Concept Check: M-estimators vs. Maximum Likelihood EstimationiidLet p denote a loss function, and let X₁,..., Xn P. Let denote the M-estimator for some unknown parameterμ* = argmin μERE [p(X₁, μ)] € R associated with P. (Here we are assuming that μ* is a one-dimensional parameter.)Consider the following functionsJ (μ) = E(μ² (X1,μ)apƏμWhich of the following statements are true? (Choose all that apply.)K(μ) = Var-(X₁₂μ)It is always true that J (u) = K (μ).J(μ) = K (μ) when p is the negative log-likelihood- in this case, both of these functions are equal to the Fisher information.Under some technical conditions, the functions J (u) and K (μ) determine the asymptotic variance of the M-estimator μ.

Given information: The information about M-estimators and Maximum Likelihood Estimation is given.

Answered: Let p denote a loss function, and let…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

solveIn random sampling from the exponential distribution, f(x) =1 θθe x− , x > 0, θ> 0, find the maximum likelihood estimator of θ and obtain the asymptotic distribution of this estimator
Let random sample from a Poisson distribution with parameter . A random variable X with Poisson distribution has a probability mass function (pmf) of p(x; lambda)=((lambda^x)(e^-lambda))/x!, x=0,1,... with E(X)=lambda and Var(X)=lambda . (d) Find the Fisher information for the random sample of the parameter lambda. (e) What is the Cramer Rao Lower bound for an unbiased estimator of the parameter lambda? (f) Is the MLE of part (c) a minimum variance unbiased estimator (MVUE) of lambda?
A random variable Y is said to have a Singh-Maddala distribution with parameter θ > 0 if the density function of Y isf(y|θ) = 2θy(1 + y2)θ+1for y > 0. Suppose that Y1,Y2,...,Yn is a random sample from a Singh-Maddala(θ) population where θ > 0 is a parameter.(a) Verify that f(y|θ) belongs to an exponential family in traditional form.(b) Compute the likelihood function L(θ) for this random sample.(c) Determine θˆMLE, the maximum likelihood estimator of θ. You may use any method that we discussed in class for finding the MLE.(d) Find the Fisher information I(θ) in a single observation from this density.(e) Using the standard normal approximation for the distribution of a maximum likelihood es- timator based on the Fisher information, construct an approximate 95% confidence interval for θ. Note that z0.025 = 1.96.Note that the Singh-Maddala distribution is also known as the Burr Type XII distribution and is used in economics and actuarial science to model income sizes.
2)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the maximum likelihood estimator (MLE) of parameter θ.
Let X1, ..., Xn be a random sample of size n from a gamma population given by the density f(x; α, β) = ( x α−1e − x β βαΓ(α) if x > 0, α > 0, β > 0 0 if x ≤ 0 with Γ(α) = R ∞ 0 x α−1 e −xdx the well-known gamma distribution. 1. If α > 0 is known for this random sample compute the maximum likelihood estimator βbML estimator of the unknown parameter β > 0. 2. Compute the moment generating function E(e sX) for every s < 1 β . 3. Compute the expectation E(X). 4. Show that the maximum likelihood estimator βbML is a unbiased estimator of β.(Hint: you may use the result of part 2)
Consider a random variable X with support [0, ∞), pdf f, and CDF F. The function h definedash(x) = f(x)/1 − F(x)for x > 0. The function h is called the failure rate or the hazard function of X. What is thefailure rate if X is exponential with parameter λ?
Suppose X and Y are continuous random variables such that the pdf is f(x,y)=x+y with 0≤x≤1,0≤y≤1. a)Draw a graph that illustrates the domain of this pdf. b)Find the marginal pdfs of X and Y.c)Compute ux,ux,var x,var y, cov(x,y) ,and pd (Rho d)) Determine the equation of the least squares regression line and draw it on your graph.
1)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]
Let X1, X2, ..., Xn be a sequence of independent and identically distributedrandom variables having the Exponential(λ) distribution, λ > 0,fXi(x) = λe−λx , x > 00 , otherwise(a) Show that the moment generating function mX(s) := E(e^sX) = λ/λ−s for s < λ;(b) Using (a) find the expected value E(Xi) and the variance Var(Xi).(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .(d) Consider a random variable X having Gamma(α, λ) distribution,fX(x) = (λαxα-1/Γ(α)) e−λx , x > 00 , otherwiseShow that the moment generating function of the random variable X is mX(s) =λα 1/(λ−s)α for s < λ, where Γ(α) isΓ(α) = (integral from 0 to inifity ) xα−1e−xdx.(e) What is the probability distribution of Y given in (c)? Explain youranswer.
A discrete random variable X has probability mass function X 0 1 2 3 4 P(x) 0.1 0.2 0.2 0.2 0.3 Use the inverse transform method to generate a random sample of size 1000 from the distribution of X. Construct a relative frequency table and compare with the theoretical values.
b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.
Let X1, . . . , Xn be iid with pdf f(x) = 1 x √ 2πθ2 e − (log(x)−θ1) 2 2θ2 , −∞ < x < ∞, and unknown parameters θ1 and θ2. Find the maximum likelihood estimators for θ1 and θ2, respectively

SEE MORE QUESTIONS

Recommended textbooks for you

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

Elementary Statistics: Picturing the World (7th E…

Statistics

ISBN:

9780134683416

Author:

Ron Larson, Betsy Farber

Publisher:

PEARSON

The Basic Practice of Statistics

Statistics

ISBN:

9781319042578

Author:

David S. Moore, William I. Notz, Michael A. Fligner

Publisher:

W. H. Freeman

Introduction to the Practice of Statistics

Statistics

ISBN:

9781319013387

Author:

David S. Moore, George P. McCabe, Bruce A. Craig

Publisher:

W. H. Freeman

SEE MORE TEXTBOOKS

Let p denote a loss function, and let X₁,..., X, id P. Let denote the M-estimator for some unknown parameter * = argmin ¹μERE [p (X₁, µ)] = R associated with P. (Here we are assuming that μ* is a one-dimensional parameter.) Consider the following functions I(u) •[0²P (X, 11]