On the Positivity of Linear Combinations of Three Order Statistics as Estimators of Scale

• Activity Number: 406
• Type: Contributed
• Date/Time: Thursday, August 15, 2002 : 10:30 AM to 12:20 PM
• Sponsor: General Methodology
• Author(s): Dale Umbach*+
• Affiliation(s): Ball State University
• Keywords: Order Statistics ; Spacings ; Scale ; Positivity

Abstract

We consider estimation of the scale parameter in a continuous location/scale model based on three order statistics, X1 < X2 < X3, from a sample of size n. One can form many reasonable linear unbiased estimators of the scale parameter by various choices of coefficients. Since the scale parameter is not negative, a desirable property of such an estimator would be that it is non-negative with probability one. However, it turns out that this property does not hold for some very reasonable choices of coefficients.

Bai, Sarkar, and Wang (1996) Statist. & Prob. Let. 32, 181-188, establish positivity for the BLUE of scale for a large class of distributions, namely the log-convex distributions. However, positivity has not been established in general for the BLUE when three or more order statistics are involved.

In this paper, we prove the following. Let CV1 be the coefficient of variation for X2 - X1, CV2 be the coefficient of variation for X3 - X2, and CORR be the correlation between X3 - X2 and X2 - X1, which are invariant under changes of location and scale. Then, the BLUE for scale is nonnegative with probability one if, and only if, CV1/CV2 > CORR and CV2/CV1 > CORR.