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For testing H_{0}: Ï€_{j}= Ï€_{j0}j = 1,. . . ,c, using sample multinomial proportions {Ï€Ì‚_{j}}, the likelihood-ratio statistic (1.17) is

Show that G^{2} â‰¥ 0, with equality if and only if Ï€Ì‚_{j }= Ï€_{j0 }for all j. (Apply Jensenâ€™s inequality to E(â€“2n log X), where X equals Ï€_{j0}/Ï€Ì‚_{j }with probability Ï€Ì‚_{j}.)

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