Simultaneous Confidence Intervals

(20 points) Researchers study how a child’s reading score \(Y\) depends on home reading time \(x_{1}\) (hours/week) and classroom support level \(x_{2}\) (a small integer score). It is believed that extra home reading time is more effective when classroom support is higher, so the mean is modeled with an interaction between \(x_{1}\) and \(x_{2}\). To make the intercept interpretable, home reading time is centered at \(1\) hour/week.

Consider the linear model \[ Y_{i} = \beta _{0} + \beta _{1}(x_{1,i} - 1) + \beta _{2}(x_{1,i} \times x_{2,i}) + \epsilon _{i}, \qquad i = 1, \ldots , 5, \] where \(\epsilon _{i} \sim \mathcal {N}(0, \sigma ^{2})\) are independent. The observed data consist of: \[ (x_{1}, x_{2}, Y) = (0, 1, 4), (1, 0, 3), (2, 1, 7), (1, 2, 6), (3, 1, 9). \]

A least squares fit yields \(\hat {\beta } = (\hat {\beta }_{0}, \hat {\beta }_{1}, \hat {\beta }_{2})^{\top } = (2.10, 0.90, 1.20)^{\top }\) and \(RSS = 1.50\).

1. Write down the design matrix \(\mathbf {X}\) for this regression model and the parameter vector \(\beta = (\beta _{0}, \beta _{1}, \beta _{2})^{\top }\). Identify the estimated variance \(S^{2}\).
2. We wish to test the joint null hypothesis that there is no effect of the predictors beyond the baseline, \[ H_{0}: \beta _{1} = \beta _{2} = 0. \] Express this hypothesis in matrix form \(A\beta = 0\) for an appropriate matrix \(A\), and state \(q = \operatorname {rank}(A)\).
3. Write down the \(F\)-statistic for testing \(H_{0}\) and state the distribution of \(F\) under \(H_{0}\). (You do not need to evaluate the formula.)
4. Using Scheffé’s method, construct simultaneous 95% confidence intervals for \(\beta _{0}, \beta _{1}, \beta _{2}\). You may express these intervals in terms of \((\mathbf {X}^{\top }\mathbf {X})^{-1}\), but plug in all other quantities.

Let \(X_1,\ldots ,X_n\) be independent and identically distributed random variables with density function \[ h(t;\theta )=\frac {3t^2}{\theta ^3}1\{0\leq t\leq \theta \}, \] where \(\theta >0\) is a parameter. Find a \(100(1-\alpha )\%\), two-sided, equal-tailed confidence interval for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be iid with density \[ f(x)=2x/\theta ^2, \qquad 0<x<\theta . \] Let \(Y=X_{(n)}=X_{n:n}\) be the largest order statistic of the \(X_i\). Show that \(Y/\theta \) is a pivotal quantity for \(\theta \) and use it to construct a 99% confidence interval for \(\theta \).

Let \(X_1\sim N(0,4\theta )\), \(X_2\sim N(0,9\theta )\), \(X_3\sim N(0,4\theta ^2)\), \(X_4\sim N(0,12\theta ^2)\), and assume that all \(X\)s are independent of each other. Using all four \(X\) variables and the \(t\) distribution, compute a \(100(1-\alpha )\%\) confidence interval for \(\theta \).

Let \(X_1,\ldots ,X_n\) be iid with the Gamma\((\theta ,\kappa )\) distribution, meaning that their common pdf is \[ f(x;\theta ,\kappa )=\frac {1}{\theta ^\kappa \Gamma (\kappa )} x^{\kappa -1}e^{-x/\theta }, \qquad x>0, \] with \(\kappa \) known but \(\theta \) unknown. Derive a \(100(1-\alpha )\%\) equal-tailed confidence interval for \(\theta \) based on the sufficient statistic.

Let \(X_1,\ldots ,X_n\) be iid Uniform\((0,\theta )\) with \(\theta >0\), and \(X_{(n)}\) be the largest order statistic.

(a) Find the probability that the random interval \((X_{(n)},2X_{(n)})\) contains \(\theta \).

