/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Let \(X_{1}, X_{2}, \ldots, X_{n... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 11

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from the Poisson distribution with \(0<\theta \leq 2\). Show that the mle of \(\theta\) is \(\widehat{\theta}=\min \\{\bar{X}, 2\\}\).

Short Answer

Expert verified
$\widehat{\theta}=\min \{\bar{X}, 2\}$

Step by step solution

01

Write down the likelihood function

Recall that if \(X_i\) for \(i=1,2,...,n\) are iid Poisson random variables, the likelihood function is given by \(L(\theta |X)=\prod_{i=1}^{n} e^{-\theta} \frac{\theta^{X_i}} {X_i!}\). Taking the natural logarithm of both sides, we get the log-likelihood function, \(l(\theta |X)= \sum_{i=1}^{n} [-\theta +X_i log(\theta) -log(X_i!)]\). Note that we can omit the last term since it does not depend on \(\theta\).
02

Maximize the log-likelihood function

Differentiate the log-likelihood function w.r.t \(\theta\) to find the maximum. Setting \(l'(\theta)\) equal to zero, the derivative is given by \(-n+\frac{\sum X_i}{\theta} = 0\). Solve this equation for \(\theta\) to get \(\theta = \frac{\sum X_i}{n} = \bar{X}\), where \(\bar{X}\) is the sample mean.
03

Show that the mle is \(\widehat{\theta}=\min \{\bar{X}, 2\}\)

Recall the conditions on \(\theta\), \(0<\theta \leq 2\). If the sample mean \(\bar{X} <= 2\), we can set \(\theta = \bar{X}\) without violating these conditions. However, if the sample mean \(\bar{X} > 2\), we must set \(\theta\) to its maximum allowed value which is 2. This leads to the result that \(\widehat{\theta}=\min \{\bar{X}, 2\}\).

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Most popular questions from this chapter

\left\\{X_{i}>\theta_{0}\right\\}\(. (b) Show that the scores test for this model is equival鈥� # Consider Example \)6.3 .4\(. (a) Show that we can write \)S^{*}=2 T-n\(, where \)T=\\#\left\\{X_{i}>\theta_{0}\right\\}\(. (b) Show that the scores test for this model is equivalent to rejecting \)H_{0}\( if \)Tc_{2}\( (c) Show that under \)H_{0}, T\( has the binomial distribution \)b(n, 1 / 2) ;\( hence, determine \)c_{1}\( and \)c_{2}\( so the test has size \)\alpha\(. (d) Determine the power function for the test based on \)T\( as a function of \)\theta$.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta \exp \left\\{-|x|^{\theta}\right\\} / 2 \Gamma(1 / \theta),-\infty0 .\) Suppose \(\Omega=\) \(\\{\theta: \theta=1,2\\} .\) Consider the hypotheses \(H_{0}: \theta=2\) (a normal distribution) versus \(H_{1}: \theta=1\) (a double exponential distribution). Show that the likelihood ratio test can be based on the statistic \(W=\sum_{i=1}^{n}\left(X_{i}^{2}-\left|X_{i}\right|\right)\).

Let \(X\) be \(N(0, \theta), 0<\theta<\infty\). (a) Find the Fisher information \(I(\theta)\). (b) If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from this distribution, show that the mle of \(\theta\) is an efficient estimator of \(\theta\). (c) What is the asymptotic distribution of \(\sqrt{n}(\widehat{\theta}-\theta) ?\)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Poisson distribution with \(\operatorname{mean} \theta>0\) (a) Show that the likelihood ratio test of \(H_{0}: \theta=\theta_{0}\) versus \(H_{1}: \theta \neq \theta_{0}\) is based upon the statistic \(Y=\sum_{i=1}^{n} X_{i} .\) Obtain the null distribution of \(Y\). (b) For \(\theta_{0}=2\) and \(n=5\), find the significance level of the test that rejects \(H_{0}\) if \(Y \leq 4\) or \(Y \geq 17\)

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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