4. In each case below, the outcome of some function, T(X1n)), of n iid random variables X1n) = (XI, ... , Xn) is being considered for providing an estimate of some function of parameters, q(B). Determine whether ET(Xlnd =

q(BJ, limn ... "" ET(Xln)) := q(B), and plim T(Xln)) := q(B).

a. X;'s ~ iid Gamma

(a, tJ) and TlX1n)) := Xn is being used to estimate q

(a, III = all.

b. X;'s ~ iid Gamma(a,tJ) and T(X1n)) = 2::7=I(Xj -

Xn)2/(n - 1) is being used to estimate q

(a, tJ):= a1l1 .

292 Chapter 5 Basic Asymptotics

c. X/s ~ iid Bernoulli(p) and llX1nl ) = Xn(I - Xn) is being used to estimate q(p) = pi I - pl.

d. X/s ~ iid N(J.L, a2 ) and llX1nl ) = (2:7=1 Xj -nl /2 )/(n+ 1)

is used to estimate J.L.