intervals. The relationship is explained in Section 2. The proof of Theorem 1 is given in Section 3.