STA215: Statistical Inference
Homework #2
Due: Thursday, Feb 10

Let X_{j} be a sequence of independent Ga(4,lambda) random
variables (see explanation of parameterization here), with pdf f(x)=(lambda^{4} x^{3}
/6)e^{x*lambda} for x>0. Then:
 Find the MLE T_{n}(X) for lambda, upon observing
X_{1},...,X_{n}.
 Is T_{n}(X) consistent? Show that it is.
 Find the limit as n increases of
n·E[(T_{n}(X)lambda)^{2}]
 You may find it helpful to compute E[X^{p}] for all p.
Please do so.

Let X_{j} have a Lognormal distribution with parameters mu and
sigma^{2}; such a random variable has a representation
X_{j} = exp(mu+sigma · Z_{j}) for a standard
N(0,1) random variable Z_{j}), and let theta be the mean
e^{mu+sigma²/2}, and let lambda be the quantity
e^{sigma²/2}.
 Find the MLE for theta and lambda.
 The Harmonic Mean is defined by H_{n} = n/(1/X_{1}
+...+1/X_{n}). Does it converge to theta as n increases?
Find the (possibly zero) asymptotic bias. [HINT: Find the limit
of 1/H_{n} first).
 OPTIONAL and HARD: Xbar is also a sensible
estimator for theta. Which is more efficient, Xbar or the MLE
you found? By how much? (This turns out to be much harder
than I'd intended; the answer only needs the asymptotoc
expectations E[n(Xbartheta)^{2}] and
E[n(MLEtheta)^{2}]. I'd skip this one if I were you)