STA215: Statistical Inference

Homework #2

Due: Thursday, Feb 10

  1. Let Xj 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 Tn(X) for lambda, upon observing X1,...,Xn.
    2. Is Tn(X) consistent? Show that it is.
    3. Find the limit as n increases of n·E[(Tn(X)-lambda)2]
    4. You may find it helpful to compute E[Xp] for all p. Please do so.

  2. Let Xj have a Lognormal distribution with parameters mu and sigma2; such a random variable has a representation Xj = exp(mu+sigma · Zj) for a standard N(0,1) random variable Zj), and let theta be the mean emu+sigma²/2, and let lambda be the quantity esigma²/2.
    1. Find the MLE for theta and lambda.
    2. The Harmonic Mean is defined by Hn = n/(1/X1 +...+1/Xn). Does it converge to theta as n increases? Find the (possibly zero) asymptotic bias. [HINT: Find the limit of 1/Hn 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)