Problem 5 Let X1, X2, ...,Xn be a random sample from a distribution with pdf T“ for r € (0, µ) |0 otherwise a+1 f(r) = where a and u are positive constants. First, suppose we know that u = 1, but we don't know a and we want to estimate it. (i) Find the maximum likelihood estimator of a. Is the estimator consistent? If yes, show it. Is the estimator biased? If yes, is the bias positive or negative? (ii) Let mi be the sample mean (m1 = X). Find a method of moments estimator of a which uses mı. Is the estimator consistent? If yes, show it. Is the estimator biased? If yes, is the bias positive or negative? Second, suppose we know neither u nor a and we want to estimate both parameters. (iii) State the minimization problem of the maximum likelihood estimators of a and of u. (You don't need to find these estimators.) (iv) Let m2 be the sample second moment (m2 = E X?). Find method of moments estimators of a and of u which are functions of m1 and m2.
Problem 5 Let X1, X2, ...,Xn be a random sample from a distribution with pdf T“ for r € (0, µ) |0 otherwise a+1 f(r) = where a and u are positive constants. First, suppose we know that u = 1, but we don't know a and we want to estimate it. (i) Find the maximum likelihood estimator of a. Is the estimator consistent? If yes, show it. Is the estimator biased? If yes, is the bias positive or negative? (ii) Let mi be the sample mean (m1 = X). Find a method of moments estimator of a which uses mı. Is the estimator consistent? If yes, show it. Is the estimator biased? If yes, is the bias positive or negative? Second, suppose we know neither u nor a and we want to estimate both parameters. (iii) State the minimization problem of the maximum likelihood estimators of a and of u. (You don't need to find these estimators.) (iv) Let m2 be the sample second moment (m2 = E X?). Find method of moments estimators of a and of u which are functions of m1 and m2.
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
Transcribed Image Text:Problem 5
Let X1, X2, ...,Xn be a random sample from a distribution with pdf
T“ for r € (0, µ)
|0 otherwise
a+1
f(r) =
where a and u are positive constants.
First, suppose we know that u = 1, but we don't know a and we want to estimate it.
(i) Find the maximum likelihood estimator of a. Is the estimator consistent? If yes, show it. Is
the estimator biased? If yes, is the bias positive or negative?
(ii) Let mi be the sample mean (m1 = X). Find a method of moments estimator of a which uses
mı. Is the estimator consistent? If yes, show it. Is the estimator biased? If yes, is the bias
positive or negative?
Second, suppose we know neither u nor a and we want to estimate both parameters.
(iii) State the minimization problem of the maximum likelihood estimators of a and of u. (You
don't need to find these estimators.)
(iv) Let m2 be the sample second moment (m2 = E X?). Find method of moments estimators
of a and of u which are functions of m1 and m2.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Similar questions
- Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
Probability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage LearningMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th…StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C…StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage Learning
Elementary Statistics: Picturing the World (7th E…StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSON
The Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. Freeman
Introduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman