Solved: Each headlight on an automobile undergoing an

Chapter 14, Problem 19E

(choose chapter or problem)

Each headlight on an automobile undergoing an annual vehicle inspection can be focused either too high (H), too low (L), or properly (N). Checking the two headlights simultaneously (and not distinguishing between left and right) results in the six possible outcomes HH, LL, NN, HL, HN, and LN. If the probabilities (population proportions) for the single headlight focus direction are \(P(H)=\theta_1, P(L)=\theta_2\), and \(P(N)=1-\theta_1-\theta_2\) and the two headlights are focused independently of one another, the probabilities of the six outcomes for a randomly selected car are the following:

\(\begin{array}{l}p_{1}=\theta_{1}^{2} \quad p_{2}=\theta_{2}^{2} \quad p_{3}=\left(1-\theta_{1}-\theta_{2}\right)^{2} \\ p_{4}=2 \theta_{1} \theta_{2} \quad p_{5}=2 \theta_{1}\left(1-\theta_{1}-\theta_{2}\right) \\ p_{6}=2 \theta_{2}\left(1-\theta_{1}-\theta_{2}\right) \end{array} \)

Use the accompanying data to test the null hypothesis

\(H_{0}: p_{1}=\pi_{1}\left(\theta_{1}, \theta_{2}\right), \ldots, p_{6}=\pi_{6}\left(\theta_{1}, \theta_{2}\right)\)

where the \(\pi_{i}\left(\theta_{1}, \theta_{2}\right)\)s are given previously.

[Hint: Write the likelihood as a function of \(\theta_1\) and \(\theta_2\), take the natural log, then compute \(\partial / \partial \theta_{1}\) and \(\partial / \partial \theta_{2}\), equate them to 0, and solve for \(\hat{\theta}_{1}, \hat{\theta}_{2}\).]