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.

Keywords:

• Average number of steps
• Average sample number
• Chi-square
• Estimation after SPRT
• Maximum likelihood
• Most powerful test
• Noncentrality
• Operating characteristic
• Random SPRT
• Sequential probability ratio test
• Termination
• Truncation
• Uniformly most powerful test

Subject Classifications:

• 62L10
• 62L12
• 62H15

ACKNOWLEDGMENT

We thank an Associate Editor for many enthusiastic comments and suggestions.

Notes

^aBold entries correspond to the alternative hypothesis.

Recommended by T. K. S. Solanky