A note on the correlation of ranges in correlated normal samples

Abstract

Eighteen years ago one of us (Hartley, 1950) considered the following problem: given a sample of n paired observations, x 1 , y 1 , i = 1,…, n, drawn from a bivariate normal population with correlation coefficient p, it is required to find the correlation coefficient between the x sample range and the y sample range. This study resulted in formula (19) of Hartley (1950), given as (1) below. Recently, Mardia (1967) claimed to have proved that this formula is in error. We show below that this claim is incorrect by establishing the identify of formula (1) with one of the formulae derived by Mardia (1967), equation. (5.2). Apparently, Mardia's faulty inference has, at least partly, been caused by the fact that the isolated evaluation of the above formula (19) made by Hartley was based on a method of numerical integration involving an unacceptably large truncation error. This fact had earlier been noticed by Kurtz et al . (1966). Further comments on the difficulty involved in evaluating the above formula (19) are given at the end of the present note.