Chapter 8 Sampling Distributions of Estimators

8.1 Sampling Distribution of a Statistic

Nothing on this section.

Suppose that \(X_1, \dots , X_n\) form a random sample from the normal distribution with mean \(\mu\) and variance \(\sigma^2\). Find the distribution of \[\frac{n (\bar{X} -\mu)^2}{\sigma^2}\]
Suppose that \(Z_1, \dots , Z_6\) form a random sample from the standard normal distribution. Let \(Y = (Z_1 + Z_2 + Z_3)^2 + (Z_4 + Z_5 + Z_6)^2.\) Find the value of \(c\) such that \(cY\) has a Chi-squared distribution.

Nothing here.

Suppose that \(X_1, \dots , X_n\) form a random sample from the normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma^2\), and let \(\hat\mu\) and \(\hat\sigma^2\) denote the M.L.E.’s of \(\mu\) and \(\sigma^2\). For the sample size \(n = 17\), find a value of \(k\) such that \[Pr(\hat\mu > \mu +k\hat \sigma ) =0.95\].
Suppose that \(Z_1, \dots , Z_5\) form a random sample from the standard normal distribution. Let \(Y = \frac{Z_1 + Z_2}{ \sqrt{Z_3^2 + Z_4^2 + Z_5^2 }}.\) Find the value of \(c\) such that \(cY\) has a \(t\) distribution.

Suppose that a sample of \(n\) observations where \(X_i \stackrel{iid}{\sim} N(\mu, \sigma^2)\) and we are interested in creating a confidence interval for \(\sigma^2\) using the MLE estimator \(\hat{\sigma}^2 = \frac{1}{n}\sum(x_i - \bar{x})^2\)
1. What function of \(\hat{\sigma}^2\) and \(\sigma^2\) has a completely known distribution (though the distribution depends on the known value of \(n\))?
2. Using quantiles from the known distribution which you can denote as \(\chi^*_{q,df}\) , create a formula for a \(95\)% confidence interval for \(\sigma^2\).
Suppose that a smaple \(n\) observations where \(X_i \stackrel{iid}{\sim} \textrm{Exponential}(\beta)\) and we wish to find a 90% Confidence interval for \(\beta\). Recall that \(\sum X_i\) was the sufficient statistic, so our confidence interval should be a function of \(\sum X_i\) and \(n\) should be appropriate.
1. What is the distribution of \(\sum{X_i}\) and \(\beta \sum X_i\)? Hint: recall that the Exponential is a special case of the Gamma distribution and we had a number of rules that we derived using the MGF method regarding the sum of Gamma random variables and a Gamma random variable multiplied by a constant.
2. Denote the \(q\)th quantile of an arbitrary Gamma distribution as \(\gamma^*_{q,\alpha,\beta}\). Use \(\beta \sum X_i\) as a pivotal quantity to create a 90% CI for \(\beta\).