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Homework 2 (due Feb. 4)
1. Which of the following statistics are unbiased estimators for the pop
ulation mean
μ
? For those which are not unbiased, ﬁnd the bias.
a)
X
1
+
X
3
2
b)
X
1
+
X
3
+
X
n
3
c)
X
1
+
X
3
+3
X
2

X
4
5
d) 3
X
1

2
X
3
e)
X
1

X
2
+
X
n/
2

X
n

1
+
X
n
2. For the estimators in problem 1, ﬁnd their mean square error. Which
appears to be the most eﬃcient one (with the smallest MSE) if the standard
deviation of the population is equal to its mean (
σ
=
μ
)?
3. Consider the Poisson distribution
f
(
x
) =
e

λ
λ
x
x
!
,
x
= 0
,
1
,
2
,...
Find the maximum likelihood estimator or
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Unformatted text preview: Î» , based on a random sample of size n . 4. Find maximum likelihood estimators for the mean and variance of a normal distribution, based on a random sample of size n. 5. Suppose the size n sample comes from a distribution uniform on the interval [ a,b ] with both a and b unknown parameters. Find the MLE for a and b . What can we say about the sign of these two estimatorsâ€™ biases? 1...
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This note was uploaded on 02/07/2011 for the course IE 121 taught by Professor Perevalov during the Spring '08 term at Lehigh University .
 Spring '08
 Perevalov

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