LetXi1#x2026;Xin, be a random sample from a symmetric distribution with absolutely continuous cumulative distribution function (cdf)Fi(x) =F(x/#x3B8;i) for some unknown distribution functionF(.) and unknown scale parameter#x3B8;i0, i = l,#x2026;,k. The problem of testing the null hypothesisHo:#x3B8;1= #x2026; =#x3B8;kagainst the ordered alternativeH1:#x3B8;1#x2264; #x2026; #x2264;#x3B8;k, with at least one strict inequality is considered. The proposed test statistics, for testingHoagainstH1, are linear combinations of Hodges-Lehmann (1963) type of estimators of#x3B8;i+1/#x3B8;i,i= l,#x2026;,k-l, given by Bhattacharyya (1977). Optimum weights of linear combinations are obtained and asymptotic relative efficiencies have been computed. A numerical example is also given to demonstrate the implementation of the proposed test.
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