Chapter 6: Problem 3
Given the pdf
$$f(x ; \theta)=\frac{1}{\pi\left[1+(x-\theta)^{2}\right]},
\quad-\infty
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Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(N\left(\mu_{0}, \sigma^{2}=\theta\right)\) distribution, where \(0<\theta<\infty\) and \(\mu_{0}\) is known. Show that the likelihood ratio test of \(H_{0}: \theta=\theta_{0}\) versus \(H_{1}: \theta \neq \theta_{0}\) can be based upon the statistic \(W=\sum_{i=1}^{n}\left(X_{i}-\mu_{0}\right)^{2} / \theta_{0}\) Determine the null distribution of \(W\) and give, explicitly, the rejection rule for a level \(\alpha\) test.
Consider a location model $$X_{i}=\theta+e_{i}, \quad i=1, \ldots, n$$ where \(e_{1}, e_{2}, \ldots, e_{n}\) are iid with pdf \(f(z)\). There is a nice geometric interpretation for estimating \(\theta .\) Let \(\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)^{\prime}\) and \(\mathbf{e}=\left(e_{1}, \ldots, e_{n}\right)^{\prime}\) be the vectors of observations and random error, respectively, and let \(\mu=\theta 1\) where 1 is a vector with all components equal to one. Let \(V\) be the subspace of vectors of the form \(\mu_{i}\) i.e, \(V=\\{\mathbf{v}: \mathbf{v}=a \mathbf{1}\), for some \(a \in R\\} .\) Then in vector notation we can write the model as $$\mathbf{X}=\boldsymbol{\mu}+\mathbf{e}, \quad \boldsymbol{\mu} \in V$$
Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are iid with pdf \(f(x ; \theta)=(1 /
\theta) e^{-x / \theta}, 0
The Pareto distribution is frequently used a model in study of incomes and has the distribution function $$F\left(x ; \theta_{1}, \theta_{2}\right)=\left\\{\begin{array}{ll} 1-\left(\theta_{1} / x\right)^{\theta_{2}} & \theta_{1} \leq x \\ 0 & \text { elsewhere }\end{array}\right.$$ where \(\theta_{1}>0\) and \(\theta_{2}>0 .\) If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from this distribution, find the maximum likelihood estimators of \(\theta_{1}\) and \(\theta_{2}\).
A machine shop that manufactures toggle levers has both a day and a night shift. A toggle lever is defective if a standard nut cannot be screwed onto the threads. Let \(p_{1}\) and \(p_{2}\) be the proportion of defective levers among those manufactured by the day and night shifts, respectively. We shall test the null hypothesis, \(H_{0}: p_{1}=p_{2}\), against a two-sided alternative hypothesis based on two random samples, each of 1000 levers taken from the production of the respective shifts. Use the test statistic \(Z^{*}\) given in Example \(6.5 .3\). (a) Sketch a standard normal pdf illustrating the critical region having \(\alpha=0.05\). (b) If \(y_{1}=37\) and \(y_{2}=53\) defectives were observed for the day and night shifts, respectively, calculate the value of the test statistic and the approximate \(p-\) value (note that this is a two-sided test). Locate the calculated test statistic on your figure in Part (a) and state your conclusion. Obtain the approximate \(p\) -value of the test.
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