#StackBounty: #mathematical-statistics #confidence-interval #maximum-likelihood #asymptotics How can I obtain an asymptotic \$1-alpha\$ …

Bounty: 50

Let $$X sim Gamma(alpha,1)$$ and $$Y|X=x sim Exp(frac{1}{theta x}), alpha >1$$ and $$theta >0$$ are unknown. Let $$tau=E(Y)$$. Suppose that based on the random sample $$Y_1,…,Y_n$$, we have MLEs, $$hat{alpha}$$ and $$hat{theta}$$. Use these MLEs to develop an asymptotic $$1-alpha$$ confidence interval for $$tau$$.

my work:

First, I need to find $$tau=E(Y)=E(frac{1}{theta x})=frac{1}{theta}E(frac{1}{x})$$. We use a transformation of $$T=frac{1}{X}$$, where $$f_T(t)=frac{1}{Gamma(alpha)t^{alpha+1}}e^{-1/t},t>0$$. However, I am having trouble evaluating $$E(T)=int^infty_0frac{1}{Gamma(alpha)t^{alpha}}e^{-1/t}dt$$.

Assuming we have $$tau$$, we can get the asymptotic $$1-alpha$$ CI by using the asymptotic property of MLE. We know that $$hat{alpha}sim AN(alpha,frac{1}{ni(alpha)})$$ and $$hat{theta} sim AN(theta,frac{1}{ni(theta)})$$. However, I am failing to see how I can obtain the asymptotic CI for $$tau$$.

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