The number of minutes past the scheduled departure time that jets with no mechanical problems leave the terminal in an overcrowded airport in the northeast are iid outcomes from a uniform population distribution of the form \(f(z ; \Theta)=\Theta^{-1} I_{(0, \Theta)}(z)\). A random sample of 1,000 departures is to be used to estimate the parameter \(\Theta\) and the expected number of minutes past the scheduled departure time that a jet will leave the terminal. Summary statistics from the outcome of the random sample include \(\min (\mathbf{x})=.1, \max (\mathbf{x})=13.8, \quad \overline{\mathbf{x}}=6.8\), \(s^{2}=15.9\).

(a) Define a MLE for \(\Theta\) and for expected number of minutes past the scheduled departure time that a jet will leave the terminal. Are these MLEs functions of minimal sufficient statistics?

(b) Use the MLEs you defined above to generate ML estimates of the respective quantities of interest.

(c) Are the estimators in

(a) unbiased? consistent? (Hint: \(\mathrm{E}(\max (X))=\Theta[n /(n+1)]\) and \(\mathrm{E}\left((\max (X))^{2}ight)=\) \(\left.\Theta^{2}[n /(n+2)]ight)\)

(d) Are the estimators in

(a) MVUES?