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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 $xi _{n}$ is made of n independent draws from a normal distribution having known mean mu and unknown variance sigma^2. Specifically, we observe n realizations $x_{1}$, ..., $x_{n}$ of n independent random variables X_1, ..., X_n, all having a normal distribution with known mean mu and unknown variance sigma^2. The sample is the n-dimensional vector [eq1] which is a realization of the random vector [eq2]

The interval estimator

The interval estimator of the variance sigma^2, is based on the following point estimator of the variance:[eq3]

The interval estimator is:[eq4]where $z_{1}$ and $z_{2}$ are strictly positive constants and $z_{1}<z_{2}$.

Coverage probability

The coverage probability of the interval estimator $T_{n}$ is:[eq5]where Z is a Chi-square random variable with n degrees of freedom.

Proof

The coverage probability can be written as: [eq6]where we have defined[eq7]In the lecture entitled Point estimation of the variance, we have demonstrated that, given the assumptions on the sample $xi _{n}$ made above, the estimator of variance [eq8] has a Gamma distribution with parameters n and sigma^2. Multiplying a Gamma random variable with parameters n and sigma^2 by [eq9] one obtains a Chi-square random variable with n degrees of freedom. Therefore, the variable Z has a Chi-square distribution with n degrees of freedom.

Confidence coefficient

Note that the coverage probability does not depend on the unknown parameter sigma^2. Therefore, the confidence coefficient of the interval estimator $T_{n}$ coincides with its coverage probability:[eq10]where Z is a Chi-square random variable with n degrees of freedom.

Size

The size of the interval estimator $T_{n}$ is:[eq11]

Expected size

Note that the size depends on [eq12] and hence on the sample $xi _{n}$. The expected size of the interval estimator $T_{n}$ is:[eq13]where we have used the fact that [eq12] is an unbiased estimator of sigma^2 (i.e. [eq15], 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 $xi _{n}$ is made of n independent draws from a normal distribution having unknown mean mu and unknown variance sigma^2. Specifically, we observe n realizations $x_{1}$, ..., $x_{n}$ of n independent random variables X_1, ..., X_n, all having a normal distribution with unknown mean mu and unknown variance sigma^2. The sample is the n-dimensional vector [eq16], which is a realization of the random vector [eq17].

The interval estimator

To construct interval estimators of the variance sigma^2, we use the sample mean Xbar_n:[eq18]

and either the unadjusted sample variance:[eq19]

or the adjusted sample variance:[eq20]We consider the following interval estimator of the variance:[eq21]where $z_{1}$ and $z_{2}$ are strictly positive constants, $z_{1}<z_{2}$.

Coverage probability

The coverage probability of the interval estimator $T_{n}$ is:[eq22]where $Z_{n-1}$ is a Chi-square random variable with $n-1$ degrees of freedom.

Proof

The coverage probability can be written as: [eq23]where we have defined[eq24]In the lecture entitled Point estimation of variance, we have demonstrated that, given the assumptions on the sample $xi _{n}$ made above, the unadjusted sample variance $S_{n}^{2}$ has a Gamma distribution with parameters $n-1$ and [eq25]. Therefore, the random variable $Z_{n-1}$ has a Gamma distribution with parameters $n-1$ and $h$ where:[eq26]But a Gamma distribution with parameters $n-1$ and $n-1$ is a Chi-square distribution with $n-1$ degrees of freedom. Therefore, $Z_{n-1}$ has a Chi-square distribution with $n-1$ degrees of freedom.

Confidence coefficient

Note that the coverage probability of $T_{n}$ does not depend on the unknown parameters mu and sigma^2. Therefore, the confidence coefficient of the confidence interval coincides with its coverage probability:[eq27]where $Z_{n-1}$ is a Chi-square distribution with $n-1$ degrees of freedom.

Size

The size of the confidence interval $T_{n}$ is:[eq28]

Expected size

The expected size of $T_{n}$ is:[eq29]where in the penultimate step we have used the fact (proved in the lecture entitled Point estimation of variance) that[eq30]

Solved exercises

Below you can find some exercises with explained solutions:

  1. Exercise set 1

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