Question

? is a Pareto rv with probability distribution function given by f(x) = bx −(b+1) ,...

? is a Pareto rv with probability distribution function given by f(x) = bx −(b+1) , x≥ 1 where b is a positive number between 0 and 1

Show that ? = ln(?) has exponential distribution, ? ∼ exp(?), and find the maximum likelihood estimator of b. Explain why the moment of method fails to work in this case.

Math   Statistics-And-Probability   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let X1, X2,..., Xn be a random sample from a population with probability density function f(x)...
Let X1, X2,..., Xn be a random sample from a population with probability density function f(x) = theta(1-x)^(theta-1), where 0<x<1, where theta is a positive unknown parameter a) Find the method of moments estimator of theta b) Find the maximum likelihood estimator of theta c) Show that the log likelihood function is maximized at theta(hat)
Consider the probability density function f(x) = (3θ + 1) x3θ,    0  ≤  x  ≤  1....
Consider the probability density function f(x) = (3θ + 1) x3θ,    0  ≤  x  ≤  1. The random sample is 0.859, 0.008, 0.976, 0.136, 0.864, 0.449, 0.249, 0.764. The moment estimator of θ based on a random sample of size n is .055. Please answer the following: a) find the maximum likelihood estimator of θ based on a random sample of size n. Then use your result to find the maximum likelihood estimate of θ based on the given random...
Let Y1, Y2, . . ., Yn be a random sample from a Laplace distribution with...
Let Y1, Y2, . . ., Yn be a random sample from a Laplace distribution with density function f(y|θ) = (1/2θ)e-|y|/θ for -∞ < y < ∞ where θ > 0. The first two moments of the distribution are E(Y) = 0 and E(Y2) = 2θ2. a) Find the likelihood function of the sample. b) What is a sufficient statistic for θ? c) Find the maximum likelihood estimator of θ. d) Find the maximum likelihood estimator of the standard deviation...
1. Remember that a Poisson Distribution has a density function of f(x) = [e^(−k)k^x]/x! . It...
1. Remember that a Poisson Distribution has a density function of f(x) = [e^(−k)k^x]/x! . It has a mean and variance both equal to k. (a) Use the method of moments to find an estimator for k. (b) Use the maximum likelihood method to find an estimator for k. (c) Show that the estimator you got from the first part is an unbiased estimator for k. (d) (5 points) Find an expression for the variance of the estimator you have...
Problem 1. The Cauchy distribution with scale 1 has following density function f(x) = 1 /...
Problem 1. The Cauchy distribution with scale 1 has following density function f(x) = 1 / π [1 + (x − η)^2 ] , −∞ < x < ∞. Here η is the location and rate parameter. The goal is to find the maximum likelihood estimator of η. (a) Find the log-likelihood function of f(x) l(η; x1, x2, ..., xn) = log L(η; x1, x2, ..., xn) = (b) Find the first derivative of the log-likelihood function l'(η; x1, x2,...
1. (a) Y1,Y2,...,Yn form a random sample from a probability distribution with cumulative distribution function FY...
1. (a) Y1,Y2,...,Yn form a random sample from a probability distribution with cumulative distribution function FY (y) and probability density function fY (y). Let Y(1) = min{Y1,Y2,...,Yn}. Write the cumulative distribution function for Y(1) in terms of FY (y) and hence show that the probability density function for Y(1) is fY(1)(y) = n{1−FY (y)}n−1fY (y). [8 marks] (b) An engineering system consists of 5 components connected in series, so, if one components fails, the system fails. The lifetimes (measured in...
About convex function: (1) Please show that f(x) = ax2 + bx + c is a...
About convex function: (1) Please show that f(x) = ax2 + bx + c is a convex function if and only if a ≥ 0. (2) Please show that f(x) = 1/2·xTQx+aTx is a convex function if and only if Q is positive semi-definite.
Show the following: a) Let there be Y with the cumulative distribution function F(y). Let F(Y)=Z....
Show the following: a) Let there be Y with the cumulative distribution function F(y). Let F(Y)=Z. Show that Z~U(0,1) for F(y). b) Let X~U(0,1), and let Y := -ln(X). Show that Y~exp(1)
Consider the random variable X with density given by f(x) = θ 2xe−θx x > 0,...
Consider the random variable X with density given by f(x) = θ 2xe−θx x > 0, θ > 0 a) Derive the expression for E(X). b) Find the method of moment estimator for θ. c) Find the maximum likelihood estimator for θ based on a random sample of size n. Does this estimator differ from that found in part (b)? d) Estimate θ based on the following data: 0.1, 0.3, 0.5, 0.2, 0.3, 0.4, 0.4, 0.3, 0.3, 0.3
Given the exponential distribution f(x) = λe^(−λx), where λ > 0 is a parameter. Derive the...
Given the exponential distribution f(x) = λe^(−λx), where λ > 0 is a parameter. Derive the moment generating function M(t). Further, from this mgf, find expressions for E(X) and V ar(X).
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT