Suppose that a sequence of mutually independent and identically distributed discrete randomvariables X₁, X₂, X3,..., Xn has the following probability density functionf(x; 0) =exe-0x!0,7for x = 0,1,2,...elsewherea) Show that for any & > 0 and S₁ = 1X₁, lim P(|S₁ - 0 ≥ ) = 0.72-00b) Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 0.

To obtain the maximum likelihood estimator of θ we have to determine the likelihood and…

Answered: Show that for any >0 and S₁ = 1X₁, lim…

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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Let X₁, X₂, X3,...,Xn be a random sample of n from population X distributed with the fol probability density function: 1 ze=20, f(x;0)=√√2n0 0, if -∞0
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Assume the data are generated from the GN-SLR setting, Y = B1Xin + ti i = 1, (a) Show that the likelihood function is where €; ~ N (0,0²). We only have the observed versions of (Y₁, X₁1), which are (x₁1,₁),..., (Xn1, Yn). We want to estimate B₁ and o² with these data using the Maximum Likelihood Method. n i=1 1 L(Bo, B₁,0²) = (2ño²)−¹/² exp{-22 (Yi - B₁x₁)²} (b) Compute the negative log-likelihood function. (c) Show that the MLE estimator for ₁ is ., n 9.9 1(Bo, B₁,0²) = -log L(Bo, B₁, 0²) ÂMLE _ Σi=1 X₁Yi = An 2 (d) Show that the MLE estimator for o2 is i=1 - (62) MLE _ Σi=1(Yi – ÎMLEƑ¡1)² n
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Show that for any >0 and S₁ = 1X₁, lim P(|S₁ - 0 ≥ ) = 0. ε n 71-00 Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 0.