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
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Please answer the Q2 and Q3.
![## Maximum Likelihood Estimation
1. **Find the MLE of \( \theta \) based on a random sample \( X_1, X_2, \ldots, X_n \) from each of the following p.d.f.'s:**
(a)
\[
f(x|\theta) = \theta x^{\theta - 1}
\]
where \( 0 < x < 1, 0 < \theta \), and 0 otherwise.
(b)
\[
f(x|\theta) = (\theta + 1) x^{-\theta - 2}
\]
where \( 1 < x, 0 < \theta \), and 0 otherwise.
(c)
\[
f(x|\theta) = \theta^2 x e^{-\theta x}
\]
where \( 0 < x, 0 < \theta \), and 0 otherwise.
(d)
\[
f(x|\theta) = \theta (1 - \theta)^{x - 1}
\]
for \( x = 1, 2, \ldots, 0 < \theta < 1 \), and 0 otherwise.
2. **Find the asymptotic variance of the MLE in each part of question 1.**
3. **Consider two independent random samples \( X_1, X_2, \ldots, X_n \sim N(\mu, \sigma_1^2) \) and \( Y_1, Y_2, \ldots, Y_m \sim N(\mu, \sigma_2^2) \).**
(a) Using the data from the two random samples find the m.l.e. of \( \mu, \sigma_1^2, \) and \( \sigma_2^2 \).
(b) Find the asymptotic variance of \( \mu \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7205a63d-a1ca-48be-852d-dfbf4d68dfc1%2Fd4d8875a-9595-43ad-8c95-db2e559f337b%2Fq4icgf_processed.png&w=3840&q=75)
Transcribed Image Text:## Maximum Likelihood Estimation
1. **Find the MLE of \( \theta \) based on a random sample \( X_1, X_2, \ldots, X_n \) from each of the following p.d.f.'s:**
(a)
\[
f(x|\theta) = \theta x^{\theta - 1}
\]
where \( 0 < x < 1, 0 < \theta \), and 0 otherwise.
(b)
\[
f(x|\theta) = (\theta + 1) x^{-\theta - 2}
\]
where \( 1 < x, 0 < \theta \), and 0 otherwise.
(c)
\[
f(x|\theta) = \theta^2 x e^{-\theta x}
\]
where \( 0 < x, 0 < \theta \), and 0 otherwise.
(d)
\[
f(x|\theta) = \theta (1 - \theta)^{x - 1}
\]
for \( x = 1, 2, \ldots, 0 < \theta < 1 \), and 0 otherwise.
2. **Find the asymptotic variance of the MLE in each part of question 1.**
3. **Consider two independent random samples \( X_1, X_2, \ldots, X_n \sim N(\mu, \sigma_1^2) \) and \( Y_1, Y_2, \ldots, Y_m \sim N(\mu, \sigma_2^2) \).**
(a) Using the data from the two random samples find the m.l.e. of \( \mu, \sigma_1^2, \) and \( \sigma_2^2 \).
(b) Find the asymptotic variance of \( \mu \).
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