Question

Q. 4 Let X1, ... , Xn be a random sample from gamma (2,0) distribution. c) Show that it is the regular case of the exponential class of distributions. d) Find a complete, sufficient statistic for 0. e) Find the unique MVUE of 0. Justify each step.

Answer 1

Solution : (h) Given that yord. Sample from Let XV, .... Xn be a random gamma (a,0) distribution le; X; IN Gamma (2,0) = 1,.....n (C) Hege fxi (27) = 02 e-Oxi x:2-1 va í Riso = 02 e-oxi xi ; X; >O NOCI L (%,0) = joint distribution of X, --..Xn = fxi (227) π xi - Bone-o* xi = exp [ing pan e-*** 1 »] = exp [an logo -o x + 2 109 xi] = exp (ACC) + MCO) TCX) + h(x)]-> . Egow is the form of excponential family

Where ACO) = an logo MCO) - - 0 T(X) = xi h(x) = 2 / logxi (d) Here 5(8,0) = gan e-o Exi{xi} = n(%) golo) Where hoxes = axi go (T= 2x1) = 6e-oToan. and This is of the form of a factorization due to Neymann- Fisher factorization Theorem · .. 't is sufficient Stastistic and the family of distribution belongs to excponential family. 'T' is aløso be complete.

(e) Here T = 3 Xi is complete sufficient Stastic and E (OCT)) = 8 then get) is UMVUE of o. Let xivind Gamma (2,0) 4. T o xiw Gamma (20, 0) Thus → frit) = oan e-ot tan-1 Van NOQ 2-1 - dt E(+) - St Pt (t) dt se po eso $ 2104 at per Se-ot tvan-1-1 dt. Van 1. pelo - sen oan en-) Va ean- 2n-1 :: E(5917 - 0 E e

.: Elget] = 0 Hence get) = ant! is UMVUE of '6 * ** Thanking you -