\, \lambda \sim \textsf{Exp}(\alpha ) \, for some \, \alpha >0 \, and conditional on \, \lambda \,, \, X_1,\ldots ,X_ n\stackrel{i.i.d.}{\sim } \textsf{Exp}(\lambda ) \,.
What is the Bayesian estimator \hat{\lambda }^{\text {Bayes}}?
(If applicable, enter barX_n for \bar{X_ n},\, max_n for \displaystyle \text {max}_{i=1\ldots n} X_ i. )
\hat{\lambda }^{\text {Bayes}}=\quad
unanswered
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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(\lambda )? If it is not asymptotically normal, type in \, 0 \,. You may use the variable \lambda.
V(\lambda )=\quad
Since we have a conjugate prior, the exponential distribution, for the likelihood, the posterior distribution of λ is also an exponential distribution with parameter α + nλ and the Bayesian estimator of λ can be found by taking the expected value of the posterior distribution.
Thus, the Bayesian estimator \hat{λ}^{Bayes} is given by:
\hat{λ}^{Bayes} = E(λ | X_1, ..., X_n) = \frac{α + n \bar{X}_n}{n + 1}
where \bar{X}_n is the sample mean.
The Bayesian estimator is consistent, as it converges to the true value of λ as the sample size increases. However, it is not asymptotically normal, as it does not follow a normal distribution as the sample size increases.
Therefore, the correct answer is:
Consistent but not asymptotically normal
The asymptotic variance V(λ) is not applicable in this case as the Bayesian estimator is not asymptotically normal. Thus, V(λ) = 0.
To find the Bayesian estimator, we need to compute the posterior distribution of λ given the observed data.
Since λ ~ Exp(α) and the observed data X1,...,Xn ~ Exp(λ), we can use a conjugate prior distribution. The conjugate prior for the exponential distribution is the Gamma distribution.
Let's assume we have a Gamma prior distribution with parameters β > 0 and γ > 0, i.e., λ ~ Gamma(β, γ).
The posterior distribution of λ given the observed data can be calculated as follows:
p(λ|X1,...,Xn) ∝ p(X1,...,Xn|λ) * p(λ)
∝ ∏(λe^(-λXi)) * λ^(β-1)e^(-γλ)
∝ λ^(n+β-1)e^(-(γ+ΣXi)λ)
This is the kernel of a Gamma distribution with parameters β' = n + β and γ' = γ + ΣXi.
Therefore, the posterior distribution of λ given the observed data is λ|X1,...,Xn ~ Gamma(n + β, γ + ΣXi).
The Bayesian estimator Λ^Bayes for λ is the mean of the posterior distribution, which is given by:
Λ^Bayes = (n + β)/(γ + ΣXi).
To determine whether the Bayesian estimator is consistent and asymptotically normal, additional information about the prior distribution and the sample size n is needed. Please provide the values of β, γ, α, and n, and I will assist you further in determining the consistency and asymptotic normality.
Regarding the asymptotic variance V(λ), it can only be determined after knowing the prior distribution, the sample size, and the observed data. Please provide the necessary information to calculate V(λ).