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# Set estimation of the variance

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

## Normal IID samples - Known mean

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

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

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

The interval estimator is:where and are strictly positive constants and .

### Coverage probability

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

Proof

The coverage probability can be written as: where 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.

### Confidence coefficient

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.

### Expected size

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

## Normal IID samples - Unknown mean

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.

### The sample

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 .

### The interval estimator

To construct interval estimators of the variance , we use the sample mean :

and either the unadjusted sample variance:

or the adjusted sample variance:We consider the following interval estimator of the variance:where and are strictly positive constants, .

### Coverage probability

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

Proof

The coverage probability can be written as: where 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 where:But 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.

### Confidence coefficient

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.

### Expected size

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

## Solved exercises

Below you can find some exercises with explained solutions:

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