On SPRT and RSPRT for the Unknown Mean in a Normal Distribution with Equal Mean and Variance

Abstract

We consider a sequence of independent observations from a N(θ, θ) distribution with 0 < θ, c < ∞. Here, θ remains unknown. Given independent observations X 1,…, X n from N(θ, θ), the statistic is complete and sufficient for θ and T has a noncentral with λ =nθ when θ is the true parameter value. The problem is one of testing H 0: θ = θ0 vs. H 1: θ = θ1 where θ0, θ1 (θ0 ≠ θ1) are specified positive numbers with target type-I and type-II error probabilities 0 < α <1 and 0 < β <1, respectively, α + β <1. We first adapt Wald's sequential probability ratio test (SPRT) and describe some crucial characteristics. However, observations may instead arrive with random group sizes sequentially rather than one at a time. Thus, as a follow-up, next we implement Mukhopadhyay and de Silva's (2008) notion of a random SPRT (RSPRT). In the case of both SPRT and RSPRT, we address the issue of truncation. The proposed methodologies are illustrated with appropriate statistical data analysis. In the end, we briefly explore the problem of confidence interval estimation for the parameter θ after termination of the SPRT.