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(ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first three distributions.

(ii) Use the Fisher-Neyman factorization theorem to show that is a sufficient statistic for each of the first 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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only part ii (i asked part i in a different question)

Exercise 1 In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p=I is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators • Poisson(X) • N(μ1,0²) â = y₁ = min(x₁, · ·, £n), b = Yn = 1 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) (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 it is a sufficient statistic for each of the first three distributions.
Transcribed Image Text:Exercise 1 In lecture (Mon 1/30), we showed that for Bernoulli (p) distribution, the MLE estimator p=I is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators • Poisson(X) • N(μ1,0²) â = y₁ = min(x₁, · ·, £n), b = Yn = 1 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) (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 it is a sufficient statistic for each of the first three distributions.
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We need to find the likelihood function of the given distributions.

Exp(λ), Geometric(p), Poisson(λ), and N(μ,σ2).

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