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238 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

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(b) Show that the log likelihood is maximized by solving the V 1a ˆ 2 a 3n1n 224 . Show that if n 1, a ˆ 2 is a better
2
equations estimator than . In what sense is it a better estimator of a?
a ˆ
7-28. Consider the probability density function
n n 1
a x i ln1x i 2 a ln1x i 2

i 1 i 1
≥ ¥ 1

n n f 1x2 xe x , 0 x , 0
a x i 2
i 1
n 1
Find the maximum likelihood estimator for .
x
a i
£ i 1 § 7-29. The Rayleigh distribution has probability density

n function
(c) What complications are involved in solving the two equa- x
2
tions in part (b)? f 1x2 e x 2 , x 0, 0

7-24. Consider the probability distribution in Exercise 7-22.
Find the moment estimator of . 2
(a) It can be shown that E1X 2 2 . Use this information to
7-25. Let X 1 , X 2 , p , X n be uniformly distributed on the in- construct an unbiased estimator for .
terval 0 to a. Show that the moment estimator of a is a ˆ 2X. (b) Find the maximum likelihood estimator of . Compare
Is this an unbiased estimator? Discuss the reasonableness of your answer to part (a).
this estimator. (c) Use the invariance property of the maximum likelihood
7-26. Let X 1 , X 2 , p , X n be uniformly distributed on the estimator to find the maximum likelihood estimator of the
interval 0 to a. Recall that the maximum likelihood estimator median of the Raleigh distribution.
of a is a ˆ max 1X i 2 . 7-30. Consider the probability density function
(a) Argue intuitively why cannot be an unbiased estimator
a ˆ
for a. f 1x2 c 11 x2, 1 x 1
(b) Suppose that E1a ˆ2 na 1n 12 . Is it reasonable that a ˆ
consistently underestimates a? Show that the bias in the
estimator approaches zero as n gets large. (a) Find the value of the constant c.
(c) Propose an unbiased estimator for a. (b) What is the moment estimator for ?
ˆ
(d) Let Y max(X i ). Use the fact that Y y if and only (c) Show that 3X is an unbiased estimator for .
if each X i y to derive the cumulative distribution func- (d) Find the maximum likelihood estimator for .
tion of Y. Then show that the probability density function 7-31. Reconsider the oxide thickness data in Exercise 7-12
of Y is and suppose that it is reasonable to assume that oxide thick-
ness is normally distributed.
n 1 (a) Use the results of Example 7-9 to compute the maximum
ny 2
, 0 y a likelihood estimates of and .
f 1 y2 • a n 2
(b) Graph the likelihood function in the vicinity of and ˆˆ ,

0 , otherwise the maximum likelihood estimates, and comment on its
shape.
Use this result to show that the maximum likelihood esti- 7-32. Continuation of Exercise 7-31. Suppose that for the
mator for a is biased. situation of Exercise 7-12, the sample size was larger (n 40)
7-27. For the continuous distribution of the interval 0 to a, but the maximum likelihood estimates were numerically
we have two unbiased estimators for a: the moment estimator equal to the values obtained in Exercise 7-31. Graph the
a ˆ 2X and a ˆ 31n 12 n4 max1X i 2 , where max(X i ) is likelihood function for n 40, compare it to the one from
1
2
the largest observation in a random sample of size n (see Exercise 7-31 (b), and comment on the effect of the larger
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Exercise 7-26). It can be shown that V1a ˆ 2 a 13n2 and that sample size.
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7-4 SAMPLING DISTRIBUTIONS

Statistical inference is concerned with making decisions about a population based on the
information contained in a random sample from that population. For instance, we may be
interested in the mean fill volume of a can of soft drink. The mean fill volume in the