Problem 4 Consider the following variation on the "German tank problem:" we observe a sample of 10 i.i.d. draws Xi ~ U[0, 0] and are interested in estimating 0. We consider estimators that are based on the sample minimum Xmin := min{X1, ..., X10} and sample maximum Xmax := max{X1, . .., X10}. (a) State the c.d.f. Fmax(x) of Xmax and derive its p.d.f. fmax(x) by taking the derivative. (b) Use your answer to (a) to calculate the expectation E[Xmax]. (c) Use your answer to (b) to construct an unbiased estimator for 0 that is of the form 0max := CmarXmax for some appropriately chosen constant Cmax. (d) Calculate the expectation of E[Xmin]. Hint: notice that min {X1, . . ., X10} = 0 - max {0 - X1, ...,0 - X10} and that the random variable 0 - X also follows the U[0, 0] distribution. (e) Use your answer to (d) to construct an unbiased estimator for 0 that is of the form 0min := CminXmin for some appropriately chosen constant Cmin. (f) Show that Var(Xmin) = Var(Xmax). Hint: as in part (d), you may use the fact that min {X1, . . ., X10} = 0 - max {0 - X1, ...,0 -X10} and that the random variable 0 - X also follows the U[0, 0] distribution. (f) Derive the efficiency ratio r := Var(0max) Var(0min) Is one of the estimators efficient relative to the other