Question

1. Let Xi, . . . , Xn be a random sample from a uniform distribution on the interval (e-1,0 + 1). (a) (10 points) Find a moment estimator for 0 (b) (10 points) Use the following data to obtain a moment estimate for 0: 11.72 12.81 12.09 13.47 12.37

Answer 1

$X_i$ is uniformly distributed on the interval $(\theta-1,\theta+1)$

The pdf of the distribution of $X_i$ is

fx(x) =

The expected value of $X_i$ is the first theoretical moment, (using the standard result for uniform distribution)

$E(X_i)=\frac{a+b}{2}=\frac{(\theta-1)+(\theta+1)}{2}=\theta$

The first sample moment is the sample mean

$\bar{X}=\frac{\sum_{i=1}^nX_i}{n}$

a) Equating the first sample moment to the first theoretical moment

$\theta=\frac{\sum_{i=1}^nX_i}{n}$

Since there is only one unknown parameter,

ans: the method of moment estimator for $\theta$ is

$\hat{\theta}_{MM}=\frac{\sum_{i=1}^nX_i}{n}$

b) Using the sample data, we can get the moment estimate as

Ση! Xi 11.72 + 12.81 + 12.09 + 13.47 + 12.37 9.1M = = 12.492

ans: A moment estimate of $\theta$ is 12.492

That means we can say that the sample is drawn from a uniform distribution in the interval (11.492,13.492)