title: “ECON 626: Problem Set 3” #author: “Paul Schrimpf” date: 2026-10-08 bibliography: ../../626.bib —

\[ \def\R{{\mathbb{R}}} \def\Er{{\mathrm{E}}} \]

Problem 1

@song2021 Chatper 4, exercise 1.3.

Consider the binary choice model in Example 1.3 and assume that \(\tilde{\beta}_0\) and \(\tilde{\beta}_1\) have appropriate estimators \(\hat{\beta}_0\) and \(\hat{\beta}_1\). Provide a sample analogue estimator of the average derivative, replacing \(\tilde{\beta}_0\) and \(\tilde{\beta}_1\) with \(\hat{\beta}_0\) and \(\hat{\beta}_1\)

Problem 2

The Rayleigh distribution has probability density function (with respect to Lebesgue measure) \[ f(x; \theta) = \frac{x}{\theta} \exp\left(-\frac{x^2}{2\theta}\right) \mathbf{1}\{x > 0\}, \]

for \(\theta > 0\). Suppose \(X_1, \dots, X_n\) are independently and identically Rayleigh(\(\theta\)) distributed.

  1. Find the maximum likelihood estimator \(\theta\).

  2. Show that the maximum likelihood estimator is unbiased.

  3. Calculate the variance of the maximum likelihood esitmator.

  4. Derive the Cramér–Rao lower bound for any unbiased estimator of \(\theta\). Determine whether \(\hat{\theta}^{MLE}\) is a minimum variance unbiased estimator.

  5. Find the most powerful test of size \(\alpha\) for testing \[ H_0: \theta = \theta_0 \quad \text{versus} \quad H_1: \theta = \theta_1, \] where \(\theta_1 > \theta_0\).