(d) Is it easy to study the finite-sample distributional property of the MLE Â, such as E(1), var(Â), the mean square error (MSE) in estimating À, and its exact sampling distribution?
(d) Is it easy to study the finite-sample distributional property of the MLE Â, such as E(1), var(Â), the mean square error (MSE) in estimating À, and its exact sampling distribution?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![### Problem 2: Exponential Distribution Sample Analysis
Suppose that \( X_1, \ldots, X_n \) form a random sample from an Exponential(\(\lambda\)) distribution, with the probability density function (PDF) given by:
\[ f(x|\lambda) = \lambda e^{-\lambda x} \quad \text{for } x > 0, \, \lambda > 0. \]
In the above form, \(\lambda\) is a rate parameter, and the expected value and variance of \(X\) are:
\[ E(X) = 1/\lambda \quad \text{and} \quad \text{var}(X) = 1/\lambda^2. \]
#### Tasks:
**(a)** Find a Method of Moments (MOM) estimator \(\tilde{\lambda}\) of \(\lambda\).
**(b)** Write down the likelihood function \(L_n(\lambda)\) and obtain the maximum likelihood estimator (MLE) \(\hat{\lambda}\) of \(\lambda\).
**(c)** Let \(\tau\) denote the population median. Obtain the MLE of \(\tau\). *(Hint: First find the specific form of the median in terms of \(\lambda\) and utilize the invariance property of the MLE.)*
**(d)** Is it easy to study the finite-sample distributional property of the MLE \(\hat{\lambda}\), such as \(E(\hat{\lambda})\), \(\text{var}(\hat{\lambda})\), the mean square error (MSE) in estimating \(\lambda\), and its exact sampling distribution?
**(e)** Compute the Fisher information \(I_n\) in the sample data. Accordingly, obtain the asymptotic distribution of \(\hat{\lambda}\).
**(f)** Obtain a sufficient statistic \(T(X_1, \ldots, X_2)\) for \(\lambda\) via the factorization theorem. Verify that the MLE \(\hat{\lambda}\) is a function of \(T\).
This problem requires understanding of the exponential distribution, estimation techniques like MOM and MLE, as well as concepts such as Fisher information and sufficient statistics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc58b2eee-5b79-4c9d-923b-d4e913d44dbe%2Faaf64239-bb97-40be-be13-85962fcce5be%2Fatprtv4_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 2: Exponential Distribution Sample Analysis
Suppose that \( X_1, \ldots, X_n \) form a random sample from an Exponential(\(\lambda\)) distribution, with the probability density function (PDF) given by:
\[ f(x|\lambda) = \lambda e^{-\lambda x} \quad \text{for } x > 0, \, \lambda > 0. \]
In the above form, \(\lambda\) is a rate parameter, and the expected value and variance of \(X\) are:
\[ E(X) = 1/\lambda \quad \text{and} \quad \text{var}(X) = 1/\lambda^2. \]
#### Tasks:
**(a)** Find a Method of Moments (MOM) estimator \(\tilde{\lambda}\) of \(\lambda\).
**(b)** Write down the likelihood function \(L_n(\lambda)\) and obtain the maximum likelihood estimator (MLE) \(\hat{\lambda}\) of \(\lambda\).
**(c)** Let \(\tau\) denote the population median. Obtain the MLE of \(\tau\). *(Hint: First find the specific form of the median in terms of \(\lambda\) and utilize the invariance property of the MLE.)*
**(d)** Is it easy to study the finite-sample distributional property of the MLE \(\hat{\lambda}\), such as \(E(\hat{\lambda})\), \(\text{var}(\hat{\lambda})\), the mean square error (MSE) in estimating \(\lambda\), and its exact sampling distribution?
**(e)** Compute the Fisher information \(I_n\) in the sample data. Accordingly, obtain the asymptotic distribution of \(\hat{\lambda}\).
**(f)** Obtain a sufficient statistic \(T(X_1, \ldots, X_2)\) for \(\lambda\) via the factorization theorem. Verify that the MLE \(\hat{\lambda}\) is a function of \(T\).
This problem requires understanding of the exponential distribution, estimation techniques like MOM and MLE, as well as concepts such as Fisher information and sufficient statistics.
Expert Solution
Step 1
Given that form a random sample from a Exponential distribution, with PDF
where is a rate parameter and and
Step by step
Solved in 3 steps
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