2. Two independent random samples X1, ..., X, and Y1,.,Ym follow Poisson distribu- tions: X; ~ Poi(X) and Y; ~ Poi(y), where A > 0 and y 2 0 are two unknown parameters. Suppose we wish to test the hypothesis Họ : A = Xo, y = %0 v.s. H1 : not Họ, where Ao and 79 are given values. Let l(A, y) denote the log-likelihood function from these two samples. (a) Show that the log-likelihood function is given by m 1(A, 7) = –n) – myd + log A ( ) X; + > _Y; ) + log y>Y;. i=1 j=1 j=1 (b) Hence show that the maximum likelihood estimators of A and y are E, Y;/m -j=1 E, X;/n° (c) Derive the likelihood ratio test statistic for testing this hypothesis. (d) Specify the rejection region given by the likelihood ratio test.

2. Two independent random samples X1, ..., X, and Y1,.,Ym follow Poisson distribu- tions: X; ~ Poi(X) and Y; ~ Poi(y), where A > 0 and y 2 0 are two unknown parameters. Suppose we wish to test the hypothesis Họ : A = Xo, y = %0 v.s. H1 : not Họ, where Ao and 79 are given values. Let l(A, y) denote the log-likelihood function from these two samples. (a) Show that the log-likelihood function is given by m 1(A, 7) = –n) – myd + log A ( ) X; + > _Y; ) + log y>Y;. i=1 j=1 j=1 (b) Hence show that the maximum likelihood estimators of A and y are E, Y;/m -j=1 E, X;/n° (c) Derive the likelihood ratio test statistic for testing this hypothesis. (d) Specify the rejection region given by the likelihood ratio test.

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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