Let X1, Xn denote a random sample from a N(μ, o²) distribution, where the mean μ and the variance σ² are both unknown. (i) Without differentiation of the log likelihood function, show that the maximum likelihood (ML) estimator of σ is given by where 82 Σ(X; -X)² n ΣΕ Χ n (ii) Consider now estimators of o² of the form where c is a constant. n T₁ = c(X; - X)², j=1 Find the estimator of σ2 having the least mean squared error in the class {Te c>0}. [NOTE: You may assume the distribution of 82, but you need to derive the expressions where necessary for the moments of 8.] (iii) Give the parametric function g(0) that defines the quantile of order a of the normal distribution in terms of 0 and the corresponding quantile % of the standard normal distribution.
Let X1, Xn denote a random sample from a N(μ, o²) distribution, where the mean μ and the variance σ² are both unknown. (i) Without differentiation of the log likelihood function, show that the maximum likelihood (ML) estimator of σ is given by where 82 Σ(X; -X)² n ΣΕ Χ n (ii) Consider now estimators of o² of the form where c is a constant. n T₁ = c(X; - X)², j=1 Find the estimator of σ2 having the least mean squared error in the class {Te c>0}. [NOTE: You may assume the distribution of 82, but you need to derive the expressions where necessary for the moments of 8.] (iii) Give the parametric function g(0) that defines the quantile of order a of the normal distribution in terms of 0 and the corresponding quantile % of the standard normal distribution.
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,
Xn denote a random sample from a N(μ, o²) distribution,
where the mean μ and the variance σ² are both unknown.
(i) Without differentiation of the log likelihood function, show that the
maximum likelihood (ML) estimator of σ is given by
where
82
Σ(X; -X)²
n
ΣΕ Χ
n
(ii) Consider now estimators of o² of the form
where c is a constant.
n
T₁ = c(X; - X)²,
j=1
Find the estimator of σ2 having the least mean squared error in the
class {Te c>0}.
[NOTE: You may assume the distribution of 82, but you need to derive
the expressions where necessary for the moments of 8.]
(iii) Give the parametric function g(0) that defines the quantile of order a
of the normal distribution in terms of 0 and the corresponding quantile
% of the standard normal distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66318fc8-0ad6-4ce3-b166-594a4cfc43a5%2F5b525acb-384b-4f50-b921-60c17f45b046%2F2db4waw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let X1,
Xn denote a random sample from a N(μ, o²) distribution,
where the mean μ and the variance σ² are both unknown.
(i) Without differentiation of the log likelihood function, show that the
maximum likelihood (ML) estimator of σ is given by
where
82
Σ(X; -X)²
n
ΣΕ Χ
n
(ii) Consider now estimators of o² of the form
where c is a constant.
n
T₁ = c(X; - X)²,
j=1
Find the estimator of σ2 having the least mean squared error in the
class {Te c>0}.
[NOTE: You may assume the distribution of 82, but you need to derive
the expressions where necessary for the moments of 8.]
(iii) Give the parametric function g(0) that defines the quantile of order a
of the normal distribution in terms of 0 and the corresponding quantile
% of the standard normal distribution.
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