Question: For testing H 0 : Ï j = Ï j0 j = 1,. . . ,c, using sample multinomial proportions {ÏÌ j }, the likelihood-ratio

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 π̂j = Ï€j0 for all j. (Apply Jensen€™s inequality to E(€“2n log X), where X equals Ï€j0/π̂j with probability π̂j.)

G = -2n L ; log(7;0/ ;). r, log(j0/ r;,

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