Question

Let \(X_{1}, \ldots, X_{n}\) and \(Y_{1}, \ldots, Y_{m}\) follow the location model $$ \begin{aligned} X_{i} &=\theta_{1}+Z_{i}, \quad i=1, \ldots, n \\ Y_{i} &=\theta_{2}+Z_{n+i}, \quad i=1, \ldots, m, \end{aligned} $$ where \(Z_{1}, \ldots, Z_{n+m}\) are iid random variables with common pdf \(f(z) .\) Assume that \(E\left(Z_{i}\right)=0\) and \(\operatorname{Var}\left(Z_{i}\right)=\theta_{3}<\infty\) (a) Show that \(E\left(X_{i}\right)=\theta_{1}, E\left(Y_{i}\right)=\theta_{2}\), and \(\operatorname{Var}\left(X_{i}\right)=\operatorname{Var}\left(Y_{i}\right)=\theta_{3}\). (b) Consider the hypotheses of Example 8.3.1, i.e., $$ H_{0}: \theta_{1}=\theta_{2} \text { versus } H_{1}: \theta_{1} \neq \theta_{2} \text { . } $$ Show that under \(H_{0}\), the test statistic \(T\) given in expression \((8.3 .4)\) has a limiting \(N(0,1)\) distribution. (c) Using part (b), determine the corresponding large sample test (decision rule) of \(H_{0}\) versus \(H_{1}\). (This shows that the test in Example \(8.3 .1\) is asymptotically correct.)

Short answer

The exercise demonstrates the relation of the expectation and variance of random variables, and statistical testing of hypotheses. The test statistic \(T\) under \(H_{0}: \theta_{1}=\theta_{2}\) follows a standard normal (N(0,1)) distribution. The large sample test (decision rule) of \(H_{0}\) versus \(H_{1}\) is developed based on this information, where the null hypothesis is rejected if the absolute value of \(T\) is sufficiently large.

Step-by-step solution

E(Xi) Calculation

We know that \(E(X_i)\) is equal to \(\theta_1 + E(Z_i)\). Since assumed that \(E(Zi) = 0\), we can infer that \(E(X_i) = \theta_1\).

E(Yi) Calculation

Similar to the calculation of \(E(X_i)\), we can perform for \(E(Y_i)\) as well. Since \(E(Yi) = \theta_2 + E(Z_i)\) and \(E(Zi) = 0\), we get \(E(Y_i) = \theta_2\).

Variance Calculation

Since variances are not affected by the change of location, we are given that \(\operatorname{Var}\left(Z_{i}\right)=\theta_{3}<\infty\). Therefore, \(\operatorname{Var}\left(X_{i}\right)=\operatorname{Var}\left(Y_{i}\right)=\theta_{3}\).

Statistics and Hypotheses

Under the null hypothesis \(H_{0}\), \(\theta_{1}=\theta_{2}\). Given this, the test statistic \(T\) simplifies to a value that follows a standard normal (N(0,1)) distribution. This is due to the Central Limit Theorem as the sample size goes to infinity.

Large Sample Test (Decision Rule)

It follows from part (b) that we reject the null hypothesis \(H_{0}: \theta_{1}=\theta_{2}\) if the absolute value of \(T\) is sufficiently large. This is because under the null hypothesis, \(T\) should follow a standard normal distribution and hence should not take on extreme values. This gives us a decision rule for testing the hypothesis based on the given location model.