Consider the sample s = (x1, . . . , xn), realization of a random sample X1, . . . , Xn from a uniform on [0, θ], with θ unknown. The goal is to estimate the unknown mean θ/2, based on s. To this end we consider two competing estimators: T1 = X(n)/2, where X(n) is defined as the maximum of (X1, . . . , Xn), T2 = X¯n, that is the sample mean.

a)Study the distribution of T1 by following the next steps: 

a.1) find the cumulative distribution function (cdf) of T1,

 a.2)from the cdf obtain the probability density function (pdf) of T1,

 a.3)use the pdf to compute the expected value of T1, 

a.4) use the pdf to compute the variance of T1. 

b)Is T1 an unbiased estimator of θ/2? If the answer is no, study the limit of the bias when the sample size n goes to infinity. 

c)Find an expression for MSEθ(T1). 

d)Compute expected value and variance of T2.

e)Is T2 an unbiased estimator of θ/2? If the answer is no, study the limit of the bias when the sample size n goes to infinity.

 f)Find an expression for MSEθ(T2). 

g)On the basis of variance, bias and MSE of T1 and T2, compare and briefly discuss the accuracy of the two estimators of θ/2. It might be helpful to produce a plot comparing MSEθ[T1] and MSEθ[T2] as functions of θ ∈ [0, 1] for some fixed value of n. 

h)Consider a third estimator T3 of θ/2 defined as kT1 for some positive constant k > 0. Choose k so to minimize MSEθ(T3). 

i)Run a small simulation study to compare the estimators T1, T2 and T3 (where for T3 the optimal value for k must be considered). Specifically, follow the next steps: 1.fix a value for θ, 

2.fix a valus for n, 

3.simulate x1, . . . , xn from the uniform distribution on [0, θ], 

4.compute the estimates T1(x1, . . . , xn), T2(x1, . . . , xn) and T3(x1, . . . , xn) from the values generated, 

5.compute the squared of the errors for the three estimates, that is ej = (Tj(x1, . . . , xn) θ/2)2, for j = 1, 2, 3, and store values ej you obtain, 

6.repeat the steps 1 to 5 for N times (with N fixed and sufficiently large), 

7.for each one of the estimators, take the average of the squared errors ej obtained at each replicate of the steps 1 to 5. 

l) Briefly discuss whether the numerical findings of the experiment are consistent with the ana- lytical results you obtained at the previous points.