Homework 1 – STAT 543
On campus: Due Friday, January 19 by 5:00 pm (TA’s office);
you also may turn in the assignment in class on the same Friday
Distance students: Due Wednesday, January 24 by 12:00 pm (TA’s email)
1. For X1 , . . . , Xn , show that µ02 − (µ01 )2 =
This result states that
moments.
1
n
Pn
i=1 (Xi
1
n
Pn
i=1 (Xi
− X̄n )2 , using the sample moments and sample mean.
− X̄n )2 is an estimator of Var(X1 ) = E(X12 ) − [E(X1 )]2 based on sample
2. Find the method of moment estimators (MMEs) of the unknown parameters based on a random sample
X1 , X2 , . . . , Xn of size n from the following distributions:
(a) Negative Binomial (3, p), unknown p
(b) Pareto (α, β), unknown α and β
See “Table of Common Distributions” in Casella & Berger (pages 623-623) for the definitions/properties of
the above distributions.
3. Problem 7.6(b)-(c), Casella & Berger
(Skip part (a).)
4. Problem 7.7, Casella & Berger
5. Problem 7.11, Casella & Berger (Assume the support of f (x|θ) is 0 < x < 1.)
P
Hint: Show − log Xi is exponential and Y = ni=1 − log Xi is gamma for the second part in (a).
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