The Exact Distributions of the Stopping Times and Their Functionals in Two-Stage and Sequential Fixed-Width Confidence Intervals of the Exponential Parameter

Abstract

The asymptotic properties of two-stage and sequential procedures for fixed-width, 2δ, confidence interval estimation of the mean β of an exponential distribution are well known. In the present paper we derive the exact operating characteristics of these procedures for any given δ > 0. The methodology is similar to that of Zacks and Mukhopadhyay (2005 Zacks , S. and Mukhopadhyay , N. ( 2005 ). Exact Risks of Sequential Point Estimators of the Exponential Parameter , Sequential Analysis 25 : 203 – 220 .[Taylor & Francis Online] , [Google Scholar]); (2006 Zacks , S. and Mukhopadhyay , N. ( 2006 ). Bounded Risk Estimation of the Exponential Parameter in a Two-Stage Sampling , Sequential Analysis 25 : 1 – 16 . [Google Scholar]). The results are, however, more explicit.     

On Two-Stage Sampling for Fixed-Width Interval Estimation of the Common Variance of Equi-Correlated Normal Distributions

Abstract

The present article presents two-stage procedures for fixed-width interval estimators of the common variance of equi-correlated normal distributions. This study is a continuation of those of Zacks and Ramig (1987 Zacks , S. and Ramig , P. ( 1987 ). Confidence Intervals for the Common Variance of Equicorrelated Normal Random Variables , in Contributions to the Theory and Applications of Statistics, A Volume in Honor of Herbert Soloman , New York : Academic Press . [Google Scholar]) and Ghezzi and Zacks (2005 Ghezzi , D. J. and Zacks , S. ( 2005 ). Inference on the Common Variance of Correlated Normal Random Variables , Communications in Statistics - Theory and Methods 34 : 1517 – 1531 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We derive formulae for the distributions of total sample sizes and their functionals.     

Modified Three-Stage Sampling for Fixed-Width Interval Estimation of the Common Variance of Equicorrelated Normal Distributions

Abstract

In this work, we present a modified three-stage sampling procedure to construct fixed-width confidence intervals of the common variance of equicorrelated normal distributions. We derive the exact distribution of the stopping variable of this sampling scheme and also the exact distribution of the estimator of the common variance at stopping. The modified three-stage sampling substantially reduces the expected sample size compared to that of the two-stage sampling scheme of Haner and Zacks (2013 Haner , D. M. and Zacks , S. ( 2013 ). On Two-Stage Sampling for Fixed-Width Interval Estimation of the Common Variance of Equi-Correlated Normal Distributions , Sequential Analysis 32 : 1 – 13 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The coverage probabilities of the proposed interval estimators are computed exactly and are compared with the coverage probabilities obtained by two-stage sampling. We derive exact formulae for the functionals of the stopping variable and the estimator of the common variance at stopping.