Suppose that X,X,X, is a random sample from a distribution with pdf:fCx; 8) = (02"x--1, if210,elsewhereFurthermore, E(X,) =;a) Find the method of moments estimator (MME) of e.b) Find the maximum likelihood estimator (MLE) of 0.c) Find the Fisher information for e in the sample.d) Find the Cramer-Rao lower bound for an unbiased estimator of 0.e) Suppose that the 10 independent observations,

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Answered: Suppose that X,X,X, is a random sample…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 25EQ

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Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.
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Q3. Suppose that X;i, i = 1,2, .,n have an Exponential distribution with parameter 2. a) Find maximum likelihood estimator (Â) for 2. b) Is â is an unbiased estimator?
Q4. Let X1,X 2,.X , be a random sample from an exponential distribution with parameter 0. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of e. (b) Find the maximum likelihood estimator (m.l.e.) of 0. (c) Is this estimator unbiased and does its variance attain the Cramer-Rao lower bound? (d) What is the method of moments estimator of 0?
The number of clients a store has during a week is Poisson distributed with - 25. The amount of money spent by each client follows a random = el,000t+250t2. Assuming the mean A variable with moment generating function o(t) independence between the number of clients, and the amount of money spent by each client. Find the mean and variance of the amount of money received by the store every week.
Suppose that Y, Y2,.-.,Y, are independent random variables from a gamma distribution of the parameter (2,6+4). a) Use the method of moment to obtain estimator of B. b) Show that the estimator B, obtained in part a) is unbiased. c) Show that the estimator B is a minimum variance unbiased estimator of ß.
The built-in data set, UKgas, is a time series recording the quarterly UK gas consumption from 1960 to 1986. Using R we can convert this data into the vector x by the assignment x30 we can create a confidence interval for u using a normal critical value. If we want the confidence interval to be at the 94% level and we use a normal critical value, then what critical value should we use? i) Calculate a 94% confidence interval(using a normal critical value) for μ. j) How long is the 94% confidence interval just created in part i?

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