STAT 411 HW #2 – Written
Due February 26, 2004
You can turn it in to my office, 116B IH, or mailbox in 101 IH.
There are also problems on Mallard.
1. Suppose X1, . . . , Xnare iid from an exponential family model with pdf (for one observa-
tion)
f(x|θ) = a(x)eθx−ψ(θ).
Let µ(θ) = ψ0(θ) and σ2(θ) = ψ00(θ).
(a) Give the following functions in terms of the functions µ(θ) and σ2(θ) (these are
expressions from STAT 410 you need to just recall):
(i) Eθ[Xi].
(ii) V arθ[Xi].
(iii) I1(θ), the Fisher Information in one observation.
(iv) The CRLB for estimates of θbased on the nobservations.
(b) Show that the MLE based on the nobservations is b
θ=µ−1(x).
(c) What is d
dw µ−1(w)?
(d) Using the ∆-method, approximate E[b
θ] and V ar[b
θ].
(e) How does the variance in (d) compare to the CRLB? What is the asymptotic efficiency
of the MLE?
2. Suppose X1, . . . , Xnare iid with mean µand variance σ2, and suppose that V ar[X2
i]<∞.
(a) Show that
S2
n−→Pσ2,
where S2
nis the sample variance based on nobservations. (You can decide whether to divide
by nor n−1.) Hint: Write
X(xi−x)2=Xx2
i−nx2,
then work on the two terms separately.
(b) Does Sn−→Pσ? Why or why not?
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