The research department of the Personal Computer Monthly magazine is analyzing the operating life of computer chips that are produced by a major manufacturer located in the Silicon Valley. The research staff postulates that a random sample of lifetimes being analyzed adheres to the statistical model

\(\mathbf{X} \sim \theta^{-n} e^{-\sum_{i=1}^{n} x_{i} / \theta} \prod_{i=1}^{n} I_{(0, \infty)}\left(x_{i}ight), \quad\) where \(\theta>0\).

The magazine publishes an index defined by \(\beta=1 / \theta\) to measure the quality of computer chips, where the closer \(\beta\) is to zero, the better the computer chip. The joint density of \(\mathbf{X}\) is reparameterized so that the density function is parameterized by \(\beta\), as

\(\mathbf{X} \sim \beta^{n} e^{-\beta \sum_{i=1}^{n} x_{i}} \prod_{i=1}^{n} I_{(0, \infty)}\left(x_{i}ight), \quad\) where \(\beta>0\).

(a) Does the reparameterized family of density functions belong to the exponential class of density functions?

(b) Define a set of minimal sufficient statistics for the reparameterized density function. Are the minimal sufficient statistics complete sufficient statistics?

(c) Does there exist an unbiased estimator of \(\beta\) whose variance achieves the Cramer-Rao lower bound?

(d) Define the maximum likelihood estimator for the parameter \(\beta\). Is the MLE a function of the complete sufficient statistic? Is the MLE a consistent estimator of \(\beta\) ?

(e) Is the MLE the MVUE of \(\beta\) ? Is the MLE asymptotically normally distributed? Asymptotically efficient?