Testing Linear Hypotheses in the Gaussian Regression Model

Construct an $F$-test of significance level $\alpha$ for testing the hypothesis $H_{0}: \beta_{2}=\beta_{1}$ in problem 2 from the self-study problems of section 9.

$X_{1}, \ldots , X_{n}$ is a sample from the distribution $\mathscr {N}\left(a_{1}, \sigma^{2}\right)$, $Y_{1}, \ldots , Y_{m}$ is a sample from the distribution $\mathscr {N}\left(a_{2}, \sigma^{2}\right)$, $Z_{1}, \ldots , Z_{k}$ is a sample from the distribution $\mathscr {N}\left(a_{3}, \sigma^{2}\right)$. Construct an $F$-test of size $\alpha$ for testing the hypothesis:

Solve problem $11.2$ using theorem $11.1$.

Let $X_{i} \sim N\left(a, i \sigma^{2}\right), i=0,1, \ldots , n, Y_{i} \sim \mathscr {N}\left(jb, \sigma^{2}\right), j=1, \ldots , m$ be independent random variables, where $a, b, \sigma^{2}$ are unknown parameters. Reduce the problem to a linear model and construct an $F$-test of size $\alpha$ for testing the hypothesis $H_{0}: a+b=1$.

Using the linear regression method, construct an approximation of the function $f(x)$ by a cubic polynomial based on the following data:

Let $X_{i j} \sim \mathscr {N}\left(\mu_{j}, \sigma^{2}\right), i=1, \ldots , n_{j}, j=1, \ldots , k$, be independent random variables, and let $\left\{ \mu_{j}\right\}_{j=1}^{k}, \sigma^{2}$ be unknown parameters. Denote $N=\sum_{j=1}^{k} n_{j}$. To test the homogeneity hypothesis $H_{0}: \mu_{1}=\ldots =\mu_{k}$ we use the $F$-test of one-way analysis of variance with the following statistic

where

Prove that when $H_{0}$ holds, $R \sim F_{k-1, N-k}$.