For testing H0: πj = πj0 j = 1,. .

For testing H0: πj= πj0j = 1,. . . ,c, using sample multinomial proportions {π̂j}, the likelihood-ratio statistic (1.17) is

G² = -2n L îî; log(7;0/ î;). îr, log(™j0/ îr;,


Show that G2 ≥ 0, with equality if and only if π̂= Ï€j0 for all j. (Apply Jensen’s inequality to E(–2n log X), where X equals Ï€j0/π̂with probability π̂j.)

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