4. Consider a random sample {Xi, i = 1, 2, ...n} of size n from a distribution with pdf f(x; 8) = 0(1+ x)-(1+0) if x > 0, and 0 otherwise. %3D (a) Show that the maximum likelihood estimator (mle) Ô of e is a function of E- In(1+ X;) (b) Let Y = In (1+X), find pdf of Y and mgf of Y. (c) Using result (b) to show that Z = E In(1 + X;) has a Gamma distribution. Specify the parameters of the gamma distribution (hint: find mgf of Z)
4. Consider a random sample {Xi, i = 1, 2, ...n} of size n from a distribution with pdf f(x; 8) = 0(1+ x)-(1+0) if x > 0, and 0 otherwise. %3D (a) Show that the maximum likelihood estimator (mle) Ô of e is a function of E- In(1+ X;) (b) Let Y = In (1+X), find pdf of Y and mgf of Y. (c) Using result (b) to show that Z = E In(1 + X;) has a Gamma distribution. Specify the parameters of the gamma distribution (hint: find mgf of Z)
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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Transcribed Image Text:4. Consider a random sample {X¡, i = 1, 2, ...n} of size n from a distribution with pdf
f(x; 8) = 0(1+ x)-(1+0) if x > 0, and 0 otherwise.
(a) Show that the maximum likelihood estimator (mle) ô of 0 is a function of
E-1 In(1 + X¡)
(b) Let Y = In (1+X), find pdf of Y and mgf of Y.
%3D
(c) Using result (b) to show that Z = E, In(1 + X;) has a Gamma distribution. Specify
the parameters of the gamma distribution (hint: find mgf of Z)
(d) For test Ho: 0=1 vs Hi : 0 > 1, we reject null hypothesis if (X1, ..., Xn) € C, where
C= {(X, .., Xn), E-, In(1 + X;) > k}. Give an expression to obtain k for given
significance level a and what is the expression of power when 0 =1 (no need to
integrate the function).
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