Question

Suppose that X1, X2, ..., Xn is an iid sample from the probability density function (pdf) given by where β > 0 is unknown and m is a known constant larger than 1. (a) Show that T-T(X)-Σ-i Xi is a complete and sufficient statistic for Ux(z|β) : β 〉 0} (b) Show that c) For t > 0, show that the conditional density of Xı, given T- t, is 「( mn 1_21) m(n-1)-1 (d) Show that m- 1 mn - 1 (e) What is the UMVUE for T()1/B?

Answer 1

$(a)~The~joint~distribution~of~X_1,X_2,...,X_n~is\\ f(x_1,x_2,..,x_n|\beta)=\frac{1}{(\Gamma(m))^n\beta^{nm}}\left( \prod_{i=1}^nx_i\right )^n\exp\left(-\frac{1}{\beta}\sum_{i=1}^nx_i \right )\\ =g_\beta(t)h(x_1,x_2,...,x_n)\\ where,~t=\sum_{i=1}^nx_i,~g_{\beta}(t)=\frac{1}{\beta^{nm}}\exp(-t/\beta),~h(x_1,...,x_n)=\frac{1}{(\Gamma(m))^n}\left( \prod_{i=1}^nx_i\right )^n\\ Hence~from~Neyman-Fisher's~factorization~theorem,~T=\sum_{i=1}^nX_i~is~sufficient~statistic~for~\theta.\\ Again~T~is~complete,~so~T~is~complete~sufficient~statistic~for~\theta.$$(b)~E_{\beta}\left(\frac{1}{X_1} \right )=\frac{1}{\Gamma(m)\beta^m}\int_0^\infty x^{m-2}e^{-x/\beta}dx=\frac{1}{\Gamma(m)\beta^m}\times \Gamma(m-1)\beta^{m-1}\\ =\frac{1}{(m-1)\beta}\Rightarrow E_{\beta}\left(\frac{m-1}{X_1} \right )=\frac{1}{\beta}.\\ (c)~Since~X_i's~are~independent~so~from~additive~property~of~gamma~distribution,\\ T=\sum_{i=1}^n X_i\sim Gamma(mn,\beta).\\ Then~f_{X_1|T}(x_1|t)=\frac{f_{X_1}(x_1)f_{T-X_1}(t-x_1)}{f_T(t)}\\~where~T-X_1\sim Gamma(m(n-1),\beta)\\ =\frac{\frac{1}{\Gamma(m)\beta^m}x_1^{m-1}e^{-x_1/\beta}\times \frac{1}{\Gamma(m(n-1))\beta^{m(n-1)}}(t-x_1)^{m(n-1)-1}e^{-(t-x_1)/\beta}}{\frac{1}{\Gamma(nm)\beta^{nm}}t^{nm-1}e^{-t/\beta}}\\ =\frac{\Gamma(nm)}{\Gamma(m)\Gamma(m(n-1))}\times \frac{1}{t}\left(\frac{x_1}{t} \right )^{m-1}\left(1-\frac{x_1}{t} \right )^{m(n-1)-1},~0<x_1<t\\ =0~otherwise\\$

$(d)~E\left(\frac{t}{X_1}|T=t \right )=\frac{\Gamma(nm)}{\Gamma(m)\Gamma(m(n-1))}\int_0^t\frac{x_1}{t}\left(\frac{x_1}{t} \right )^{m-1}\left(1-\frac{x_1}{t} \right )^{m(n-1)-1}dx_1\\ =\frac{1}{t}\times\frac{\Gamma(nm)}{\Gamma(m)\Gamma(m(n-1))}\int_0^1\left(y\right )^{m-2}\left(1-y \right )^{m(n-1)-1}dy;~take~y=x_1/t\\ =\frac{1}{t}\times\frac{\Gamma(nm)}{\Gamma(m)\Gamma(m(n-1))}\times \frac{\Gamma(m-1)\Gamma(m(n-1))}{\Gamma(mn-1)}=\frac{mn-1}{(m-1)t}\\ So,~E\left(\frac{m-1}{X_1}|T=t \right )=\frac{mn-1}{t}.\\ (e)~Since~\frac{m-1}{X_1}~is~an~unbiased~estimator~of~\frac{1}{\beta}~and~T~is~complete\\ ~sufficient~statistic~then~from~Lehmann~Scheff� ~theorem,~\frac{mn-1}{T}~is~UMVUE~for~\frac{1}{\beta}.$