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Statistics / Mathematical Statistics and Data Analysis 3 / Chapter 8 / Problem 53

Mathematical Statistics and Data Analysis | 3rd Edition | ISBN: 9788131519547 | Authors: John A. Rice

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Let X1, . . . , Xn be i.i.d. uniform on [0, ]. a. Find the

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QUESTION:

Let \(X_{1}, \ldots, X_{n}\) be i.i.d. uniform on \([0, \theta]\).

a. Find the method of moments estimate of \(\theta\) and its mean and variance.

b. Find the mle of \(\theta\).

c. Find the probability density of the mle, and calculate its mean and variance. Compare the variance, the bias, and the mean squared error to those of the method of moments estimate.

d. Find a modification of the mle that renders it unbiased.

Questions & Answers

QUESTION:

Let \(X_{1}, \ldots, X_{n}\) be i.i.d. uniform on \([0, \theta]\).

a. Find the method of moments estimate of \(\theta\) and its mean and variance.

b. Find the mle of \(\theta\).

c. Find the probability density of the mle, and calculate its mean and variance. Compare the variance, the bias, and the mean squared error to those of the method of moments estimate.

d. Find a modification of the mle that renders it unbiased.

ANSWER:

Step 1 of 4

Given that

are i.i.d. uniformly distributed on .

(a)

The first moment of each is obtained as:

So, the method of moments provides, .

Thus, the method of moments estimate of is:

Now,

The mean of is already computed to be:

Now, the variance of is obtained as:

This implies the following results:

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