Maximum Likelihood Estimation
Maximum Likelihood EstimationEric ZivotMay 14, 2001This version: November 15, 20091 Maximum Likelihood The Likelihood FunctionLetX1,...,Xnbe an iid sample with probability density function (pdf)f(xi; ),where is a(k 1)vector of parameters that characterizef(xi; ).For example, ifXi N( , 2)thenf(xi; )=(2 2) 1/2exp( 12 2(xi )2)and =( , 2) densityof the sample is, by independence, equal to the product of the marginaldensitiesf(x1,...,xn; )=f(x1; ) f(xn; )=nYi=1f(xi; ).The joint density is anndimensional function of the datax1,...,xngiven the para-meter vector .The joint density1satisfiesf(x1,...,xn; ) 0Z Zf(x1,...,xn; )dx1 dxn= Likelihood function is defined as the joint density treated as a functions of theparameters :L( |x1,...,xn)=f(x1,...,xn; )=nYi=1f(xi; ).
Figure xxx illustrates the normal likelihood for a representative sample of size n=25. Notice that the likelihood has the same bell-shape of a bivariate normal density Suppose σ 2 =1.Then
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