(a) Consider X₁,..., X₂ to be a random sample from the geometric distribution, with probability mass function: P(X = x) = p(1 − p)ª, with x = 0, 1,2,3,..., and p € (0, 1]. (i) Using the MGF (M(t) = 1-(1-p)et (ii) Find the Maximum Likelihood Estimator (MLE) for p. derive E[X] and Var[X].
(a) Consider X₁,..., X₂ to be a random sample from the geometric distribution, with probability mass function: P(X = x) = p(1 − p)ª, with x = 0, 1,2,3,..., and p € (0, 1]. (i) Using the MGF (M(t) = 1-(1-p)et (ii) Find the Maximum Likelihood Estimator (MLE) for p. derive E[X] and Var[X].
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![Hint: Use substitutions y = 1 - x-(μ-n)
and y' =
x-(μ-n)
21
21
(e) Describe the criteria which can be used to choose between these two estimates.
B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with
probability mass function: P(X = x) = p(1 − p), with x = 0, 1, 2, 3,..., and
p € (0, 1].
(i) Using the MGF M(t) = 1-1-p)et derive E[X] and Var[X].
(ii) Find the Maximum Likelihood Estimator (MLE) for p.
(b) Suppose X₁,..., Xn is a random sample from a Beta(01, 1) population, and Y₁,..., Ym
is an independent random sample from a Beta(02, 1) population. We want to find
the approximate Likelihood Ratio Test for Ho: 01 = 0₂ = 0o, versus H₁ : 0₁ 0₂.
To this aim:
(i) Under the alternative hypothesis H₁ 0₁ 02, show that the MLE for ₁ and
0₂ are:
0₁
Page 5 of 6
n
Σ log(x;)
Recall, that the PDF of Beta(a, b) is fy (y):
m
00=
En log(yi)
[(a+b)
r(a) (b)
I(a) = (a − 1)! for all positive integers a and I(1) = 1
-
(ii) Under the null hypothesis Ho: 0₁ = 0₂ = 0o, show that the MLE for 0 is:
-¹ (1-y)-1 and
n + m
Σ log(x;) + Σ log(yi)
Turn the page over](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3011c556-643e-4a01-b0e0-55d8cf24eddf%2F660064c9-1bfa-4150-bfe4-bece2d118275%2F47xlmw5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Hint: Use substitutions y = 1 - x-(μ-n)
and y' =
x-(μ-n)
21
21
(e) Describe the criteria which can be used to choose between these two estimates.
B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with
probability mass function: P(X = x) = p(1 − p), with x = 0, 1, 2, 3,..., and
p € (0, 1].
(i) Using the MGF M(t) = 1-1-p)et derive E[X] and Var[X].
(ii) Find the Maximum Likelihood Estimator (MLE) for p.
(b) Suppose X₁,..., Xn is a random sample from a Beta(01, 1) population, and Y₁,..., Ym
is an independent random sample from a Beta(02, 1) population. We want to find
the approximate Likelihood Ratio Test for Ho: 01 = 0₂ = 0o, versus H₁ : 0₁ 0₂.
To this aim:
(i) Under the alternative hypothesis H₁ 0₁ 02, show that the MLE for ₁ and
0₂ are:
0₁
Page 5 of 6
n
Σ log(x;)
Recall, that the PDF of Beta(a, b) is fy (y):
m
00=
En log(yi)
[(a+b)
r(a) (b)
I(a) = (a − 1)! for all positive integers a and I(1) = 1
-
(ii) Under the null hypothesis Ho: 0₁ = 0₂ = 0o, show that the MLE for 0 is:
-¹ (1-y)-1 and
n + m
Σ log(x;) + Σ log(yi)
Turn the page over
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 28 images
Similar questions
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman