#### week407

East Los Angeles College, MAS 2302
Excerpt: ... Maximum likelihood estimation MAS2302/MAS3302: Introduction to Statistical Inference: Week 4 Dr. David Walshaw November 5, 2007 Dr. David Walshaw MAS2302/MAS3302:Introduction to Statistical Inference:Week 4 Maximum likelihood estimation Example ...

#### lecture4 notes

Wisconsin, STAT 312
Textbook: Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )
Excerpt: ... Stat 312: Lecture 4 Maximum Likelihood Estimation Moo K. Chung mchung@stat.wisc.edu January 30, 2003 Concepts 1. For a random sample X1 , , Xn the likelihood function is given as the product of probability or density functions, i.e. L() = f (x1 ; ...

#### problems107

East Los Angeles College, MAS 2302
Excerpt: ... 1 MAS2302/MAS3302: Introduction to Statistical Inference Problems 1 (Not assessed) A. Computing project 1. In this question you will investigate the behaviour of the maximum likelihood estimator for the Geometric parameter p. Using similar commands to those used in Exercises 1, simulate m data sets of size n from the Geometric(0.01) distribution, and study the distribution of the maximum likelihood estimator (obtained in Question 2), by simulating 1000 samples of size: i) 5; ii) 50 and iii) 500. Describe the shape of the distribution of maximum likelihood estimates in each case, and say what appears to be happening to the shape of this distribution as the sample size increases. (N.B. you will need to modify the commands which specify the probability distribution, and which calculate the maximum likelihood estimate.) N.B. In this course we use the version of the Geometric distribution favoured by R, with probability mass function: Pr(X = x) = (1 p)x p, x = 0, 1, 2, . . . Here X can be thought of as the ...

#### exercises107

East Los Angeles College, MAS 2302

#### handout407

East Los Angeles College, MAS 2302
Excerpt: ... 1 MAS2302/MAS3302 Introduction to Statistical Inference Handout 4 Example 2.4.2 Now consider the m.l.e. X as an estimator for in Example 2.4.1. What are its properties? Solution: 2 2.5 The asymptotic distribution of the maximum likelihood estima ...

#### stat400lec24

UIllinois, STAT 400
Excerpt: ... Statistics 400 Chapter 7 Estimation Maximum likelihood estimates Example: If X1, X2, ., X16 are observations of a random sample of size 16 from a normal distribution N(50,100), Find Blah blah blah How do we know that the mean is 50 and the variance is 100? In this chapter, we consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Example: f( x; )=(1/ ) e-x/ 0 < x < Objective: Choose one member of How? = { :0 < < } as the most likely value of Take a random sample from the distribution to elicit some information about the unknown parameter of X1, X2,., Xn is a random sample of size n from the distribution x1, x2,., xn are observed values Estimator: The function of X1, X2,., Xn used to estimate , say the statistics u(X1, X2,., Xn), is called an estimator of . There are two general methods (1) Maximum likelihood (2) Method of moments Ping Ma Lecture 24 Fall 2005 -1- M ...

#### lecture05 notes

Wisconsin, STAT 312
Textbook: Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )
Excerpt: ... Stat 312: Lecture 05 Maximum Likelihood Estimation Moo K. Chung mchung@stat.wisc.edu September 14, 2004 ^ 1. (Invariance Principle) If is the MLE's of ^ parameter then the MLE of h() is h() for some function h. Proof (partial). Consider likelihood ...

#### PS4

Cornell, ECON 614
Excerpt: ... ECON 614 MACROECONOMIC THEORY II: Problem Set 4 due: Friday April 4 Karel Mertens, Cornell University For the estimation of a VAR as in the lecture notes, show that 1. The GLS estimator is identical to the multivariate OLS estimator obtained from min ...

#### solutionsex107

East Los Angeles College, MAS 2302
Excerpt: ... e contents of the matrix psim. Now, add the following commands below those already entered into your Notepad window: pmle<-vector() for (i in 1:m) { pmle[i]<-mean(psim[,i]) } hist(pmle) At this point you may like to save the contents of your Notepad window! Now copy and paste these commands into R, and press the return key. You should now have a histogram of maximum likelihood estimates for . Investigate the mean and variance of your sample of 1000 m.l.e.s using the R commands mean() and var(). 2 Leaving R and Notepad open, and ensuring that you have a version of the Notepad window saved, answer the following questions on paper. (a) Describe the sampling distribution of the maximum likelihood estimator for the Poisson parameter for samples of size 5 obtained from a Poisson(1) distribution. Solution: The sampling distribution is positively skewed (or right-skewed). It has a mean approximately equal to 1.0 and a variance of approximately 0.2. [6 Marks] (b) By changing the value for n in ...

#### problems107solutions

East Los Angeles College, MAS 2302
Excerpt: ... 1 MAS2302/MAS3302: Introduction to Statistical Inference Problems 1: Solutions A. Computing project 1. In this question you will investigate the behaviour of the maximum likelihood estimator for the Geometric parameter p. Using similar commands to those used in Exercises 1, simulate m data sets of size n from the Geometric(0.01) distribution, and study the distribution of the maximum likelihood estimator (obtained in Question 2), by simulating 1000 samples of size: i) 5; ii) 50 and iii) 500. Describe the shape of the distribution of maximum likelihood estimates in each case, and say what appears to be happening to the shape of this distribution as the sample size increases. (N.B. you will need to modify the commands which specify the probability distribution, and which calculate the maximum likelihood estimate.) N.B. In this course we use the version of the Geometric distribution favoured by R, with probability mass function: Pr(X = x) = (1 - p)x p, x = 0, 1, 2, . . . Here X can be thought of as the number ...

