Let X1, …, Xn and Y1, …, Ym follow the location model Xi =θ1+Zi, i=1, …, n Yi =θ2+Zn+

step 1 To solve this question, we have to first write PDF of x, which is f of x, is equals to 2x sigma

Let X1, …, Xn and Y1, …, Ym follow the location model Xi ...

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.)

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.)

Key Concepts

Location Models

Location models describe data that can be expressed as a constant (the location parameter) added to a random error term. This framework is useful for modeling shifts in the central tendency of the distribution, where the variability of the data remains constant.

Expectation and Variance in Shifted Distributions

When a random variable is shifted by a constant, its expectation is increased by that constant while its variance remains unchanged. This property is essential in the analysis of location models as it allows for simple derivations of the mean and variability of the observed data.

Hypothesis Testing for Equality of Means

This involves comparing the location parameters (means) of two populations. The hypothesis test assesses whether the shift in location parameters is statistically significant, which is a fundamental question in inferential statistics.

Central Limit Theorem (CLT)

The CLT is a key concept that explains why, under certain conditions, the distribution of the sample mean approximates a normal distribution as the sample size increases. This is critical for justifying the use of normal-based tests for hypothesis testing in large samples.

Asymptotic Distribution of Test Statistics and Decision Rules

In large sample theory, test statistics often converge in distribution to a standard normal distribution under the null hypothesis. This asymptotic behavior facilitates the creation of decision rules—by comparing the test statistic to critical values from the normal distribution, one can decide whether to reject or not reject the null hypothesis.

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