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(v) Show that I, s² are also a pair of jointly sufficient statistics for N(,0²). (Hint: relate s² to v.)

(v) Show that I, s² are also a pair of jointly sufficient statistics for N(,0²). (Hint: relate s² to v.)

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 v. ( i already asked other parts)

In lecture (Mon 1/30), we showed that • for Bernoulli (p) distribution, the MLE estimator p=x Exercise 1 is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators Geometric (p) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(x) • Poisson(X) • N(14,0²) â= y₁ = min(x₁,...,xn), b = Yn :=: max(T₁,...,n) (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(μ, 02). (Hint: relate s² to v.)
Transcribed Image Text:In lecture (Mon 1/30), we showed that • for Bernoulli (p) distribution, the MLE estimator p=x Exercise 1 is sufficient for the parameter p; • for Uniform([a, b]), the MLE estimators Geometric (p) are jointly sufficient for the parameters a, b. In this exercise, you will deduce similar results for the following four distributions: • Exp(x) • Poisson(X) • N(14,0²) â= y₁ = min(x₁,...,xn), b = Yn :=: max(T₁,...,n) (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(μ, 02). (Hint: relate s² to v.)
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