1. (20) Suppose X1,..., X, are i.i.d. with common probability density function: fo(x) = *€ (-1,1), Ger 2sinh(0)' where sinh(x) = is the hyperbolic sine function and > 0 is the distribution parameter. In addition, let Y₁ = I{X > 0}, where 1{} is the indicator function. In other words, Y; = 1 if Xi > 0 and Yi 0 otherwise. (a) [5] Give an equation for the maximum likelihood estimator MLE, X based on X1,..., Xn- (Give the equation only, no need to solve ÔMLE. X). (b) [5] Give an equation for the method of moments estimator MOM, X based on X1,..., Xn. (Give the equation only, no need to solve мOM, X). (c) [5] Find the maximum likelihood estimator OMLE, Y based on Y₁,..., Y. (Hint: maxi- mize the likelihood function of Y₁...., Yn). (d) [5] Find the method of moments estimator MOM, Y based on Y₁,..., Y.
1. (20) Suppose X1,..., X, are i.i.d. with common probability density function: fo(x) = *€ (-1,1), Ger 2sinh(0)' where sinh(x) = is the hyperbolic sine function and > 0 is the distribution parameter. In addition, let Y₁ = I{X > 0}, where 1{} is the indicator function. In other words, Y; = 1 if Xi > 0 and Yi 0 otherwise. (a) [5] Give an equation for the maximum likelihood estimator MLE, X based on X1,..., Xn- (Give the equation only, no need to solve ÔMLE. X). (b) [5] Give an equation for the method of moments estimator MOM, X based on X1,..., Xn. (Give the equation only, no need to solve мOM, X). (c) [5] Find the maximum likelihood estimator OMLE, Y based on Y₁,..., Y. (Hint: maxi- mize the likelihood function of Y₁...., Yn). (d) [5] Find the method of moments estimator MOM, Y based on Y₁,..., Y.
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Related questions
Question
![1. (20) Suppose X1,..., X, are i.i.d. with common
probability density function:
fo(x) =
*€ (-1,1),
Ger
2sinh(0)'
where sinh(x) = is the hyperbolic sine function and > 0 is the distribution parameter.
In addition, let Y₁ = I{X > 0}, where 1{} is the indicator function. In other words, Y; = 1
if Xi > 0 and Yi
0 otherwise.
(a) [5] Give an equation for the maximum likelihood estimator MLE, X based on X1,..., Xn-
(Give the equation only, no need to solve ÔMLE. X).
(b) [5] Give an equation for the method of moments estimator MOM, X based on X1,..., Xn.
(Give the equation only, no need to solve мOM, X).
(c) [5] Find the maximum likelihood estimator OMLE, Y based on Y₁,..., Y. (Hint: maxi-
mize the likelihood function of Y₁...., Yn).
(d) [5] Find the method of moments estimator MOM, Y based on Y₁,..., Y.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F350f1267-e26c-49e6-8aa4-88cf8bdf6f07%2F117425b3-4da1-4e76-a66d-15ac33c7fbce%2Fdfwn7ow_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. (20) Suppose X1,..., X, are i.i.d. with common
probability density function:
fo(x) =
*€ (-1,1),
Ger
2sinh(0)'
where sinh(x) = is the hyperbolic sine function and > 0 is the distribution parameter.
In addition, let Y₁ = I{X > 0}, where 1{} is the indicator function. In other words, Y; = 1
if Xi > 0 and Yi
0 otherwise.
(a) [5] Give an equation for the maximum likelihood estimator MLE, X based on X1,..., Xn-
(Give the equation only, no need to solve ÔMLE. X).
(b) [5] Give an equation for the method of moments estimator MOM, X based on X1,..., Xn.
(Give the equation only, no need to solve мOM, X).
(c) [5] Find the maximum likelihood estimator OMLE, Y based on Y₁,..., Y. (Hint: maxi-
mize the likelihood function of Y₁...., Yn).
(d) [5] Find the method of moments estimator MOM, Y based on Y₁,..., Y.
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