Maximum Likelihood Method: Estimators and Properties

method of maximum likelihood"

Consider x distributed according to

f (x; q)

where q is a parameter (or vector of parameters) unknown

is a technique to estimate the values ​​of θ given a finite sample of data

Likelihood function

P(all xi in [xi , xi + dxi])= f(xi i=1 n ∏ ;θ)dxi

q is treated as a parameter (which the pdf depends on)

the xi are fixed (the data has already been acquired)

Maximum likelihood estimators

are defined as those that maximize the likelihood function:

∂L /∂θi = 0, i = 1,2,...m

these are usually the most acceptable approximation to the problem of estimating parameters.

The idea of ​​this method is to first find the joint density function of all the observations,

Properties of the maximum likelihood estimator

consistency

asymptotic normality

efficiency