Concept Check Test Statistic for the Likelihood Ratio Test 0 1 point graded Suppose we are hypothesis testing between a null and alternative of the form l H0 ft r 1 d ft r 1 0 d 0 H1 ft r 1 d q ft r 1...

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Concept Check: Test-Statistic for the Likelihood Ratio Test 0/1 point (graded) Suppose we are hypothesis testing between a null and alternative of the form \[ \begin{array}{l} H_{0}:\left(\theta_{r+1}^{*}, \ldots, \theta_{d}^{*}\right)=\left(\theta_{r+1}^{(0)}, \ldots, \theta_{d}^{(0)}\right) \\ H_{1}:\left(\theta_{r+1}^{*}, \ldots, \theta_{d}^{*}\right) \neq\left(\theta_{r+1}^{(0)}, \ldots, \theta_{d}^{(0)}\right) . \end{array} \] Above, $\theta^{*} \in \mathbb{R}^{d}$ is an unknown parameter while the values $\theta_{r+1}^{(0)}, \ldots, \theta_{d}^{(0)}$ are known. To perform the likelihood ratio test, we define the test statistic \[ T_{n}=2\left(\ell_{n}\left(\widehat{\theta_{n}^{M L E}}\right)-\ell_{n}\left(\widehat{\theta_{n}^{c}}\right)\right) \] Assume that the technical conditions needed for the MLE to be a consistent estimator are satisfied, and assume that the null-hypothesis is true. Which of the following are true about the above test statistic $T_{n}$ ? (Choose all that apply. Refer to the slides.) $T_{n}$ is a pivotal statistic; i.e., it converges to a pivotal distribution. $T_{n}$ is asymptotically normal. \[ T_{n} \xrightarrow{(d)} \chi_{d-r}^{2} \] \[ T_{n} \xrightarrow{(d)} \chi_{r}^{2} \]

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#### Solution By Steps ***Step 1: Understanding the Test Statistic*** The test statistic $T_{n}=2\left(\ell_{n}\left(\widehat{\theta_{n}^{MLE}}\right)-\ell_{n}\left(\widehat{\theta_{n}^{c}}\right)\right)$ measures the difference between the maximum likelihood estimate under the full model and the constrained model (under $H_0$). ***Step 2: Identifying the Distribution*** Under the null hypothesis and given the conditions for the MLE are met, the distribution of $T_{n}$ converges to a chi-square distribution. The degrees of freedom are determined by the difference in the number of parameters estimated under the null and alternative hypotheses. ***Step 3: Calculating Degrees of Freedom*** The degrees of freedom for the chi-square distribution are given by $(d - r)$, where $d$ is the total number of parameters and $r$ is the number of parameters specified under the null hypothesis. #### Final Answer - $T_{n}$ converges to a pivotal distribution. - $T_{n} \xrightarrow{(d)} \chi_{d-r}^{2}$ #### Key Concept Chi-Square Distribution #### Key Concept Explanation In the context of the likelihood ratio test, the test statistic $T_{n}$, under the null hypothesis and certain regularity conditions, converges in distribution to a chi-square distribution. The degrees of freedom for this chi-square distribution are determined by the difference in the number of free parameters under the null and alternative hypotheses. This property allows for the construction of a test that can determine whether there is significant evidence to reject the null hypothesis in favor of the alternative.

Follow-up Knowledge or Question

What is the significance of a test statistic being pivotal in hypothesis testing?

How does the asymptotic normality of a test statistic impact the likelihood ratio test?

Explain the interpretation and implications of $T_{n} \xrightarrow{(d)} \chi_{d-r}^{2}$ in the context of the likelihood ratio test.

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Suppose we are hypothesis testing between a null and alternative of the form l H0 ft( r+1 d )= ft( r+1(0) d(0) ) H1 ft( r+1 d ) q ft( r+1(0) d(0) ) Above Rd is an unknown parameter while the values r+1(0) d(0) are known To perform the likelihood ratio test we define the test statistic Tn=2 ft( n ft(widehat nM L E )- n ft(widehat nc ) ) Assume that the technical conditions needed for the MLE to be a consistent estimator are satisfied and assume that the null-hypothesis is true Which of the following are true about the above test statistic Tn (Choose all that apply Refer to the slides) Tn is a pivotal statistic ie it converges to a pivotal distribution Tn is asymptotically normal Tn xrightarrow(d) d- 2 Tn xrightarrow(d) r2

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