Sufficient and Consistent estimators. Method of Moments and Method of Maximal Likelihood

Homework 7. Sufficient and Consistent estimators. Methodof Moments and Method of Maximal Likelihood
Due Thursday, 10/27, by 11 am
(1) To show that an estimator can be consistent without being unbiased or
even asymptotically unbiased, consider the following estimation procedure:
To estimate the mean of a population with the finite variance σ 2 , we first
take a random sample of size n. Then we randomly draw one of n slips of
paper numbered from 1 through n, and if the number we draw is 2, 3, . . . ,
or n, we use as our estimator the mean of the random sample; otherwise,
we use the estimate n2 . Show that this estimation procedure is consistent
and neither unbiased nor asymptotically unbiased.
(2) (a) (X1 , X2 , . . . Xn ) constitute a random sample of size n from a geometric
distribution with parameter p. Show that Y = X1 + X2 + · · · + Xn is
a sufficient statistic of p.
(b) (X1 , X2 , . . . Xn ) constitute a random sample of size n from a Poisson
distribution with parameter λ. Show that Y = X1 + X2 + · · · + Xn is
a sufficient statistic of λ.
(3) A random sample of size 10 from a beta population with parameters α, β
has sample mean 1/3 and second sample moment 1/7. Use the method of
moments to find estimates for α, β. You can use without proof formulas
α
, V ar[X] =
for mean and variance of the beta distribution: E[X] = α+β
αβ
(α+β)2 (α+β+1)
(4) Given a random sample of size n from a Poisson population with parameter
λ, use the method of moments to obtain an estimator for the parameter λ.
Then use the method of maximal likelihood to also obtain an estimator of
λ.
(5) Revisit problem from the first lecture. Alice has found typos 1, 2, . . . , 20
out of typos 1, 2, . . . , n in a draft. Bob has found 15 typos. Assume that
he is equally likely to find any 15 typos out of n.
(a) Find the probability p(n) that Bob has found exactly 10 out typos
found by Alice.
(b) Find values of n that maximize p(n). Hint: Consider p(n+1)
p(n) .
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