7.4.7 SS Reconsider the oxide thickness data in Exercise 7.3.7 and suppose that it is reasonable to assume that oxide thickness is normally distributed. a. Compute the maximum likelihood estimates of μ and o². b. Graph the likelihood function in the vicinity of Ĥ and 82, the maximum likelihood estimates, and comment on its shape. c. Suppose that the sample size was larger (n = 40) but the maximum likelihood estimates were numerically equal to the values obtained in part (a). Graph the likelihood function for n = 40, compare it to the one from part (b), and comment on the effect of the larger sample size. n 7.4.6 Let X₁, X2, ..., X, be uniformly distributed on the inter- val 0 to a. Recall that the maximum likelihood estimator of a is â = max(X;). a. Argue intuitively why a cannot be an unbiased estimator for a. b. Suppose that E(â) = na/(n+1). Is it reasonable that a consistently underestimates a? Show that the bias in the esti- mator approaches zero as n gets large. c. Propose an unbiased estimator for a. d. Let Y = max(X;). Use the fact that Y ≤ y if and only if each X; 1, â is a better estimator than â. In what sense is it a better estimator of a?
7.4.7 SS Reconsider the oxide thickness data in Exercise 7.3.7 and suppose that it is reasonable to assume that oxide thickness is normally distributed. a. Compute the maximum likelihood estimates of μ and o². b. Graph the likelihood function in the vicinity of Ĥ and 82, the maximum likelihood estimates, and comment on its shape. c. Suppose that the sample size was larger (n = 40) but the maximum likelihood estimates were numerically equal to the values obtained in part (a). Graph the likelihood function for n = 40, compare it to the one from part (b), and comment on the effect of the larger sample size. n 7.4.6 Let X₁, X2, ..., X, be uniformly distributed on the inter- val 0 to a. Recall that the maximum likelihood estimator of a is â = max(X;). a. Argue intuitively why a cannot be an unbiased estimator for a. b. Suppose that E(â) = na/(n+1). Is it reasonable that a consistently underestimates a? Show that the bias in the esti- mator approaches zero as n gets large. c. Propose an unbiased estimator for a. d. Let Y = max(X;). Use the fact that Y ≤ y if and only if each X; 1, â is a better estimator than â. In what sense is it a better estimator of a?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
Related questions
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Answer questions 7.4.6 and 7.4.7 respectively
Transcribed Image Text:7.4.7 SS Reconsider the oxide thickness data in Exercise 7.3.7
and suppose that it is reasonable to assume that oxide thickness
is normally distributed.
a. Compute the maximum likelihood estimates of μ and o².
b. Graph the likelihood function in the vicinity of Ĥ and
82, the maximum likelihood estimates, and comment on its
shape.
c. Suppose that the sample size was larger (n = 40) but the
maximum likelihood estimates were numerically equal to the
values obtained in part (a). Graph the likelihood function for
n = 40, compare it to the one from part (b), and comment on
the effect of the larger sample size.
![n
7.4.6 Let X₁, X2, ..., X, be uniformly distributed on the inter-
val 0 to a. Recall that the maximum likelihood estimator of a is
â = max(X;).
a. Argue intuitively why a cannot be an unbiased estimator
for a.
b. Suppose that E(â) = na/(n+1). Is it reasonable that a
consistently underestimates a? Show that the bias in the esti-
mator approaches zero as n gets large.
c. Propose an unbiased estimator for a.
d. Let Y = max(X;). Use the fact that Y ≤ y if and only if
each X; <y to derive the cumulative distribution function of
Y. Then show that the probability density function of Y is
ny
n-1
f(y) =
0 ≤ y ≤ a
an
0,
otherwise
Use this result to show that the maximum likelihood estima-
tor for a is biased.
e. We have two unbiased estimators for a: the moment esti-
mator â₁ = 2X and â₂ = [(n + 1)/n] max(X;), where max(X;)
is the largest observation in a random sample of size n.
It can be shown that V(â₁) = a²/(3n) and that V(a2) =
a²/[n(n+2)]. Show that if n > 1, â is a better estimator than
â. In what sense is it a better estimator of a?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63e1cfc5-d743-4910-a579-2d48ed6c9d1a%2F58067391-64ee-404e-9c8a-bb306620caff%2Fof6lurl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:n
7.4.6 Let X₁, X2, ..., X, be uniformly distributed on the inter-
val 0 to a. Recall that the maximum likelihood estimator of a is
â = max(X;).
a. Argue intuitively why a cannot be an unbiased estimator
for a.
b. Suppose that E(â) = na/(n+1). Is it reasonable that a
consistently underestimates a? Show that the bias in the esti-
mator approaches zero as n gets large.
c. Propose an unbiased estimator for a.
d. Let Y = max(X;). Use the fact that Y ≤ y if and only if
each X; <y to derive the cumulative distribution function of
Y. Then show that the probability density function of Y is
ny
n-1
f(y) =
0 ≤ y ≤ a
an
0,
otherwise
Use this result to show that the maximum likelihood estima-
tor for a is biased.
e. We have two unbiased estimators for a: the moment esti-
mator â₁ = 2X and â₂ = [(n + 1)/n] max(X;), where max(X;)
is the largest observation in a random sample of size n.
It can be shown that V(â₁) = a²/(3n) and that V(a2) =
a²/[n(n+2)]. Show that if n > 1, â is a better estimator than
â. In what sense is it a better estimator of a?
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