Math Problems Vault

Problem 2.14.1

Let $X = (X_1, \dots , X_n)$ be a sample from $\mathrm{Exp}(\theta )$ , $\theta \sim \Gamma (\alpha , \lambda )$ . Find the Bayesian estimator and check its consistency.

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Problem 2.14.2

Given a sample $X_1, \ldots , X_n$ from the Poisson distribution with parameter $\theta$ , where $\theta \sim \Gamma (\alpha , \lambda )$ , construct the best estimator in the Bayesian approach with quadratic loss function.

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Problem 2.14.3

Let $X_1, \ldots , X_n$ be a sample from the normal distribution with parameters $(\theta ,1)$ . Find the Bayesian estimator of the parameter $\theta$ , if the prior distribution of $\theta$ is $\operatorname {Ber}(p)$ . Will the resulting estimator be a consistent estimator of the parameter $\theta$ ?

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Problem 2.14.4

(a)(b)

Let $X_1, \ldots , X_n$ be a sample from the distribution $U[0,\theta ]$ . Find the Bayesian estimator of the parameter $\theta$ , if $\theta$ has prior distribution

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(a)

$U[0,1]$ ;

(b)

with density $q(t) = \frac{1}{t^2} \; \mathbb {1}_{t \geq 1}$ .

Check the resulting estimators for consistency.

Problem 2.14.5

Let $X_1, \dots , X_n$ be a sample from the normal distribution with parameters $(\theta ,1)$ . Find the Bayesian estimator of the parameter $\theta$ , if the prior distribution of $\theta$ is $\mathscr {N}\left(b, \sigma^2\right)$ .

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Problem 2.14.6

(a)(b)(c)

Let $X_1, \dots , X_n$ be a sample from the distribution

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(a)

$\mathscr {N}\left(\theta , 1\right)$ ;

(b)

$\mathscr {N}\left(0, \theta \right)$ ( $\theta$ is the standard deviation, $\theta^2$ is the variance);

(c)

$\operatorname {Bin}(m,\theta )$ , $m \in \mathbb {N}$ is known.

Choose a conjugate distribution for the parameter $\theta$ and find the Bayesian estimator.