Let X₁,,X, be a random sample from the exponential distribution with parameter 8, i.e., f(x | 0) = 0e0x, x>0,0> 0. 1. (a) Derive 6, the MLE of 0. (b) Construct the approximate equal-tailed 95% confidence intervals for using the expected information and the observed information. (c) Using the Neyman-Pearson lemma, specify which of the following is the form of the critical region for the most powerful test of size a of Ho: 0 = 0 versus H₁ : 0 = 0₁, where 8 < 0₁: A. > K; B.
Let X₁,,X, be a random sample from the exponential distribution with parameter 8, i.e., f(x | 0) = 0e0x, x>0,0> 0. 1. (a) Derive 6, the MLE of 0. (b) Construct the approximate equal-tailed 95% confidence intervals for using the expected information and the observed information. (c) Using the Neyman-Pearson lemma, specify which of the following is the form of the critical region for the most powerful test of size a of Ho: 0 = 0 versus H₁ : 0 = 0₁, where 8 < 0₁: A. > K; B.
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 be a random sample from the exponential distribution with parameter 8, i.e.,
f(x | 0) = 8e-9x, x>0,0>0.
1. (a) Derive ê, the MLE of 0.
(b) Construct the approximate equal-tailed 95% confidence intervals for using the expected information
and the observed information.
(c) Using the Neyman-Pearson lemma, specify which of the following is the form of the critical region for
the most powerful test of size a of
Ho: 0 = 0 versus H₁ : 0 = 0₁,
where 8 < 0₁:
A.
x > K;
B. X <K
for some constant K.
(d) Determine the value of K in part 1(c) such that the size of the test is a, in terms of the quantile of
Chi-squared distribution.
2. It is known that
1
~ N(,) approximately for large n,
where I (0) is the expected information. According to the results you obtained in part 1(a)-(b),
(a) write an R function
D
Sim_exponential <- function(theta, size, n_sim)
in order to
i) fully investigate the accuracy of the approximation (1);
ii) estimate the coverage probability of the approximate 95% confidence interval for using the
observed information,
where theta represents the true value of 0, size stands for the size of each simulated random sample
and n_sim is the number of simulated random samples.
(b) run the command
Sim_exponential (theta-x, size=y, n_sim=z)
and comment on your results in detail.
[Please refer to the cover page for your values of x, y and z.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0dc7fdfc-2d05-4936-8b9d-5ee1ad16be99%2F9faa4e5f-4d09-4e98-b65b-f02f0c9f2e64%2F9gac8hd_processed.png&w=3840&q=75)
Transcribed Image Text:Let X₁,..., Xn be a random sample from the exponential distribution with parameter 8, i.e.,
f(x | 0) = 8e-9x, x>0,0>0.
1. (a) Derive ê, the MLE of 0.
(b) Construct the approximate equal-tailed 95% confidence intervals for using the expected information
and the observed information.
(c) Using the Neyman-Pearson lemma, specify which of the following is the form of the critical region for
the most powerful test of size a of
Ho: 0 = 0 versus H₁ : 0 = 0₁,
where 8 < 0₁:
A.
x > K;
B. X <K
for some constant K.
(d) Determine the value of K in part 1(c) such that the size of the test is a, in terms of the quantile of
Chi-squared distribution.
2. It is known that
1
~ N(,) approximately for large n,
where I (0) is the expected information. According to the results you obtained in part 1(a)-(b),
(a) write an R function
D
Sim_exponential <- function(theta, size, n_sim)
in order to
i) fully investigate the accuracy of the approximation (1);
ii) estimate the coverage probability of the approximate 95% confidence interval for using the
observed information,
where theta represents the true value of 0, size stands for the size of each simulated random sample
and n_sim is the number of simulated random samples.
(b) run the command
Sim_exponential (theta-x, size=y, n_sim=z)
and comment on your results in detail.
[Please refer to the cover page for your values of x, y and z.]
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