Simultaneous one-sided confidence intervals for the ordered pairwise differences of exponential location parameters

Abstract

We consider k(k>=3) exponential populations such that an observation from ith population has the probability density function (pdf) f(x|μi, θi)=1/θi exp{-(x-μi)/θi} |(μi, ∞) (x), where μi>0, θi0 and I(.) is the indicator function, i =1,…,k. Simple test procedures for testing the null hypothesis H0:μ1= …=μk against the alternative hypothesis H1:μ1≤…≤μk with at least one strict inequality, are proposed in two situations : (i) θ1=…. =θk=θ (unknown) and (ii) all θ's equal to unity. For some significance levels α∞(0,1), exact critical points of each test procedure are tabulated for k=3,…, 9 by solving two or three dimensional integral equations. Simultaneous one-sided confidence intervals for all ordered pair wise differences μj-μi - (1≤i≤j≤k) and all nonnegative contrasts of μs, obtained by simple inversion of these test procedures, is discussed using these critical points. Chen (1982) proposed a test procedure for this problem in situation (i) and discussed simultaneous confidence intervals (SCIs) of linear contrasts of us. Our critical points are substantially smaller than the critical points proposed by Chen (1982). Statistical simulation, used to check the performance of the proposed critical points and the computation of powers, revealed that (i)the actual size levels of our critical points are almost conservative and (ii) the power of the proposed test relative to Chen's test is larger particularly for small sample size and tight slippage parameter configurations. An application of these results to Pareto family of distributions is also discussed.