Estimating the Variance of the MLE for a Normal Distribution with Known Median

To understand the variance of the Maximum Likelihood Estimator (MLE) for the variance of a normal distribution when the median is known, we can follow a series of steps. This article will break down the process, providing a clear and comprehensive explanation of the calculations involved.

Understanding the Problem

In a normal distribution N(, ^2), if the median is known, it is equal to the mean , due to the symmetry of the normal distribution. Hence, we can consider the mean as a known parameter. The goal is to find the variance of the MLE of the variance of a normal distribution when the median is known.

MLE of the Variance

Given a sample X_1, X_2, ..., X_n drawn from N(, ^2), the likelihood function for the sample is:

L(^2) prod_{i1}^n frac{1}{sqrt{2pi ^2}} exp left(-frac{(X_i - )^2}{2^2}right)

Taking the logarithm of the likelihood function gives:

log L(^2) -frac{n}{2} log 2pi - frac{n}{2} log ^2 - frac{1}{2^2} sum_{i1}^n (X_i - )^2

To find the MLE of ^2, we differentiate the log-likelihood function with respect to ^2 and set it to zero:

frac{partial log L}{partial ^2} -frac{n}{2^2} - frac{1}{2^4} sum_{i1}^n (X_i - )^2 0

Solving this equation, we get:

hat{}^2 frac{1}{n} sum_{i1}^n (X_i - )^2 S^2

Variance of the MLE

The variance of the MLE hat{}^2 can be derived using the properties of the sample variance. For a normal distribution, the variance of the sample variance estimator is given by:

text{Var}(hat{}^2) frac{2^4}{n - 1}

However, since ^2 is unknown, we can estimate it using the MLE hat{}^2:

text{Var}(hat{}^2) approx frac{2(hat{}^2)^4}{n - 1}

Conclusion

Thus, the variance of the MLE of the variance of a normal distribution with a known median and hence known mean is approximately:

text{Var}(hat{}^2) approx frac{2(hat{}^2)^4}{n - 1}

This formula provides an estimate of the variability of the MLE of the variance based on the sample size n.

Further Explanation

Since the median of a normal distribution is equal to the mean, we can say that knowing the median implies the mean is also known. Therefore, the MLE of the variance of a normal distribution when the mean is known can be derived by considering the likelihood function and the log-likelihood function, as shown in the previous steps. The variance of this MLE is an important measure of how accurate the estimate is, and it is derived using the Cramér-Rao Lower Bound (CRLB).

Using the CRLB, we can derive the following expression for the variance of the MLE:

text{Var}(S^2) frac{2^4}{n}

This shows that the variance of the MLE hat{}^2 is closely related to the true variance ^2 and the sample size n.

In summary, understanding the MLE for the variance of a normal distribution with a known median involves several mathematical steps, including the likelihood function, the log-likelihood function, and the CRLB. The variance of the MLE provides a measure of how well the estimate of the variance is likely to be.