Homework Help is Here – Start Your Trial Now!
Math
Statistics
for Bernoulli (p) distribution, the MLE estimator is sufficient for the parameter p; p=i

for Bernoulli (p) distribution, the MLE estimator is sufficient for the parameter p; p=i

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
See similar textbooks
icon
Related questions
Question
Exercise 1 In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p=ī is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators • N(μ,0²) ô =! :=min(T ,xn), b = Yn := max(x₁,...,xn) are jointly sufficient for the parameters a, b . In this exercise, you will deduce similar results for the following four distributions: Exp(X) • Geometric (p) Poisson(X) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/2 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that I, v are a pair of jointly sufficient statistics for N(μ,0²). (v) Show that I, s² are also a pair of jointly sufficient statistics for N(,0²). (Hint: relate s² to v.)
Transcribed Image Text:Exercise 1 In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p=ī is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators • N(μ,0²) ô =! :=min(T ,xn), b = Yn := max(x₁,...,xn) are jointly sufficient for the parameters a, b . In this exercise, you will deduce similar results for the following four distributions: Exp(X) • Geometric (p) Poisson(X) (i) For each of these four distributions, write down their likelihood functions. (Hint: the log-likelihood functions for these distributions were computed in previous lecture and homework.) (ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions. (iii) Show that 1/2 is also a sufficient statistic for each of these three distributions. (iv) Use the Fisher-Neyman factorization theorem to show that I, v are a pair of jointly sufficient statistics for N(μ,0²). (v) Show that I, s² are also a pair of jointly sufficient statistics for N(,0²). (Hint: relate s² to v.)
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
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 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…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
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
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS