#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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