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Set estimation of the variance - Exercise set 1

This exercise set contains some solved exercises on set estimation of the variance. The theory needed to solve these exercises is introduced in the lecture entitled Set estimation of the variance.

Exercise 1.1

Suppose you observe a sample of $100$ independent draws from a normal distribution having known mean $mu =0$ and unknown variance sigma^2. Denote the $100$ draws by X_1, ..., $X_{100}$. Suppose that:[eq1]

Find a confidence interval for sigma^2, using a set estimator of sigma^2 having $90%$ coverage probability.

Hint: a Chi-square random variable Z with $100$ degrees of freedom has a distribution function [eq2] such that:[eq3]

nav_button Solution

For a given sample size n, the interval estimator[eq4]has coverage probability[eq5]where Z is a Chi-square random variable with n degrees of freedom and [eq6] are strictly positive constants. Thus, if we set[eq7]then:[eq8]which is equal to our desired coverage probability. Thus, the confidence interval for sigma^2 is:[eq9]

Exercise 1.2

Suppose you observe a sample of $100$ independent draws from a normal distribution having unknown mean mu and unknown variance sigma^2. Denote the $100$ draws by X_1, ..., $X_{100}$. Suppose that their adjusted sample variance $s_{100}^{2}$ is equal to $5$, i.e.:[eq10]

Find a confidence interval for sigma^2, using a set estimator of sigma^2 having $99%$ coverage probability.

Hint: a Chi-square random variable Z with $99$ degrees of freedom has a distribution function [eq11] such that:[eq12]

nav_button Solution

For a given sample size n, the interval estimator[eq13]has coverage probability[eq14]where Z is a Chi-square random variable with $n-1$ degrees of freedom and [eq15] are strictly positive constants. Thus, if we set[eq16]then:[eq17]which is equal to our desired coverage probability. Thus, the confidence interval for sigma^2 is:[eq18]

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