Homework 5

Homework 5 is due 10/21/2022 at 11:59 PM. Submit your homework on Canvas as one PDF document.

The PDF version of this assignment can be found here.

1. Let X_1, X_2, \ldots, X_n be iid with the following density function

   f(x) = \left\{\begin{array}{cc} (\theta + 1)x^\theta& 0\le x\le 1;\theta>-1 \\ 0 & \mathrm{elsewhere} \end{array} \right.

   Find the MLE for \theta.

2. Let X_1, X_2, \ldots, X_n be iid with the following density function

   f(x) = \left\{\begin{array}{cc} \frac{1}{\Gamma(\alpha)\theta^\alpha}x^{\alpha-1}e^{-x/\theta} & 0<x;0< \theta \\ 0 & \mathrm{elsewhere} \end{array} \right.

   where \alpha>0 is known. Find the MLE for \theta.

3. Let X_1, X_2, \ldots, X_n be iid with the following density function

   f(x) = \left\{\begin{array}{cc} e^{-(x-\theta)} & x>\theta \\ 0 & \mathrm{elsewhere} \end{array} \right.

   Find the method of moment estimator for \theta.