Question

On this page, you will be given a distribution and another distribution conditional on the first one. Then, you will find the posterior distribution in a Bayesian approach. You will compute the Bayesian estimator, which is defined in lecture as the mean of the posterior distribution. Then, determine if the Bayesian estimator is consistent and/or asymptotically normal.

We recall that the Gamma distribution with parameters \, q>0 \, and \, \lambda >0 \, is the continuous distribution on \, (0,\infty ) \, whose density is given by \, \displaystyle f(x)=\frac{\lambda ^ q x^{q-1}e^{-\lambda x}}{\Gamma (q)} \,, where \, \Gamma \, is the Euler Gamma function \, \Gamma (q)=\int _0^\infty t^{q-1} e^{-t} dt \,, and its mean is \, q/\lambda \,.

We also recall that the \, \textsf{Beta}(a,b) \, distribution has the density \, \displaystyle f(x) = \frac{x^{a-1}(1-x)^{b-1}}{B(a,b)} \, and expectation \, a/(a+b) \,, where \, \displaystyle B(a,b) = \frac{\Gamma (a) \Gamma (b)}{\Gamma (a+b)} \,.

\, p\sim \textsf{Beta}(a,b) \, for some \, a,b>0 \, and conditional on \, p \,, \, X_1,\ldots ,X_ n\stackrel{i.i.d.}{\sim } \textsf{Ber}(p) \,.

What is the Bayesian estimator \hat{p}^{\text {Bayes}}?

(If applicable, enter barX_n for \bar{X_ n},\, max_n for \displaystyle \text {max}_{i=1\ldots n} X_ i. )

\hat{p}^{\text {Bayes}}=\quad
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Determine whether the Bayesian estimator is consistent, and whether it is asymptotically normal.



Consistent and asymptotically normal

Consistent but not asymptotically normal

Asymptotically normal but not consistent

Neither consistent nor asymptotically normal
unanswered
If it is asymptotically normal, what is its asymptotic variance V(a,b,p)? If it is not asymptotically normal, type in \, 0 \,.

V(a,b,p)=\quad