Question

1. Let Xi...., X, be a random sample from a distribution with pdf f(x;0) = 030-11(0 < x < 1), where 0 > 0. Find the maximum likelihood estimator of u = 8/1 b) Find a sufficient statistic and check completeness. (c) Find the UMVUE(uniformly minimum variance unbiased estimator of each of the following : 0,1/0,4 = 0/(1+0).

Answer 1

f (x; 0) =ox olaci Likelihood 1/2; 2) = oth (audz...an) GOLXicisti = 1cion 12 en L (20) = n en o + (0-1). Izn xi [ en L (2;o)] = 2 + lnxe), 1 = [enL (2; 0)]=0 n + Sensizo > ô =-n I en xi

2 [ en 2 (2 : 0)] - 1 do {en L 12:07] en se lozo Hence MLE walue at o=0; we get our as o=0 in the maximum n Elnai & MLE of o

If ô in MLE of o; then & Co is MLE of 4 (0) where 4(0) is a for of o. Soz MLE of ito = t õ I enxi Ienai Ienain Elnai . n-Elnai I e an si n MLE of M= lisensul

d? pact FC2,0) = 0x8-1 = o eco-Denx luns lifts = aco) box) ecco) dias. ph b(x) = 1 (e) - 0-1 a. lx) = nx- 30, The Family belongs to one parameter exponential Family of distributzon so, Idcxi) - Žencxi) is complete and sufficient For o XN Beta (6, 1) -goenx ~ ² =>-20 Eenxi nx²en. Ixi-- tede y=-202&nxinxen. Elintan E(Y)= an z) E ( – 20 Imxi) = 2n => FC - Zenxi) =% Ti(x) = - Eenxi and E(%) = 1 an - 2 => E(-20 Zenxi) - 2n-2 → E ) = o ; Te (X) -- Iinxi Ticx) and Telx) both are Fonction of complete suft stat Ienxi. so, By Lehmann scheffe theoren TICX) and Tz(x) sre UMVUE For Yo and o

E(X)= - Ite => ECExi) = no ito Ito enx and x are one to one Fonction so X és a Fonction of comple sufficient statistie non so, By Lehman scheffe theorem T3 (X) = 7 ir UMUOE for T3 (4) Ito :=
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