Download Random Sample - Introductory Statistics - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity!
Stat 366 Lab 2 Problems (September 21, 2006) page 1
TA: Yury Petrachenko, CAB 484, [email protected], http://www.ualberta.ca/∼yuryp/
Review Questions, Chapters 8, 9
8.15 Suppose that Y 1 , Y 2 ,... , Yn denote a random sample of size n from a population with an
exponential distribution whose density is given by
f (y) =
(1/θ)e
−y/θ
, y > 0
0 , elsewhere.
If Y (1)
= min(Y 1
, Y
2
,... , Y
n
) denotes the smallest-order statistic, show that
θ = nY (1)
is an
unbiased estimator for θ and find MSE(
θ).
9.7 Suppose that Y 1
, Y
2
,... , Y
n
denote a random sample of size n from an exponential distribution
with density function given by
f (y) =
(1/θ)e
−y/θ
, y > 0
0 , elsewhere.
In Exercise 8.15 we determined that
θ 1
= nY (1)
is an unbiased estimator of θ with MSE(
θ)= θ
2 .
Consider the estimator
θ 2
Y , and find the efficiency of
θ 1
relative to
θ 2
9.61 Let Y 1
, Y
2
,... , Y
n
denote a random sample from the probability density function
f (y) =
(θ + 1)y
θ
, 0 < y < 1; θ > − 1
0 , elsewhere.
Find an estimator for θ by the method of moments.
Stat 366 Lab 2 Problems (September 21, 2006) page 2
9.72 Suppose that Y 1
, Y
2
,... , Y
n
denote a random sample from the Poisson distribution with
mean λ.
(a) Find the maximum-likelihood estimator
λ for λ.
(d) What is the MLE for P (Y = 0) = e
−λ
?
9.75a Suppose that Y 1 , Y 2 ,... , Yn constitute a random sample from a uniform distribution with
probability density function
f (y) =
2 θ + 1
, 0 ≤ y ≤ 2 θ + 1
0 , elsewhere.
Obtain the maximum-likelihood estimator of θ.
9.80 Let Y 1
, Y
2
,... , Y
n
denote a random sample from the probability density function
f (y) =
(θ + 1)y
θ
, 0 < y < 1; θ > − 1
0 , elsewhere.
Find the maximum-likelihood estimator for θ. Compare your answer to the method of mo-
ments estimator found in Exercise 9.61.
