Bayesian estimator
On this page, you will be given a distribution and another distribution conditional on the first one. There, you will find the posterior distribution in Bayesian approach. You will compute the Bayesian estimator, which is defined 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 α and β > 0 is the continuous distribution on (0, ∞) whose density is given by f(x) = (1/β^α) * (x^(α-1)) * e^(-x/β), where Γ(α) is the Euler Gamma function [Γ(α) = ∫(0 to ∞) t^(α-1) * e^(-t) dt], and its mean is αβ.
We also recall that the Beta (α, β) distribution has the density f(x) = (1/B(α, β)) * (x^(α-1)) * ((1-x)^(β-1)), where B(α, β) = Γ(α) * Γ(β) / Γ(α+β).
(a)
1. X, Ber(p)
Beta(a, b) for some α, β > 0 and conditional on P, X1 ≤ S
Now, what is the Bayesian estimator p?
(If applicable, enter barX_n for Xn; max(X_i) for max,X; Don't worry if the parser does not render properly; the grader works independently; If you wish to have proper rendering, enclose max(X_i) by brackets: [max(X_i)])
2. Determine whether the Bayesian estimator is consistent and whether it is asymptotically normal.
a. Consistent and asymptotically normal
b. Consistent but not asymptotically normal
c. Asymptotically normal but not consistent
d. Neither consistent nor asymptotically normal