/*! 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 5 Let \(X_{1}, X_{2}, \ldots, X_{n... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 5

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.

Short Answer

Expert verified
The null distribution of the likelihood ratio test statistic \(W\) is \(\chi^{2}_{n}\). The rejection rule for a level \(\alpha\) test is that \(H_{0}\) is rejected if \(W>n + \chi^{2}_{n,\alpha/2}\) or \(W<n-\chi^{2}_{n,\alpha/2}\).

Step by step solution

01

Calculate the Maximum Likelihood Estimates

Begin with the normal distribution \(N\left(\mu_{0}, \sigma^2=\theta\right)\) and calculate the likelihood function \(L(\theta;\mathbf{x})=\left(\frac{1}{\sqrt{2\pi\theta}}\right)^n \exp\left(-\frac{\sum_{i=1}^{n}\left(X_{i}-\mu_{0}\right)^{2}}{2\theta}\right)\). Under \(H_{0}\), \(\hat{\theta}_0=\frac{\sum_{i=1}^{n}\left(X_{i}-\mu_{0}\right)^{2}}{n}\). Under \(H_{1}\), the maximum likelihood estimator \(\hat{\theta}_1\) would take the same form as \(\hat{\theta}_0\) because the estimator does not depend on the value of \(\theta\).
02

Define the Likelihood Ratio Test and \(\lambda (\mathbf{x})\)

The likelihood ratio test statistic is given by \(\lambda(\mathbf{x})=\frac{\sup_{\theta= \theta_0}L(\theta;\mathbf{x})}{\sup_{\theta}L(\theta;\mathbf{x})} = \frac{L(\theta_0;\mathbf{x})}{L(\hat{\theta}_1;\mathbf{x})}\). The definitions of our maximum likelihood estimates under \(H_{0}\) and \(H_{1}\) make it clear that in this case \(\lambda(\mathbf{x})= \left(\frac{\theta_0}{\hat{\theta}_1}\right)^{n/2} \exp\left( \frac{n}{2}\left(1-\frac{\theta_0}{\hat{\theta}_1}\right) \right)\).
03

Find the Null Distribution and the Rejection Rule

The null distribution is given by twice the natural logarithm of \(\lambda(\mathbf{x})\), or \(2\ln(\lambda(\mathbf{x}))\), and is distributed as \(\chi^{2}_{n}\). The likelihood ratio test rejects \(H_{0}\) if \(2\ln(\lambda(\mathbf{x}))\) falls in the rejection region of the \(\chi^{2}_{n}\) distribution. This means our test statistic \(W=\sum_{i=1}^{n}\left(X_{i}-\mu_{0}\right)^{2} / \theta_{0}\) falls in the rejection region if \(W>n + \chi^{2}_{n,\alpha/2}\) or \(W<n-\chi^{2}_{n,\alpha/2}\), where \(\chi^{2}_{n,\alpha/2}\) is the \(\alpha/2\) quantile of the \(\chi^{2}_{n}\) distribution.

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Most popular questions from this chapter

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 \(\boldsymbol{\mu}=\theta \mathbf{1}\), where \(\mathbf{1}\) is a vector with all components equal to \(1 .\) Let \(V\) be the subspace of vectors of the form \(\mu\); 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 $$ \mathbf{X}=\boldsymbol{\mu}+\mathbf{e}, \quad \boldsymbol{\mu} \in V $$ Then we can summarize the model by saying, "Except for the random error vector e, \(\mathbf{X}\) would reside in \(V . "\) Hence, it makes sense intuitively to estimate \(\boldsymbol{\mu}\) by a vector in \(V\) which is "closest" to \(\mathbf{X}\). That is, given a norm \(\|\cdot\|\) in \(R^{n}\), choose $$\widehat{\boldsymbol{\mu}}=\operatorname{Argmin}\|\mathbf{X}-\mathbf{v}\|, \quad \mathbf{v} \in V$$ (a) If the error pdf is the Laplace, \((2.2 .1)\), show that the minimization in \((6.3 .27)\) is equivalent to maximizing the likelihood when the norm is the \(l_{1}\) norm given by $$\|\mathbf{v}\|_{1}=\sum_{i=1}^{n}\left|v_{i}\right|$$ (b) If the error pdf is the \(N(0,1)\), show that the minimization in \((6.3 .27)\) is equivalent to maximizing the likelihood when the norm is given by the square of the \(l_{2}\) norm $$\|\mathbf{v}\|_{2}^{2}=\sum_{i=1}^{n} v_{i}^{2}$$

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Bernoulli distribution with parameter \(p .\) If \(p\) is restricted so that we know that \(\frac{1}{2} \leq p \leq 1\), find the mle of this parameter.

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.

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 \(\widehat{\theta}=\min \\{\bar{X}, 2\\}\).

Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) be a random sample from a bivariate normal distribution with \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}, \rho=\frac{1}{2}\), where \(\mu_{1}, \mu_{2}\), and \(\sigma^{2}>0\) are unknown real numbers. Find the likelihood ratio \(\Lambda\) for testing \(H_{0}: \mu_{1}=\mu_{2}=0, \sigma^{2}\) unknown against all alternatives. The likelihood ratio \(\Lambda\) is a function of what statistic that has a well- known distribution?
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