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(iii) Show that 1/7 is also a sufficient statistic for each of these three distributions.

(iii) Show that 1/7 is also a sufficient statistic for each of these three distributions.

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
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part iii (i already asked part i and part ii)

In lecture (Mon 1/30), we showed that • for Bernoulli (p) distribution, the MLE estimator p=ī Exercise 1 is sufficient for the parameter p; for Uniform([a, b]), the MLE estimators Geometric (p) Poisson(X) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(x) • N(μ1,0²) â = y₁ = min(x₁, · · , £n), b = Yn :=: max(F₁,..., In) (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 1/7 is also a sufficient statistic for each of these three distributions.
Transcribed Image Text:In lecture (Mon 1/30), we showed that • for Bernoulli (p) distribution, the MLE estimator p=ī Exercise 1 is sufficient for the parameter p; for Uniform([a, b]), the MLE estimators Geometric (p) Poisson(X) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(x) • N(μ1,0²) â = y₁ = min(x₁, · · , £n), b = Yn :=: max(F₁,..., In) (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 1/7 is also a sufficient statistic for each of these three distributions.
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