QUESTION 2Let X₁, X2, X3,..., Xn denote a random sample of size n from the population distributed with thefollowing probability density function:a)b)c)d)e)f)g)f(x; 0) = {0, +((0+1)x, if 0

It is given that the probability density function of X is f(x; θ) = { (θ + 1)•xθ, If 0 < x <…

Answered: a) b) c) f(x; 0) = 0, What is the…

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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B2. (a) Consider X₁,..., Xn to be a random sample from the geometric distribution, with probability mass function: P(X= x) = p(1-p), with x = : 0, 1, 2, 3,..., and p € (0, 1]. (i) Using the MGF (M(t) == (ii) Find the Maximum Likelihood Estimator (MLE) for p. -(1-p)et (b) Suppose X₁,..., X₁ is a random sample from a Beta(01, 1) population, and Y₁,..., Ym is an independent random sample from a Beta(02, 1) population. We want to find the approximate Likelihood Ratio Test for Ho: 01 02 00, versus H₁: 01 02. To this aim: = = (i) Under the alternative hypothesis H₁0₁02, show that the MLE for ₁ and 02 are: 0₁ 02 derive E[X] and Var[X]. n Σlog(x)' Recall, that the PDF of Beta(a, b) is fy (y) 00 = - m Σ log(yi) = ['(a) = (a − 1)! for all positive integers a and I (1) = 1) [(a+b) F(a) (b)-1(1-y)b-1 and (ii) Under the null hypothesis Ho: 0₁ 02 = 0o, show that the MLE for 00 is: 01 = n + m E log(xi) + log(yi)
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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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a) b) c) f(x; 0) = 0, What is the parameter space of 0? Find the maximum likelihood estimator (MLE) of 0. What is the MLE of g(0) ==? 1+8 elsewhere I