Just G please 2. Suppose that X₁,..., Xn form a random sample from a Exponential(\) distribution, with PDFgiven byf(x|λ) = Ae¯λx for x > 0, λ > 0.In the above form, À is a rate parameter and E(X) = 1/λ and var(X) = 1/λ².(a) Find a MOM (method of moment) estimator Ã of X.(b) Write down the likelihood function Ln (A) and obtain the maximum likelihood estimator(MLE) Â of A.(c) Let 7 denote the population median. Obtain the MLE of T. (Hint: First find the specificform of the median in terms of λ and utilize the invariance property of MLE)(d) Is it easy to study the finite-sample distributional property of the MLE Â, such as E(Â),var(Â), the mean square error (MSE) in estimating \, and its exact sampling distribution?(e) Compute the Fisher information In in the sample data. Accordingly, obtain the asymp-totic distribution of Â.(f) Obtain a sufficient statistic T(X₁,..., X₂) for À via the factorization theorem. Verify thatthe MLE Â is a function of T. (g) The PDF of exponential distribution can be parameterized differently by the mean1/λ as follows1fƒ(x|ß) ===-e-/8 for 3 > 0, x > 0.Bе=i. Find the MLE of B, 3.ii. Obtain the Fisher information In for ß. Is it the same as the Fisher informationfor X in Part 2(e)? Obtain the Cramer-Rao lower bound (CRLB) for any unbiasedestimator of B.iii. Note that, with this form of parameterization, it is much easier to explore the samplingdistribution of . Is the MLE Ô the UMVUE (uniformly minimum variance unbiasedestimator)? (Hint: Is ß unbiased for ß? Find var(8) as well.)

g) From the given information, The density function is, fx|β=1βe-xβ, β>0, x>0 i) MLE of β:…

Answered: (g) The PDF of exponential 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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Question

Just G please

2. Suppose that X₁,..., Xn form a random sample from a Exponential(\) distribution, with PDF
given by
f(x|λ) = Ae¯λx for x > 0, λ > 0.
In the above form, À is a rate parameter and E(X) = 1/λ and var(X) = 1/λ².
(a) Find a MOM (method of moment) estimator Ã of X.
(b) Write down the likelihood function Ln (A) and obtain the maximum likelihood estimator
(MLE) Â of A.
(c) Let 7 denote the population median. Obtain the MLE of T. (Hint: First find the specific
form of the median in terms of λ and utilize the invariance property of MLE)
(d) Is it easy to study the finite-sample distributional property of the MLE Â, such as E(Â),
var(Â), the mean square error (MSE) in estimating \, and its exact sampling distribution?
(e) Compute the Fisher information In in the sample data. Accordingly, obtain the asymp-
totic distribution of Â.
(f) Obtain a sufficient statistic T(X₁,..., X₂) for À via the factorization theorem. Verify that
the MLE Â is a function of T.

(g) The PDF of exponential distribution can be parameterized differently by the mean
1/λ as follows
1
fƒ(x|ß) =
==-e-/8 for 3 > 0, x > 0.
B
е
=
i. Find the MLE of B, 3.
ii. Obtain the Fisher information In for ß. Is it the same as the Fisher information
for X in Part 2(e)? Obtain the Cramer-Rao lower bound (CRLB) for any unbiased
estimator of B.
iii. Note that, with this form of parameterization, it is much easier to explore the sampling
distribution of . Is the MLE Ô the UMVUE (uniformly minimum variance unbiased
estimator)? (Hint: Is ß unbiased for ß? Find var(8) as well.)

Expert Solution

Step 1

From the given information,

The density function is,

$f (x | β) = \frac{1}{β} e^{- \frac{x}{β}}, β > 0, x > 0$

MLE of $β$ :

Consider, the likelihood function,

$L = \prod_{i = 1}^{n} f (x_{i} | β) = \frac{1}{β} e^{- \frac{x_{1}}{β}} * \frac{1}{β} e^{- \frac{x_{2}}{β}} * . . . . . * \frac{1}{β} e^{- \frac{x_{n}}{β}} = \frac{1}{β^{n}} e^{- \frac{1}{β} \sum_{i = 1}^{n} x_{i}}$

Apply 'ln' on both sides,

$\ln L = \ln [\frac{1}{β^{n}} e^{- \frac{1}{β} \sum_{i = 1}^{n} x_{i}}] \ln L = \ln (\frac{1}{β^{n}}) + \ln (e^{- \frac{1}{β} \sum_{i = 1}^{n} x_{i}}) \ln L = - n \ln β - \frac{1}{β} \sum_{i = 1}^{n} x$

MLE of $β$ obtained by using the condition, $\frac{d \ln L}{d β} = 0$ .

Therefore,

$\frac{d \ln L}{d β} = 0 \frac{- n}{β} + \frac{1}{β^{2}} \sum_{i = 1}^{n} x = 0 \frac{1}{β^{2}} \sum_{i = 1}^{n} x_{i} = \frac{n}{β} \hat{β} = \frac{\sum_{i = 1}^{n} x_{i}}{n} = \bar{x}$