#### 10a_maximumlikelihood

Uni. Worcester, ECE 531
Excerpt: ... ECE531 Lecture 10a: Maximum Likelihood Estimation ECE531 Lecture 10a: Maximum Likelihood Estimation D. Richard Brown III Worcester Polytechnic Institute 02-Apr-2009 Worcester Polytechnic Institute D. Richard Brown III 02-Apr-2009 1 / 23 ECE531 ...

#### lab5f05

Iowa State, STAT 601
Excerpt: ... be considered data from Group 1 and the second column data from Group 2. 1. Find maximum likelihood estimates for gamma models formulated for these data. Test whether the groups should be considered significantly different. Produce Wald theory intervals for model parameters. Display your results. 2. Find your own example of data to which it might be reasonable to fit a generalized linear model (published examples are fine). Using any computational tool you wish (the functions provided, built-in S or R functions, SAS, whatever) estimate models, summarize results, and depict the data and model fit graphically. ...

#### stat400lec30

University of Illinois, Urbana Champaign, STAT 400
Excerpt: ... Statistics 400 Chapter 6 Estimation Maximum likelihood estimates Example: If X1, X2, ., X16 are observations of a random sample of size 16 from a normal distribution N(50,100), Find Blah blah blah How do we know that the mean is 50 and the variance is 100? In this chapter, we consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Example: f( x; )=(1/ ) e-x/ 0 < x < = {:0 < < } Objective: Choose one member of as the most likely value of How? Take a random sample from the distribution to elicit some information about the unknown parameter of X1, X2,., Xn is a random sample of size n from the distribution x1, x2,., xn are observed values Estimator: The function of X1, X2,., Xn used to estimate , say the statistics u(X1, X2,., Xn), is called an estimator of . There are two general methods (1) Maximum likelihood (2) Method of moments Ping Ma Lecture 30 -1- Maximum likelihood Conside ...

#### stat400lec30

University of Illinois, Urbana Champaign, STAT 400
Excerpt: ... Statistics 400 Chapter 6 Estimation Maximum likelihood estimates Example: If X1, X2, ., X16 are observations of a random sample of size 16 from a normal distribution N(50,100), Find Blah blah blah How do we know that the mean is 50 and the variance is 100? In this chapter, we consider random variables for which the function form of pdf is known, but of the parameter of the pdf, say , is unknown. Parameter space : all possible values of . Example: f( x; )=(1/ ) e-x/ 0 < x < = {:0 < < } Objective: Choose one member of as the most likely value of How? Take a random sample from the distribution to elicit some information about the unknown parameter of X1, X2,., Xn is a random sample of size n from the distribution x1, x2,., xn are observed values Estimator: The function of X1, X2,., Xn used to estimate , say the statistics u(X1, X2,., Xn), is called an estimator of . There are two general methods (1) Maximum likelihood (2) Method of moments Ping Ma Lecture 30 -1- Maximum likelihood Consid ...

#### CAT_Part_2_Class_Handouts_Contents

Berkeley, SOC 271
Excerpt: ... To: From: RE: Topic: II. Course Participants Trond Petersen Content of Part II of Reader Categorical Dependent Variables Class Handouts The Class Handouts are excerpts from the Lecture Notes. They give the main equations from the Lecture Notes. The ...

#### 304_Hw8_Estimators

Cornell, CEE 3040
Excerpt: ... CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING Homework #8 Due: Monday, October 22, 2007. Read: Sections 6.1 and 6.2, and prob. 34 on p. 252 Devore6 [p. 279 Devore6] for definition of MSE. For confidence intervals please read 7.1-7.2. Goal: We are now ...

#### tutw13

Allan Hancock College, STAT 378
Excerpt: ... STAT378 Tutorial Exercise Week 13 Question 1 This is Question 2 of Assignment 4. The question considers a Poisson regression with log-linear model for the mean. The data set is exhibited below: X: Y: 2 19 5 17 7 25 10 26 20 40 25 41 27 35 29 42 30 39 ...

#### lecture13

Wisconsin, ECE 07
Excerpt: ... ECE901 Spring 2007 Statistical Learning Theory Instructor: R. Nowak Lecture 13: Maximum Likelihood Estimation 1 Summary of Lecture 12 In the last lecture we derived a risk (MSE) bound for regression problems; i.e., select an f F so that E[(f (X ...

#### ChapE_Slide

UNC, BIOS 760
Excerpt: ... CHAPTER 5 MAXIMUM LIKELIHOOD ESTIMATION ' \$ 1 Introduction to Efficient Estimation Goal MLE is asymptotically efficient estimator under some regularity conditions. & % CHAPTER 5 MAXIMUM LIKELIHOOD ESTIMATION ' \$ 2 Basic setting Suppose X ...

#### notes1

UWO, SS 2858
Excerpt: ... Lecture Notes SS2858B Outline Jan 5, 2009 Review of Discrete Random Variables Review of Expectation and Variance Special discrete variable distributions Maximum Likelihood Estimation 1 Review of Discrete Random Variables (section 2.1 in th ...