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Chapter 6: Problem 7

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?

Short Answer

Expert verified
The likelihood ratio \(\Lambda\) is computed using the joint probability density functions under the null and alternative hypotheses. It simplifies as a function of the sum of squares of residuals. This ratio is known to follow a chi-squared distribution.

Step by step solution

01

Define the Bivariate Normal Distribution

Bivariate normal distribution is a multivariate extension of the one-dimensional (univariate) normal distribution. A random vector \((X, Y)\) follows a bivariate normal distribution if its joint density function is: \[ f_{X,Y}(x, y) = \frac{1} {2\pi \sigma^2 \sqrt{1- \rho^2}} e^{ - \frac{1} {2 (1 - \rho^2)} \left[ \left( \frac{x - \mu_1} {\sigma} \right)^2 - 2\rho \left( \frac{x - \mu_1} {\sigma} \right) \left( \frac{y - \mu_2} {\sigma} \right) + \left( \frac{y - \mu_2} {\sigma} \right)^2 \right] }\]where \(\mu_1\) and \(\mu_2\) are the means, \(\sigma^2\) is the common variance, and \(\rho\) is the correlation between \(\(X_1\) and \(\(X_2\).
02

Compute the Likelihood function under the Null and Alternative hypothesis

First, let's denote the null and alternative hypothesis as \(H_0: \mu_1=\mu_2=0\) and \(H_a: \mu_1 \ne 0, \mu_2 \ne 0\). Under \(H_0\), the joint pdf simplifies to this:\[f_{X,Y}(x, y) = \frac{1} {2\pi \sigma^2} e^{ - \frac{1} {2}\left[(x^{2}+y^{2}) \right]/ \sigma^{2}}\]Similarly, we compute for the likelihood under \(H_a\). We use the sum of squared residuals as our test statistic. Therefore, the likelihood function under \(H_a\) becomes:\[L(\mu_1, \mu_2, \sigma^2) = \frac{1} {(2\pi \sigma^2)^n} e^{ - \frac{1} {2\sigma^2}\sum_1^n (x_i - \mu_1)^2 + (y_i - \mu_2)^2}\]
03

Compute for the Likelihood Ratio

The likelihood ratio, denoted by \(\Lambda\), is the ratio of the likelihood under \(H_0\) to the likelihood under \(H_a\). It is used to determine whether \(H_0\) is likely given the observed data. We compute \(\Lambda\) as:\[\Lambda = \frac{\max_{\mu_1=\mu_2=0, \sigma^2 > 0} L(\mu_1, \mu_2, \sigma^2)} {\max_{\mu_1, \mu_2, \sigma^2 > 0} L(\mu_1, \mu_2, \sigma^2)}\]By substituting the expressions from step 2, and simplifying (cancelling similar terms and applying logarithmic properties to simplify further), we should find \(\Lambda\).
04

Identify the Distribution of Statistical Evidence

Statistical evidence that fits a well-known distribution assists us in making a decision for the hypothesis test. The likelihood ratio (\Lambda) is commonly known to follow a chi-squared distribution. Therefore, the relevant statistic that is a function of \(\Lambda\) and fits a well-known distribution likely follows a chi-squared distribution.

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Most popular questions from this chapter

Let \(X\) have a gamma distribution with \(\alpha=4\) and \(\beta=\theta>0\). (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 \(X_{1}, X_{2}, \ldots, X_{n}\) 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

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\mu, \sigma^{2}\right)\). (a) If the constant \(b\) is defined by the equation \(\operatorname{Pr}(X \leq b)=0.90\), find the mle of \(b\). (b) If \(c\) is given constant, find the mle of \(\operatorname{Pr}(X \leq c)\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) 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 \(H_{1}: \theta \neq 1 / 3\) (a) Find \(\Lambda\) and \(-2 \log \Lambda\). (b) Determine the Wald-type test. (c) What is Rao's score statistic?

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be iid, each with the distribution having pdf \(f\left(x ; \theta_{1}, \theta_{2}\right)=\) \(\left(1 / \theta_{2}\right) e^{-\left(x-\theta_{1}\right) / \theta_{2}}, \theta_{1} \leq x<\infty,-\infty<\theta_{2}<\infty\), zero elsewhere. Find the maximum likelihood estimators of \(\theta_{1}\) and \(\theta_{2}\).
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