Selection Procedures For Scale Parameters When The Populations Have A Common Known Quantile
by
Narinder Kumar
Department of Statistics, Panjab University, Chandigarh-160 014
Coauthors: Amar Nath Gill, G.P. Mehta

Consider k independent populations P1,…,Pk and let G((x-mi)/qi) be the cumulative distribution function (cdf) associated with population Pi for for some unknown absolutely continuous cdf G(.) and unknown location(scale) parameter mi Î (¥,¥) (qi > 0), i=1,…,k. We assume that these k populations have the same known quantile of order p (0<p<1), not necessarily equal to ½, i.e., for each i, the known quantile xp = mi + qI G-1(p), where G-1(p) Î (-¥,¥) is some unknown constant. Without loss of generality we assume that the common known quantile xp = 0 and the k distributions differ only in their scale parameters, i.e,, observations from Pi with xp = 0 have cdf Fi(x)=G((x+qiG-1(p))/ qi) = F(x/qi), and Fi(0)=p, i=1,…,k. If xp is different from zero, one may subtract it from all the observations so that the known quantile of order p of transformed observations, with cdf G((z + qi G-1(p))/ qi) = F(z/qi), i=1,…,k, is zero.

Date received: October 28, 2002