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We revisit fixed-width (= 2d) confidence interval procedures with a preassigned confidence coefficient (≥1 - α) for the mean μ of a normal distribution when its variance σ2 is unknown. Had σ been known, the required optimal fixed sample size would be C ≡ a2σ2/d2 where a ≡ aα is the upper 50α% point of N(0, 1). In his fundamental two-stage procedure, Stein (1945, 1949) estimated C by replacing σ2 with a sample variance from pilot data of size m(≥2) and a with tm-1, α/2. Mukhopadhyay (1982) opened the possibility of incorporating less traditional estimators of σ2.We focus on estimating σ2 by a statistic defined via Gini's mean difference (GMD), mean absolute deviation (MAD), and range. Such modifications are warranted especially when we suspect one or more outliers but normality of the data may not be questioned by a standard test for normality assumption. Obviously, then, tm-1, α/2 must be replaced by the upper 50α% point corresponding to a pivotal distribution of the sample mean standardized appropriately by Um. This way, we explore the role of Mukhopadhyay (1982) two-stage confidence interval procedure when the requisite sample size is determined through GMD, MAD, or range. Associated exact and some asymptotic first-order properties are developed first. Next, we revisit Mukhopadhyay and Duggan (1997) updated two-stage methodology that was proposed when a known positive lower bound was available for σ2 in the present light. We highlight associated (i) first-order efficiency properties for all proposed fixed-width confidence interval procedures and (ii) second-order efficiency property of the GMD-based procedure. These are accompanied with extensive data analyses via simulations as well as real data. © 2013 Copyright Taylor and Francis Group, LLC.

School of Management & Entrepreneurship

World University root node

Sequential Analysis

Two-stage fixed-width confidence intervals for a normal mean in the presence of suspect outliers

Journal of the Japan Statistical Society

Asymptotic expansion of the percentiles for a sample mean standardized by GMD in a normal case with applications

It may not be an overstatement that one of the most widely reported measures of variation involves S 2, the sample variance, which is also well-known to be alternatively expressed in the form of an U statistic with a symmetric kernel of degree 2 whatever be the population distribution function. We propose a very general new approach to construct unbiased estimators of a population variance by U statistics with symmetric kernels of degree higher than two. Surprisingly, all such estimators ultimately reduce to S 2 (Theorem 3.1). While Theorem 3.1 is interesting and novel in its own right, it leads to a newer interpretation of S 2 that is much broader than what is known in the statistical literature including economics, actuarial mathematics, and mathematical finance. © 2012 Springer-Verlag.

Statistical Papers

On a new interpretation of the sample variance

Journal of Combinatorics, Information \& System Sciences 38 (1-4)

On the class of U-statistics based two-sample test for location

We pay tribute to the remarkable work of Frank Anscombe published in 1952, who invented the random central limit theorem (random CLT). This path-breaking contribution subsequently influenced practically the whole spectrum of sequential analysis very heavily. It is a humble attempt on our part to briefly sketch the breadth of its impact by connecting Anscombe's work with some of the major developments in sequential analysis that came afterwards. © 2012 Copyright Taylor and Francis Group, LLC.

A tribute to Frank Anscombe and random central limit theorem from 1952

University of Connecticut

Performance of U-Statistics Having Kernels of Degree Higher Than Two in Inference Problems with Applications

Statistical methodologies and their practice often rely upon tests and confidence interval procedures based on Studentized sample means of independent observations from a normal parent population and their Student's t distributions. This is specially so when the sample size n is small. An unmistakable impression one is left with, whether implied or not, is that such exact Student's t distributions may not be valid when the observations are dependent or non-normal. We show that one cannot discard the possibility of an exact Student's t distribution for a Studentized sample mean, or its suitable multiple, simply because the observations may be dependent or non-normal. In arriving at this conclusion, we have uncovered a very interesting and seemingly unknown feature (Theorem 2.1) of an n-dimensional multivariate t distribution with equi-correlation p, arbitrary degree of freedom v, and arbitrary n.

Metron

Examples of non-normal observations for which a studentized sample mean has a student's T-distribution

{Supplementary Material for “Estimation of the Coefficient of Variation with Minimum Risk: A Sequential Method for Minimizing Sampling Error and Study Cost”(Multivariate {\ldots}}