Question

7.77. If X1, X2,.., X, is a random sample from a distribution with p.d.f. f(x;0)=0*xe-, 0 <x< 00, zero elsewhere, where 0 e< ao: (a) Find the m.l.e., 6. of 0. Is 6 unbiased? X and then compute E(0). Hint: First find the p.d.f. of Y = (b) Argue that Y is a complete sufficient statistic for 8. (c) Find the unbiased minimum variance estimator of 0. (d) Show that X/Y and Y are (e) What is the distribution of X,/Y? independent.

Answer 1

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E() EC Bn-1 Simee E(6)D therfore is not untiase o b The plf 마 7 is 3m-1 -0y 3n 3n Whore a(e) 3 clo)-0 Therefore the distoibution of belonx to one parameter expaneneial faray of Listribuution; which means dl= 3 is O, but also not only uffici ent for minimal affieiedt & complete yis complete suffieient sttistiefon. Aacording to Leymann- Sehefe Thae O baged e Knovsthat ant completz sufficietd atis tiebr is the UMVUE u.e On 3n Now here, El6). 3n-1 3n-1 is unbased for 0 2n-1

that yjs eemplete have seen Vve sufficient tatistic bor Thrafore by Legronn Scheffe thesren we can eoneud e thatB2-1)s theorrers eymonn Schffe UMVUE fOr & e It the distribution of X P (3, 3(n-1) early shaer that Can be