Problem 1. Assume a SRS X1,··· ,Xn from the population random variable X having uniform distribution U(0,θ) with some θ 0.
(i) Find the moment estimator θˆM;
(ii) Find the maximum likelihood estimator θˆL;
(iii) For the real data (X1,··· ,X7) = (1.0,2.4,3.2,1.2,0.5,3.1,6.8), evaluate the observed values of θˆM and θˆL. Which one is better? Why?
(iv) For θ = 7, generate 100 SRS’s of size n = 30, evaluate the observed values of θˆM and θˆL. Produce box plot and mark the sample mean of the corresponding observed values of θˆM and θˆL, respectively.
(v) For θ = 7, generate a SRS of size n = 20,30,50,100,150, evaluate the observed values of θˆM and θˆL. Plot the corresponding observed values of θˆM and θˆL, respectively.
Problem 2. Assume a SRS X1,··· ,Xn from the population random variable X ∼ N(µ,σ2).
(i) Find the moment estimator (ˆµ,
(ii) Find the maximum likelihood estimator (˜µ,
Problem 3. Finish the following problems in the textbook:
