Chapter 6

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 $Z^{\ast }$ 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.

Let $X_1,X_2,\dots ,X_n$ be a random sample from the beta distribution with $\alpha =\beta =\theta$ and $$\begin{gathered} \mathrm{\Omega }= \\ \theta :\theta =1,2 \\ . \end{gathered}$$ Show that the likelihood ratio test statistic $\mathrm{\Lambda }$ for testing $H_0:\theta =1$ versus $H_1:\theta =2$ is a function of the statistic $W=$ $\sum _{i=1}^n\log X_i+\sum _{i=1}^n\log (1-X_i)$

Recall that $\hat{\theta }=-n/\sum _{i=1}^n\log X_i$ is the mle of $\theta$ for a $\mathrm{Beta}(\theta ,1)$ distribution. Also, $W=-\sum _{i=1}^n\log X_i$ has the gamma distribution $\mathrm{\Gamma }(n,1/\theta )$. (a) Show that $2\theta W$ has a ${\chi }^2(2n)$ distribution. (b) Using Part (a), find $c_1$ and $c_2$ so that $$P\left(c_{1}<\frac{2 \theta n}{\hat{\theta}}

Let $X_1,X_2,\dots ,X_n$ ba random sample from a distribution with one of two pdfs. If $\theta =1$, then \(f(x ; \theta=1)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2},-\infty

Consider a location model $$X_i=\theta +e_i,i=1,\dots ,n$$ where $e_1,e_2,\dots ,e_n$ are iid with pdf $f(z)$. There is a nice geometric interpretation for estimating $\theta .$ Let $\mathbf{X}={(X_1,\dots ,X_n)}^{\prime }$ and $\mathbf{e}={(e_1,\dots ,e_n)}^{\prime }$ 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, $$\begin{gathered} V= \\ \mathbf{v}:\mathbf{v}=a\mathbf{1}\text{)},forsome\text{(}a\in R \\ . \end{gathered}$$ Then in vector notation we can write the model as $$\mathbf{X}=\mathit{\mu }+\mathbf{e},\mathit{\mu }\in V$$

Let $S^2$ be the sample variance of a random sample of size $n>1$ from $N(\mu ,\theta ),0<\theta <\mathrm{\infty }$, where $\mu$ is known. We know $E\left(S^2\right)=\theta$ (a) What is the efficiency of $S^2?$ (b) Under these conditions, what is the mle $\hat{\theta }$ of $\theta$ ? (c) What is the asymptotic distribution of $\sqrt{n}(\hat{\theta }-\theta )?$

Let $X_1,X_2,\dots ,X_n$ be a random sample from a distribution with pmf $p(x;\theta )={\theta }^x(1-\theta {)}^{1-x},x=0,1$, where $0<\theta <1.$ We wish to test $H_0:\theta =1/3$ versus $H_1:\theta \ne 1/3$ (a) Find $\mathrm{\Lambda }$ and $-2\log \mathrm{\Lambda }$. (b) Determine the Wald-type test. (c) What is Rao's score statistic?

Let $X_1,X_2,\dots ,X_n$ be a random sample from a Poisson distribution with mean $\theta >0.$ Test $H_0:\theta =2$ against $H_1:\theta \ne 2$ using (a) $-2\log \mathrm{\Lambda }$. (b) a Wald-type statistic. (c) Rao's score statistic.

Let $X_1,X_2,\dots ,X_n$ be a random sample from a $\mathrm{\Gamma }(\alpha ,\beta )$ -distribution where $\alpha$ is known and $\beta >0$. Determine the likelihood ratio test for $H_0:\beta ={\beta }_0$ against $H_1:\beta \ne {\beta }_0$

Let \(Y_{1}0\). (a) Show that $\mathrm{\Lambda }$ for testing $H_0:\theta ={\theta }_0$ against $H_1:\theta \ne {\theta }_0$ is $\mathrm{\Lambda }={\left(Y_n/{\theta }_0\right)}^n$, $Y_n\le {\theta }_0$, and $\mathrm{\Lambda }=0$, if $Y_n>{\theta }_0$ (b) When $H_0$ is true, show that $-2\log \mathrm{\Lambda }$ has an exact ${\chi }^2(2)$ distribution, not ${\chi }^2(1).$ Note that the regularity conditions are not satisfied.