Stein (1945: Annals of Mathematical Statistics; 1949: Econometrica) developed a fundamental two-stage procedure for constructing a fixed-width (=2d) confidence interval with a confidence coefficient at least (1 − α) for mean μ of a normal distribution when its variance σ2 is also unknown. We suppose that 0 < σ < ∞.
Had sigma been known, the required sample size would have been, C = a2σ2/d2, where (i) 2d (>0) is the preassigned length of the confidence interval centered at the sample mean, (ii) 0 < α < 1 is preassigned, and (iii) a is the upper 50α% point of a standard normal distribution. But, since σ was unknown, Stein estimated C by incorporating the sample variance from pilot data in place of σ2 and replacing a with am−1, the upper 50α% point of tm−1 distribution where m (≥2) is the pilot sample size.
We revisit Mukhopadhyay's (1982: Scandinavian Actuarial Journal) generalization of the Stein procedure to construct fixed-width confidence intervals of μ where we incorporate less traditional unbiased and/or consistent estimators, g2(U), of σ2 instead of the sample variance. Here, we focus on the idea of a basic statistic U defined via mean absolute deviation, range, Gini's mean difference while estimating or equivalently C. Obviously, then, am−1 would have to be replaced by an upper 50α% point corresponding to the pivotal distribution of the sample mean standardized by g(Um), n≥m.
In the face of observing some possible outliers when random samples are drawn from a normal population, we will explore the role of Mukhopadhyay's (1982) two-stage fixed-width confidence interval procedure for the normal mean when the requisite sample size is determined using the mean absolute deviation, range, or Gini's mean difference. We will summarize selected theoretical properties of such methodologies along with some interesting data analyses.
Keywords: Fixed-width confidence interval; Gini's mean difference; mean absolute deviation; range; sample variance; Pivotal distribution
Biography: Bhargab is a graduate student at the Department of Statistics, University of Connecticut, USA. He is pursuing his Phd under the supervision of Prof. Nitis Mukhopadhyay. He received both BSc and MSc degrees in Statistics from University of Calcutta, India in 2005 and 2007 respectively.