STA215: Statistical Inference

Homework #2

Due: Thursday, Feb 10

1. Let X_j be a sequence of independent Ga(4,lambda) random variables (see explanation of parameterization here), with pdf f(x)=(lambda⁴ x³ /6)e^-x*lambda for x>0. Then:
   1. Find the MLE T_n(X) for lambda, upon observing X₁,...,X_n.
   2. Is T_n(X) consistent? Show that it is.
   3. Find the limit as n increases of n·E[(T_n(X)-lambda)²]
   4. You may find it helpful to compute E[X^p] for all p. Please do so.

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2. Let X_j have a Lognormal distribution with parameters mu and sigma²; 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.
   1. Find the MLE for theta and lambda.
   2. The Harmonic Mean is defined by H_n = n/(1/X₁ +...+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).
   3. 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(Xbar-theta)²] and E[n(MLE-theta)²]. I'd skip this one if I were you)