Page 289 - Applied statistics and probability for engineers

Section 7-4/Methods of Point Estimation 267

if the sample results are accepted as being correct, the prior information must be incorrect. The
maximum likelihood estimate would then be the better estimate to use.
If the sample results are very different from the prior information, the Bayes estimator
will always tend to produce an estimate that is between the maximum likelihood estimate and
the prior assumptions. This was illustrated in Example 7-16. If there is more inconsistency
between the prior information and the sample, there will be more difference between the two
estimates.

Exercises FOR SECTION 7-4

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion

=
7-44. Let X be a geometric random variable with param- (a) It can be shown that E X( 2 ) 2θ. Use this information to
eter p. Find the maximum likelihood estimator of p based on a construct an unbiased estimator for θ.
random sample of size n. (b) Find the maximum likelihood estimator of θ. Compare
7-45. Consider the Poisson distribution with parameter λ. your answer to part (a).
Find the maximum likelihood estimator of λ, based on a ran- (c) Use the invariance property of the maximum likelihood
dom sample of size n. estimator to ind the maximum likelihood estimator of the
7-46. Let X be a random variable with the following prob- median of the Raleigh distribution.
ability distribution: 7-52. Let X , X , … , X be uniformly distributed on the interval
1
2
n
θ
( ⎧ ⎪ θ + ) 1 x , 0 ≤ x ≤ 1 0 to a. Recall that the maximum likelihood estimator of a is
(
f x) = ⎨
⎩ ⎪ 0 , otherwise ˆ a = max( ).
X i
Find the maximum likelihood estimator of θ based on a random (a) Argue intuitively why ˆ a cannot be an unbiased estimator
sample of size n. for a.
(
7-47. Consider the shifted exponential distribution (b) Suppose that E a( ) ˆ = na / n +1 ). Is it reasonable that ˆ a con-
sistently underestimates a? Show that the bias in the esti-
(
f x) = λ e −λ( x−θ) , x ≥ θ mator approaches zero as n gets large.
When θ = 0, this density reduces to the usual exponential dis- (c) Propose an unbiased estimator for a.
tribution. When θ > 0, there is positive probability only to the (d) Let Y = max( ). Use the fact that Y ≤ y if and only if
X i
right of θ. each X i ≤ y to derive the cumulative distribution function
(a) Find the maximum likelihood estimator of λ and θ based of Y . Then show that the probability density function of
on a random sample of size n. Y is
(b) Describe a practical situation in which one would suspect ⎧ ny n−1
that the shifted exponential distribution is a plausible model. f y ( ) = ⎨ ⎪ a n , 0 ≤ y ≤ a
7-48. Consider the probability density function ⎪ ⎩ , 0 otherwise
θ
∞
f x) = 1 xe − x / θ , 0 ≤ x < , 0 < < ∞ Use this result to show that the maximum likelihood esti-
(
θ 2 mator for a is biased.
Find the maximum likelihood estimator for θ. (e) We have two unbiased estimators for a: the moment
7-49. Let X X 2 , ,… X n be uniformly distributed on the interval estimator ˆ a 1 = 2 X and ˆ a 2 = [( n 1+ ) / n]max ( X i ), where
1 ,
0 to a. Show that the moment estimator of a is ˆ a = 2 X. Is this an max( ) is the largest observation in a random sample
X i
unbiased estimator? Discuss the reasonableness of this estimator. of size n. It can be shown that V a( ) = a / n) and that
2
ˆ
(3
1
7-50. Consider the probability density function V a ) = a / n n 2 )]. Show that if n >1, ˆ a 2 is a better
+
[
(
2
( ˆ
2
c + θ )
(
f x) = (1 x , − Ð ≤1 x 1 estimator than ˆ a. In what sense is it a better estimator
(a) Find the value of the constant c. of a?
(b) What is the moment estimator for θ? 7-53. Consider the Weibull distribution
ˆ
(c) Show that θ = 3X is an unbiased estimator for θ.
(d) Find the maximum likelihood estimator for θ. ⎧ ⎪ β ⎛ x⎞ β−1 − ⎛ ⎜ ⎟ β
x ⎞
δ

⎜ ⎟
7-51. The Rayleigh distribution has probability density function f x ( ) = ⎨ ⎪ δ δ ⎠ e ⎝ ⎠ , 0 < x
⎝
2
θ
f x ( ) = x e − x / θ2 , x > , 0 0 < < ∞ ⎪
θ ⎩ ⎪ 0, otherwise