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## BASIC STA THEORY I

by: Helga Torp Sr.

41

0

4

# BASIC STA THEORY I STA 321

Helga Torp Sr.
UK
GPA 3.87

Mai Zhou

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COURSE
PROF.
Mai Zhou
TYPE
Class Notes
PAGES
4
WORDS
KARMA
25 ?

## Popular in Statistics

This 4 page Class Notes was uploaded by Helga Torp Sr. on Friday October 23, 2015. The Class Notes belongs to STA 321 at University of Kentucky taught by Mai Zhou in Fall. Since its upload, it has received 41 views. For similar materials see /class/228269/sta-321-university-of-kentucky in Statistics at University of Kentucky.

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Date Created: 10/23/15
Example of MLE Computations using R First of all7 do you really need R to compute the MLE Please note that MLE in many cases have explicit formula Second of all7 for some common distributions even though there are no explicit formula7 there are standard existing routines that can compute MLE Example ofthis catergory include Weibull distribution with both scale and shape parameters7 logistic regres sion7 etc If you still cannot nd anything usable then the following notes may be useful We start with a simple example so that we can cross check the result Suppose the observations X17X27Xn are from Nu702 distribution 2 parameters u and 02 The log likelihood function is 2709 2 2 2 71210g27T712log02longi 0 actually we do not have to keep the terms 712 log 27139 and log Xm since they are constants In R software we rst store the data in a vector called xvec xvec lt C2537 3 20 or some other numbers then de ne a function which is negative of the log lik fn lt functiontheta sum O5Xvec theta1 2theta2 05 logtheta2 where there are two parameters theta1 and theta2 They are compo nents of a vector theta then we try to nd the max actually the min of negative log lik nlmfn theta lt C O 1 hessianTRUE OI optimtheta lt CO1 fn hessianTRUE You may need to try several starting values here we used CO 1 for thethetaietheta10 theta21 Actual R output session gt xvec lt C2537 3 20 you may try other values gt fn I have pre defined fn functiontheta sum O5Xvec theta1 2theta2 05 logtheta2 gt nlmfn theta lt CO2 hessianTRUE minimization minimum 1 1200132 estimate 1 1714284 11346933 gradient 1 3709628eO7 5166134eO9 hessian 1 2 1 6169069e01 4566031eO6 2 4566031eO6 2717301eO2 Code 1 1 iterations 1 12 gt meanXveC 1 1714286 this Checks out with estimate1 gt sum xvec meanxvec 2 7 1 1134694 this also Checks out w estimate2 gt outputl lt nlmfn theta lt C210 hessianTRUE gt solveoutput1hessian to compute the inverse of hessian which is the approx var cor matrix 2 1 16209919201 3028906eO4 2 00003028906 3680137e01 gt sqrt diagsolveoutput1hessian 1 1273182 6066413 gt 11346947 1 1620991 gt sqrt11346947 1 1273182 st dev of mean Checks out gt optim theta lt C29 fn hessianTRUE minimization diff R function par 1 1713956 11347966 va1ue 1 1200132 Counts function gradient 45 NA Convergence 1 O message NULL hessian 1 2 1 6168506e01 1793543e05 2 1793543e05 2717398eO2 Comment We know long ago the variance of i can be estimated by 5271 or replace 52 by the MLE of 02 may be even this is news to you then you need to review some basic stat But how many of you know or remember the variancestandard devia tion of the MLE of 02 or 52 by above calculation we know its standard

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