Mathematical Statistics Practice 1

Let $X_1, X_2, \ldots, X_n$ be i.i.d. from $\text{Uniform}(0, \theta)$ where θ > 0 is unknown.

(1) Find a sufficient statistic for θ.

(2) Show that $X_{(n)} = \max(X_1, \ldots, X_n)$ is sufficient for θ.

(3) Is $\bar{X}$ sufficient for θ?

Let $X_1, \ldots, X_n$ be i.i.d. from a distribution with density:

(1) Find the MLE of θ.

(2) Is the MLE unbiased? If not, find an unbiased estimator based on the MLE.

(3) Find the variance of the MLE.

Let $X_1, \ldots, X_n$ be i.i.d. Poisson(λ).

(1) Find the Fisher information $I(\lambda)$ for a single observation.

(2) Find the Cramér-Rao lower bound for unbiased estimators of λ.

(3) Show that $\bar{X}$ attains the CRLB.

(4) Find the CRLB for estimating $g(\lambda) = e^{-\lambda}$.

Let $X_1, \ldots, X_n$ be i.i.d. from Gamma(α, β) with density:

Find the method of moments estimators for α and β.

Let $X_1, \ldots, X_n$ be i.i.d. N(μ, σ²) where σ² is known.

Test $H_0: \mu = \mu_0$ vs $H_1: \mu \neq \mu_0$.

(1) Derive the likelihood ratio test statistic.

(2) Find the rejection region for significance level α.

(3) Show the LRT is equivalent to the z-test.

Let $X_1, \ldots, X_n$ be i.i.d. from Exponential(λ).

(1) Find a pivot based on $\sum X_i$.

(2) Construct a 95% confidence interval for λ.

(3) Find the expected length of this confidence interval.

Let $X_1, \ldots, X_n$ be i.i.d. Bernoulli(p).

(1) Show that $T = \sum X_i$ is complete and sufficient.

(2) Find the UMVUE of p.

(3) Find the UMVUE of p(1-p).

Let $X_1, \ldots, X_n$ be i.i.d. Poisson(λ). Assume a Gamma(α, β) prior for λ.

(1) Find the posterior distribution of λ.

(2) Find the Bayes estimator under squared error loss.

(3) Find a 95% Bayesian credible interval for λ.