1. Find the MLE of based on a random sample X₁, X₂, ..., Xn from each of the following p.d.f.'s. (a) where 0 < x < 1, 0 < 0, and 0 otherwise. (b) f(x|0) = 0x0-1 f(x|0) = (0+1)x-0-2 where 1 < x, 0 <0, and 0 otherwise. (c) where 0 < x, 0 <0, and 0 otherwise. (d) f(z|0) = 02re-e f(x|0) = 0(1-0)²-1 for x = 1,2,..., 0 < 0 < 1, and 0 otherwise 2. Find the asymptotic variance of the MLE in each part of question 1. 3. Consider two independent random samples X₁, X2, ..., X₂ ~ N(μ, o?) and Y₁, Y2, ..., Ym N(μ, σ2). (a) Using the data from the two random samples find the m.l.e. of μ, of, and 02. (b) Find the asymptotic variance of μ.
1. Find the MLE of based on a random sample X₁, X₂, ..., Xn from each of the following p.d.f.'s. (a) where 0 < x < 1, 0 < 0, and 0 otherwise. (b) f(x|0) = 0x0-1 f(x|0) = (0+1)x-0-2 where 1 < x, 0 <0, and 0 otherwise. (c) where 0 < x, 0 <0, and 0 otherwise. (d) f(z|0) = 02re-e f(x|0) = 0(1-0)²-1 for x = 1,2,..., 0 < 0 < 1, and 0 otherwise 2. Find the asymptotic variance of the MLE in each part of question 1. 3. Consider two independent random samples X₁, X2, ..., X₂ ~ N(μ, o?) and Y₁, Y2, ..., Ym N(μ, σ2). (a) Using the data from the two random samples find the m.l.e. of μ, of, and 02. (b) Find the asymptotic variance of μ.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Please answer the Q2 and Q3.
Transcribed Image Text:Maximum Likelihood Estimation
1. Find the MLE of based on a random sample X₁, X₂, ..., X₁ from each of the following
p.d.f.'s.
(a)
where 0 < x < 1, 0 < 0, and 0 otherwise.
(b)
f(x0) = 0x0-1
f(x|0) = (0+1)x-⁰-2
where 1 < x, 0 <0, and 0 otherwise.
(c)
where 0 < x, 0 <0, and 0 otherwise.
(d)
f(c\0) = 02re-tr
xe
f(x0) = 0(1-0)-1
for x = 1, 2, ..., 0 < 0 < 1, and 0 otherwise
2. Find the asymptotic variance of the MLE in each part of question 1.
Xn
3. Consider two independent random samples X₁, X2, ..., X₁
N(μ, σ2).
~ N(µ, o²) and Y₁, Y2, ..., Ym~
(a) Using the data from the two random samples find the m.l.e. of µ, o², and o².
(b) Find the asymptotic variance of μ.
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