1. Suppose X = (X₁,..., Xn) is a random sample from the Gamma distribution withshape a and rate 0, i.e.,fx(x) =Jar(a)X²--x0Each X, has expectation E(X)generating function Mx (t) = (1 – t/0)¯¤.2a/0, variance Var(X)x>0, a > 0, 0 >0.S(b) Verify that each score function has zero expectation.(c) Assuming a to be known:(a) Assuming both a and to be unknown, write down the log likelihood functionl(0, a; X) and the corresponding score functions andдеде20θα=(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of anunbiased estimator of 0.(ii) Is there any unbiased estimator of whose variance attain the CRLB?111Show thata1=- -'Σ(;)i=1a/0², and momentis an unbiased estimator for 0. What is the MVU estimator for 0?(iv) Identify a change of parameter n = n(0) for which there exists an unbiasedestimator with variance attained the CRLB.(v) For the parameter in the previous part, identify the MVU estimator whosevariance attains the CRLB. Compute this variance.

Answered: 1. Suppose X = (X₁,..., Xn) is a random…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Question

1. Suppose X = (X₁,..., Xn) is a random sample from the Gamma distribution with
shape a and rate 0, i.e.,
fx(x) =
Ja
r(a)
X²
-
-x0
Each X, has expectation E(X)
generating function Mx (t) = (1 – t/0)¯¤.
2
a/0, variance Var(X)
x>0, a > 0, 0 >0.
S
(b) Verify that each score function has zero expectation.
(c) Assuming a to be known:
(a) Assuming both a and to be unknown, write down the log likelihood function
l(0, a; X) and the corresponding score functions and
де
де
20
θα
=
(i) Calculate the Cramer Rao Lower Bound (CRLB) for the variance of an
unbiased estimator of 0.
(ii) Is there any unbiased estimator of whose variance attain the CRLB?
111
Show that
a
1
=
- -'Σ(;)
i=1
a/0², and moment
is an unbiased estimator for 0. What is the MVU estimator for 0?
(iv) Identify a change of parameter n = n(0) for which there exists an unbiased
estimator with variance attained the CRLB.
(v) For the parameter in the previous part, identify the MVU estimator whose
variance attains the CRLB. Compute this variance.

Expert Solution

Step 1: Write the given information

Since you have posted a question with multiple sub-parts, we will solve the first two sub-parts for you. To get the remaining sub-part solved please repost the complete question and mention the sub-parts to be solved.”

Given $X equals open parentheses X subscript 1 comma space. space. space. comma space X subscript n close parentheses$ be a random sample from the Gamma distribution with shape $alpha$ and rate $theta$ .

$f subscript X open parentheses x close parentheses equals fraction numerator theta to the power of alpha over denominator capital gamma open parentheses alpha close parentheses end fraction x to the power of alpha minus 1 end exponent e to the power of negative x theta end exponent comma space x greater than 0 comma space alpha greater than 0 comma space theta greater than 0.$

$E open parentheses X close parentheses equals alpha over theta$ , $V a r open parentheses X close parentheses equals alpha over theta squared$ and $M subscript X open parentheses t close parentheses equals open parentheses 1 minus t over theta close parentheses to the power of negative alpha end exponent$ .