Two-Stage Fixed-Width Confidence Intervals for a Normal Mean in the Presence of Suspect Outliers

Abstract

Abstract 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 ≡ a 2σ2/d 2 where a ≡ a α is the upper 50α% point of N(0, 1). In his fundamental two-stage procedure, Stein (1945 Stein , C. ( 1945 ). A Two Sample Test for a Linear Hypothesis Whose Power Is Independent of the Variance , Annals of Mathematical Statistics 16 : 243 – 258 .[Crossref] , [Google Scholar], 1949 Stein , C. ( 1949 ). Some Problems in Sequential Estimation , Econometrica 17 : 77 – 78 . [Google Scholar]) estimated C by replacing σ2 with a sample variance from pilot data of size m(≥2) and a with t m−1, α/2. Mukhopadhyay (1982 Mukhopadhyay , N. ( 1982 ). Stein's Two-Stage Procedure and Exact Consistency, Skandinavisk Aktuarietdskrift pp. 110 – 122 . [Google Scholar]) 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, t m−1, α/2 must be replaced by the upper 50α% point corresponding to a pivotal distribution of the sample mean standardized appropriately by U m . This way, we explore the role of Mukhopadhyay (1982 Mukhopadhyay , N. ( 1982 ). Stein's Two-Stage Procedure and Exact Consistency, Skandinavisk Aktuarietdskrift pp. 110 – 122 . [Google Scholar]) 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 Mukhopadhyay , N. and Duggan , W. T. ( 1997 ). Can a Two-Stage Procedure Enjoy Second-Order Properties? Sankhya, Series A 59 : 435 – 448 . [Google Scholar]) 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.