Problem set 8: Confidence Intervals

Let \(X_1,\ldots,X_n\) be a sample from \(\operatorname{U}(0,\theta)\). Construct an exact confidence interval for the parameter \(\theta\) with confidence level \(\gamma\), whose endpoints are functions of:

  1. \(\overline{X}\);

  2. \(X_{(1)}\);

  3. \(X_{(n)}\).

Find the asymptotic behavior of the length of each interval as \(n\to\infty\).

Let \(X_1,\ldots,X_n\) be a sample from a distribution with density \[ p_\theta(x) = \frac{3x^2}{8\theta^3}\mathbf{1}(x\in[0,2\theta]). \]

Using the statistic \(X_{(1)}\), construct an exact confidence interval for the parameter \(\theta\) with confidence level \(\gamma\).

Let \(X_1,\ldots,X_n\) be a sample from the Cauchy distribution with density \[ p_\theta(x) = \frac{1}{\pi\bigl(1+(x-\theta)^2\bigr)}. \]

Construct an exact asymptotic confidence interval for \(\theta\) with confidence level \(\gamma\).

Let \(X_1,\ldots,X_n\) be a sample from \(\operatorname{Pois}(\theta)\). Construct an asymptotic confidence interval for the parameter \(\theta\) with confidence level \(\gamma\).

Let \(X_1,\ldots,X_n\) be a sample from the \(\Gamma(\alpha,\lambda)\) distribution. Construct an asymptotic confidence interval for the parameter \(\lambda\) with confidence level \(\gamma\):

  1. if \(\alpha\) is known;

  2. if \(\alpha\) is an unknown parameter.

Let \(X_1,\ldots,X_n\) be a sample from the \(\operatorname{Pareto}(\theta,1)\) distribution, where \(\theta>0\).

Construct an exact confidence interval for the parameter \(\theta\) with confidence level \(\gamma\).