(b) Find the constant \(c\) such that \((X_{(n)},cX_{(n)})\) is a \(100(1-\alpha )\%\) confidence interval for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be independent and identically distributed random variables with density function \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\lambda }\exp (-t/\lambda ), & \text {if } t\geq 0. \end {cases} \] Construct an equal tail \(1-\alpha \) confidence interval for \(\lambda \). Compute the expected value of the length of the confidence interval.

Let \(X_1,\ldots ,X_n\) be an i.i.d. sample from a \(\operatorname {Uniform}(0,\theta )\) distribution, where \(\theta >0\) is unknown. Find a 95% confidence interval for \(\theta \).

Let \(X\) be a random variable with values in an interval \(I\), and suppose that \(f\) and \(g\) are nondecreasing functions on \(I\) such that \(f(X)\) and \(g(X)\) have finite variance. Show that \(\operatorname {Cov}(f(X),g(X))\geq 0\).

Hint: If \(X_1\) and \(X_2\) are i.i.d. as \(X\), then \((f(X_1)-f(X_2))(g(X_1)-g(X_2))\geq 0\).

Assume a sample of size \(n\) from \(\operatorname {Uniform}[\eta -\theta ,\eta +\theta ]\), which is a location-scale family. Here \(\eta \) is real and \(\theta >0\).

(a) Find a \(100(1-\alpha )\%\) confidence interval for \(\theta \).

(b) Find a \(100(1-\alpha )\%\) confidence interval for \(\eta \).

It is not necessary to find the explicit distribution of the pivotal quantities; just denote the needed quantiles by \(x_p\) and \(y_p\) for an appropriate \(p\).

Let \(X_1,\ldots ,X_n\) be independent identically distributed random variables with distribution function \(F\). We assume that \(F\) is strictly increasing and continuous on its support. Let \(x_{1/2}\) denote the median.

a. Find a \(1-\alpha \) confidence interval for \(x_{1/2}\). (5 points)

b. Provide a large sample approximation for the confidence interval. (5 points)

Find a confidence interval for \(\theta \) that has asymptotic level \((1-\alpha )\%\).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\leq \theta ,\\ e^{-(t-\theta )}, & \text {if } t>\theta . \end {cases} \] (a) Find two \(1-\alpha \) equal-tail confidence intervals for \(\theta \). One should be based on the smallest order statistic and the other should be based on the sample mean.

(b) Compare the expected lengths of the confidence intervals.

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed uniform \((0,\theta )\) random variables with common density function \[ f(t)= \begin {cases} 0, & \text {if } t\notin [0,\theta ],\\ 1/\theta , & \text {if } t\in [0,\theta ]. \end {cases} \] (a) Find the maximum likelihood estimator for \(\theta \).

(b) Find a \(1-\alpha \) equal-tail confidence interval for \(\theta \).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential \((\lambda )\) random variables. Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed exponential \((\mu )\) random variables. The two samples are independent. We wish to test \(H_0:\theta =\mu \) against the alternative that \(H_0\) is not true.

(a) Find the likelihood ratio test.

(b) Find the asymptotic distribution of the likelihood ratio under \(H_0\).

(c) Find an equal-tail confidence interval for \(\tau =\lambda /\mu \).

Let \(X_1,\ldots ,X_n\) be a random sample from \(N(\theta ,\theta )\) (i.e., normal with mean and variance both equal to \(\theta \)), where \(\theta >0\).

(a) Give three pivotal quantities (or pivots) involving the entire sample, which respectively have a standard normal distribution, a \(t\)-distribution, and a chi-squared distribution, and indicate the numbers of degrees of freedom. (5 points)

(b) Use the normal pivotal quantity in part (a) to obtain a \(100(1-\alpha )\) percent confidence interval for \(\theta \). (5 points)

Let \(X_1,\ldots ,X_n\) be an i.i.d. sample from a uniform \((0,\theta )\) distribution, where \(\theta >0\).

(a) Find the maximum likelihood estimator \(\hat {\theta }\) for \(\theta \);

(b) Construct a \(100(1-\alpha )\%\) confidence interval for \(\theta \) based on \(\hat {\theta }\).

Construct a \((1-\alpha )\) two-sided confidence interval for the correlation coefficient of a bivariate normal distribution. (HINT. This is an exponential family.)

Construct a \((1-\alpha )\) two-sided confidence interval for the correlation coefficient of a bivariate normal distribution. (HINT. This is an exponential family.)