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 ; ...

East Los Angeles College, MAS 2302

Excerpt: ... 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. (b) By changing the value for n in your notepad window and recopying into R, describe the sampling distribution of the maximum likelihood estimator for the Poisson parameter for samples of size 500 obtained from a Poisson(1) d ...

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 ...

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 ...

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 ...

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. ...

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 ...

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 ...

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 ...

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 ...