Maximum Likelihood Estimation - Maple Programming Help

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Maximum Likelihood Estimation

 

Introduction

Theory

Load Packages

Generate Data Set of Random Numbers

Maximizing the Likelihood Function

Histogram

Introduction

The likelihood function describes how closely a probability distribution describes a data set.  This application will demonstrate how to:

 

• 

Generate a data set according to a Weibull distribution with a specified scale and shape parameter.

• 

Backsolve the scale and shape parameters for this data set by maximizing the likelihood function using the Global Optimization Toolbox.

• 

Compare the resulting probability distribution against a histogram of the data set.

Theory

The Weibull probability distribution has the following density function:

Pt | α, β = βtβ1ⅇtαβαβ,

 

where α and β are the scale and shape parameters.  The likelihood that a data set fits this distribution is:

i=1nPti | α, β

 

Taking the log of this gives:

lni=1nPti | α, β 

 

Hence we derive the following expression describing the likelihood function:

i=1nlnPti | α, β

Load Packages

restart:

withStatistics:withGlobalOptimization:withplots:

Generate Data Set of Random Numbers

Number of random numbers to generate

n10000:

Define the desired scale and shape parameters for the data set.

α73:β17:

Generate a sample of random numbers according to a Weibull distribution

Data:=SampleRandomVariableWeibullα,β,n:

Maximizing the Likelihood Function

The probability density of a Weibull distribution

Pt,α, ββtβ1ⅇtαβαβ:

The likelihood function

maxLike:=α,β→addlnPDatai,α,β,i=1..n:

Find the values of α and β that maximize the likelihood function with the Global Optimization Toolbox

results:=GlobalSolvemaxLikealpha_ML,beta_ML,alpha_ML=1..100,beta_ML=1..100,maximize2;

results:=beta_ML=17.1301969320734,alpha_ML=72.9747976582517

(5.1)

α__GOTrhsselecthas,results,alpha_ML1; β__GOTrhsselecthas,results,beta_ML1;

α__GOT:=72.9747976582517

β__GOT:=17.1301969320734

(5.2)

These values match those used to generate the data set.

 

You can also match moments to find the shape and scale parameters

fsolveMomentData,1=MomentWeibulla,b,1,MomentData,2=MomentWeibulla,b,2 

a=72.98206445,b=17.00542373

(5.3)

Histogram

plotopts:=axesfont=Calibri,labelfont=Calibri,labels=t,Probability Density,labeldirections=horizontal,vertical:p1HistogramData,color=Gainsboro,plotopts:p2:=plotPt,α,β,t=sortData1..sortDatan,color=Black,legend=Sample Parameters,plotopts:p3:=plotPt,α__GOT,β__GOT,t=sortData1..sortDatan,color=Red,legend=Estimated Parameters,plotopts:displayp1,p2,p3,size=800,golden

In the above plot, the red line corresponds to the estimated parameters based on the sample population and the black line corresponds to the desired sample parameters.