This lecture presents some examples of set
estimation problems, focusing on **set estimation of the
variance**, that is, on using a sample to produce a set estimate of the
variance of an unknown distribution.

In this example we make the same assumptions we made in the example of point estimation of the variance entitled Normal IID samples - Known mean. The reader is strongly advised to read that example before reading this one.

The sample is made of independent draws from a normal distribution having known mean and unknown variance . Specifically, we observe realizations , ..., of independent random variables , ..., , all having a normal distribution with known mean and unknown variance . The sample is the -dimensional vector which is a realization of the random vector

The interval estimator of the variance is based on the following point estimator of the variance:

The interval estimator iswhere and are strictly positive constants and .

The coverage probability of the interval estimator iswhere is a Chi-square random variable with degrees of freedom.

Proof

The coverage probability can be written aswhere we have definedIn the lecture entitled Point estimation of the variance, we have demonstrated that, given the assumptions on the sample made above, the estimator of variance has a Gamma distribution with parameters and . Multiplying a Gamma random variable with parameters and by one obtains a Chi-square random variable with degrees of freedom. Therefore, the variable has a Chi-square distribution with degrees of freedom.

Note that the coverage probability does not depend on the unknown parameter . Therefore, the confidence coefficient of the interval estimator coincides with its coverage probability:where is a Chi-square random variable with degrees of freedom.

The size of the interval estimator is

Note that the size depends on and hence on the sample . The expected size of the interval estimator iswhere we have used the fact that is an unbiased estimator of (i.e., , see the lecture entitled Point estimation of the variance).

This example is similar to the previous one. The only difference is that we now relax the assumption that the mean of the distribution is known.

In this example, the sample is made of independent draws from a normal distribution having unknown mean and unknown variance . Specifically, we observe realizations , ..., of independent random variables , ..., , all having a normal distribution with unknown mean and unknown variance . The sample is the -dimensional vector , which is a realization of the random vector .

To construct interval estimators of the variance , we use the sample mean :and either the unadjusted sample varianceor the adjusted sample varianceWe consider the following interval estimator of the variance:where and are strictly positive constants, .

The coverage probability of the interval estimator iswhere is a Chi-square random variable with degrees of freedom.

Proof

The coverage probability can be written aswhere we have definedIn the lecture entitled Point estimation of variance, we have demonstrated that, given the assumptions on the sample made above, the unadjusted sample variance has a Gamma distribution with parameters and . Therefore, the random variable has a Gamma distribution with parameters and whereBut a Gamma distribution with parameters and is a Chi-square distribution with degrees of freedom. Therefore, has a Chi-square distribution with degrees of freedom.

Note that the coverage probability of does not depend on the unknown parameters and . Therefore, the confidence coefficient of the confidence interval coincides with its coverage probability:where is a Chi-square distribution with degrees of freedom.

The size of the confidence interval is

The expected size of iswhere in the penultimate step we have used the fact (proved in the lecture entitled Point estimation of variance) that

Below you can find some exercises with explained solutions.

Suppose you observe a sample of independent draws from a normal distribution having known mean and unknown variance . Denote the draws by , ..., . Suppose that:

Find a confidence interval for , using a set estimator of having coverage probability.

Hint: a Chi-square random variable with degrees of freedom has a distribution function such that

Solution

For a given sample size , the interval estimatorhas coverage probabilitywhere is a Chi-square random variable with degrees of freedom and are strictly positive constants. Thus, if we setthenwhich is equal to our desired coverage probability. Thus, the confidence interval for is

Suppose you observe a sample of independent draws from a normal distribution having unknown mean and unknown variance . Denote the draws by , ..., . Suppose that their adjusted sample variance is equal to , that is,

Find a confidence interval for , using a set estimator of having coverage probability.

Hint: a Chi-square random variable with degrees of freedom has a distribution function such that

Solution

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