Question

Cirele out your Cast Mon& Wed or Mon. Evening. am (lan 15 that X,, X2, .,X Question 7. Suppose distribution with the following p.d.f. ,X is a simple random sample from th 0, otherwise where θ > 0, a random sample of size 10 yields data 0.92 0.79 0.9 0.65 0.86 0.47 0.73 0.97 0.94 0.77 1) (6 points) Get the moment estimator of θ, and compute the estimate for this data;

Answer 1

We will get the first moment, the expected value of X

$\begin{align*} E(X)&=\int_{-\infty}^{\infty}xf(x;\theta)dx\\ &=\int_0^1x\theta x^{\theta-1}dx\\ &=\int_0^1\theta x^{\theta}dx\\ &=\left.\theta \frac{x^{\theta+1}}{\theta+1}\right|_0^1\\ &=\frac{\theta}{\theta+1} \end{align*}$

Let $\begin{align*}\bar{x}=\frac{\sum x_i}{n} \end{align*}$ be the sample mean.

The method of moment estimator calculated by equating the sample mean to the expectation

$\begin{align*}&\frac{\theta}{\theta+1}=\bar{x}\\ \implies &\theta=\bar{x}(\theta+1)\\ \implies &\theta(1-\bar{x})=\bar{x}\\ \implies &\theta=\frac{\bar{x}}{1-\bar{x}} \end{align*}$

The method of moment estimator is

$\begin{align*} \hat{\theta}=\frac{\bar{x}}{1-\bar{x}} \end{align*}$

We will get the mean of the given sample

$\begin{align*} \bar{x}=\frac{\sum x_i}{n}=\frac{0.92+ 0.79+ 0.9+ 0.65+ 0.86+ 0.47+ 0.73+ 0.97+ 0.94+ 0.77}{10}=0.80 \end{align*}$

the estimate is

$\begin{align*} \hat{\theta}=\frac{\bar{x}}{1-\bar{x}}=\frac{0.80}{1-0.80}=4 \end{align*}$