The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: f (x|x0,θ) = θ xθ 0 x−θ−1 , x ≥ x0, θ> 1 Assume that x0 > 0 is given and that X1, X2,..., Xn is an i.i.d. sample. a. Find the method of moments estimate of θ. b. Find the mle of θ. c. Find the asymptotic variance of the mle.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: f (x|x0,θ) = θ xθ 0 x−θ−1 , x ≥ x0, θ> 1 Assume that x0 > 0 is given and that X1, X2,..., Xn is an i.i.d. sample. a. Find the method of moments estimate of θ. b. Find the mle of θ. c. Find the asymptotic variance of the mle.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 32EQ

See similar textbooks

Similar questions

Suppose that we want to estimate the parameter θ of the geometric distribution on the basis of a single obser-vation. If the loss function is given by L[d(x), θ] = c{d(x) − θ}2 and is looked upon as a random variable having the uniform density h(θ ) = 1 for 0 <θ< 1 and h(θ ) = 0 else-where, duplicate the steps in Example 9 to show that (a) the conditional density of given X = x isφ(θ|x) = x(x + 1)θ (1 − θ )x−1 for 0 <θ< 10 elsewhere(b) the Bayes risk is minimized by the decision function d(x) = 2x + 2
Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.
Find the moment generating function for the random variable X whose density function is f(x). fx = 2x 0<x<1 MX(t) = _______________________________________
Suppose that two continuous random variables X and Y have joint probability density function fxy = A( ex+y + e2x+y) , 1 ≤ x ≤ 2 ,0≤ y≤3 0 elsewhere a. P ( 3/2 ≤ X ≤ 2, 1 ≤ Y ≤ 2) b. Are the random variables X and Y independent? c. find the conditional density X given Y = 0
For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
A. Find the Cramér–Rao lower bound for the variance of an unbiased estimator of θ,based on a sample of size n for the following pdf: f(x)=θxθ−1, 0<x<1, θ > 0. B. Find the Method of Moment estimator of θ in the above pdf.C. Consider the following density function:X -2 -1 0 1 2f(x) 1/8 2/8 2/8 2/8 1/8Find P(-1≤X≤1) (ii) F(X) (iii) P(X≤1.25) (iv) P(X>0)
: Calculate the coefficients of skewness and kurtosis of the following frequency distribution. X 0 1 2 3 4 5 Frequency 2 2 4 6 8 3
Find the critical value of t for a t-distribution with 30 degrees of freedom such that the area between −t and t is 99%.
Reflux 80 0.65 0.69 0.63 0.63 0.66 0.85 0.71 0.65 0.92 0.58 0.69 0.76 0.59 0.88 0.66 0.66 0.65 0.76 0.72 0.69 0.69 0.67 0.68 0.69 0.83 Assuming the yield data obtained at a reflux rate of 80 has an Weibull distribution, what is the method of moments estimator (method 2) for the population parameter β.
Let X and Y be discrete random variables with joint pdf f(x,y) given by the following table: y = 1 y = 2 y = 3 x = 1 0.1 0.2 0 x = 2 0 0.167 0.4 x = 3 0.067 0.022 0.033 Find the marginal pdf’s of X and Y. Are X and Y independent?
The proportion of people who respond to a certain mail-order solicitation is a random variable X having the following density function. f(x) = 2(x+1)/3, 0<x<1, 0, elsewhere Find σ2g(X) for the function g(X)=5X2+4. σ2g(X)= ?
Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)