Testing Parametric Hypotheses

Generalize the result of Example 6.1 to the case where the variance $\sigma^{2}$ is unknown.

The sample $Z_{n}$ corresponds to the distribution $\mathcal{N}\left(\theta_{1} ; \theta_{2}\right)$. Test, at significance level $p$, the hypothesis $H_{0}: \theta_{2} = \sigma^{2}$ against $H_{1}: \theta_{2} \neq \sigma^{2}$.

Let $\left\{ X_{k}, k = 1, \ldots , n\right\}$ and $\left\{ Y_{m}, m = 1, \ldots , n\right\}$ be independent samples generated by the random variables $X \sim \mathcal{N}\left(\theta_{1} ; \sigma_{1}^{2}\right)$ and $Y \sim \mathcal{N}\left(\theta_{2} ; \sigma_{2}^{2}\right)$, where $\sigma_{1}$ and $\sigma_{2}$ are known. At significance level $p$, test the hypothesis $H_{0}: \theta_{1} = \theta_{2}$ against $H_{1}: \theta_{1} \neq \theta_{2}$.

Under the conditions of Example 6.4, find the probability $\beta$ of a type II error.

Under the conditions of the previous problem, determine the minimum sample size $n_{0}$ for which the quantity $\beta$ will be at most 0,01 , if $\theta_{0} = 0, \gamma = 1$, $\sigma^{2} = 1, p = 0,05$.

Under the conditions of Example 6.4, find the minimum sample size $n_{0}$ for which the probabilities of type I and type II errors do not exceed the given values $a > 0$ and $b > 0$, respectively.

The sample $Z_{n} = \left\{ X_{k}, k = 1, \ldots , n\right\}$ corresponds to the distribution $\mathcal{N}\left(0 ; \theta^{2}\right), \theta > 0$. Construct the most powerful test for testing, at significance level $p$, the hypothesis $H_{0}: \theta = \theta_{0}$ against the alternative $H_{1}: \theta = \sigma > \theta_{0}$.