A note on the asymptotic efficiency of the sobel–wald test

Abstract

Let X1,X2, … be iid random variables with the pdf f(x,θ)=exp(θx−b(θ)) relative to a σ-finite measure μ, and consider the problem of deciding among three simple hypotheses Hi:θ=θi (1⩽i⩽3) subject to P(accept Hi|θi)=1−α (1⩽i⩽3). A procedure similar to Sobel–Wald procedure is discussed and its asymptotic efficiency as compared with the best nonsequential test is obtained by finding the limit lima→0(EiN(a)/n(a)), where N (a) is the stopping time of the proposed procedure and n(a) is the sample size of the best non-sequential test. It is shown that the same asymptotic limit holds for the original Sobel–Wald procedure. Specializing to N(θ,1) distribution it is found that lima→0 (EiN(α)/n(α))=14 (i=1,2) and lima→0 (E3N(α)n(α))=δ21/4δ, where δi=(θi+1−θi) with 0<δ1⩽δ2. Also, the asymptotic efficiency evaluated when the X's have an exponential distribution.