@MISC{_maximumlikelihood, author = {}, title = {MAXIMUM LIKELIHOOD ESTIMATION}, year = {} }

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Abstract

Maximum likelihood is by far the most pop-ular general method of estimation. Its wide-spread acceptance is seen on the one hand in the very large body of research dealing with its theoretical properties, and on the other in the almost unlimited list of applications. To give a reasonably general definition of maximum likelihood estimates, let X = (X1,...,Xn) be a random vector of observa-tions whose joint distribution is described by a density fn(x|) over the n-dimensional Euclidean spaceRn. The unknown parameter vector is contained in the parameter space ⊂ Rs. For fixed x define the likelihood∗ function of x as L() = Lx() = fn(x|) con-sidered as a function of ∈ . Definition 1. Any ̂ = ̂(x) ∈ which maximizes L() over is called a maximum likelihood estimate (MLE) of the unknown true parameter . Often it is computationally advantageous to derive MLEs by maximizing logL() in place of L(). Example 1. Let X be the number of suc-cesses in n independent Bernoulli trials with success probability p ∈ [0, 1]; then Lx(p) = f (x|p) = P(X = x|p)