Question

1. Let Y1 < Y2 < · · · Ym be the order statistics of m independent observations X1, · · · , Xm from a uniform distribution on the interval [θ, θ + 1].

(a) (5 points) Find the distribution of Yr, where r is a integer and 1 ≤ r ≤ m.

(b) (5 points) Calculate V ar(Ym) if θ = 0.

(c) (5 points) Suppose θ is unknown, m = 5 and we have observed that x1 = 3.1 x2 = 3.3 x3 = 3.15 x4 = 3.6 x5 = 4

Does the M.L.E of θ exist? If yes, what is the M.L.E.? Justify the uniquensess of M.L.E of θ.[HINT: use the joint p.d.f. of X1, · · · X5, we don’t need order statistics here.]

(d) (5 points) Use the observed values x1, · · · , x5 in (c), find the method of moments estimation of θ.[HINT: find E(X) at first, we don’t need any order statistics here.]

Answer 1