Interval Estimates of Parameters

The sample $\left\{ X_{1}, \ldots , X_{n}\right\} , n \gg 1$ corresponds to the Poisson distribution $\operatorname {Pois}\left(\theta \right), \theta > 0$. Construct an asymptotic confidence interval for $\theta$ of confidence level $q$.

The sample $\left\{ X_{1}, \ldots , X_{n}\right\}$ corresponds to the distribution $\mathcal{N}(\mu ; \theta ), \mu$ is known. Construct a central confidence interval of confidence level $q$ for the variance $\theta$.

The sample $\left\{ X_{1}, \ldots , X_{n}\right\} , n \gg 1$ corresponds to the distribution $\operatorname {Exp}\left(\frac{1}{\theta }\right)$, $\theta > 0$. Construct an asymptotic confidence interval of confidence level $q$ for the parameter $\theta$.

From the sample $z_{n} = \left\{ -0,26 ;-0,36 ; 1,83 ; 0,54 ;-2,06\right\}$ corresponding to the distribution $\mathcal{N}\left(\theta_{1} ; \theta_{2}^{2}\right)$, find confidence intervals of confidence level $q = 0,9$ for $\theta_{1}$ and $\theta_{2}$.

The sample $Z_{n} = \left\{ X_{k}, k = 1, \ldots , n\right\} , n \gg 1$ corresponds to the uniform distribution $R[0 ; a], a > 0$. Construct an asymptotic confidence interval of confidence level $q$ for the parameter $\theta = \mathbb {E}\left[X_{1}\right]$.

The sample $Z_{n} = \left\{ X_{1}, \ldots , X_{n}\right\} , n \gg 1$ corresponds to the distribution with probability density $p(x ; \theta ) = \exp \left\{ \theta -x\right\} , x \geqslant \theta$, where $\theta > 0$. Show that $\left[X_{(1)}+\frac{\ln (1-q)}{n} ; X_{(1)}\right]$ is a confidence interval of confidence level $q$ for $\theta$.

Under the conditions of Problem 6, construct a central confidence interval of confidence level $q$ for $\theta$.

Let the sample $\left\{ X_{1}, \ldots , X_{n}\right\} , n \gg 1$ correspond to the distribution $R[0 ; \theta ], \theta > 0$. Show that $\left(\frac{X_{(n)}}{\theta }\right)^{n}$ is a pivotal statistic, and construct a confidence interval for $\theta$ of minimum length with confidence level $q$.

Let $Z_{n}^{(1)} = \left\{ X_{1}, \ldots , X_{n}\right\}$ and $Z_{n}^{(2)} = \left\{ Y_{1}, \ldots , Y_{n}\right\}$ be two independent samples, where $Z_{n}^{(1)}$ corresponds to the distribution $\mathcal{N}\left(\theta_{1} ; \sigma_{1}^{2}\right)$, and $Z_{n}^{(2)}$ to $\mathcal{N}\left(\theta_{2} ; \sigma_{2}^{2}\right)$ ($\sigma_{1}$ and $\sigma_{2}$ are known). It is required to construct a confidence interval of confidence level $q$ for the parameter $\theta = \theta_{1}-\theta_{2}$.