a.
Provide an estimator
a.
Answer to Problem 104SE
The method of moments estimator of
Explanation of Solution
Calculation:
The expectation of the given random variable Y is obtained as follows:
The method of moments estimator of
Now, to obtained the method of moments estimator of
Hence, the method of moments estimator of
b.
Provide an estimator
b.
Answer to Problem 104SE
An estimator
Explanation of Solution
Calculation:
Consider an indicator
Now, the likelihood function of the given random variable Y is defined as follows:
To maximize the likelihood it is needed to choose
So, it is needed to choose
Hence,
Hence, an estimator
c.
Make some adjustment to make
Also find the efficiency of the adjusted
c.
Explanation of Solution
Calculation:
From Part (a), it is obtained that method of moments estimator of
It is also known that
That is,
That is,
The distribution function of
The probability density function of
Hence,
Thus,
Thus, the unbiased estimator of
Now,
This, the variance of
Thus,
Now, the expected value of
Thus, the variance of Y is,
Thus,
Thus, the efficiency of the adjusted
Thus, the efficiency of the adjusted
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Chapter 9 Solutions
Mathematical Statistics with Applications