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
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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.
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