Maximum Likelihood Estimation

Sun, 28 Oct 2018 17:11:47 +0100

### Introduction :

Maximum Likelihood Estimation is the method of estimating the parameters of a statistical model, given the observations. It attempts to find the parameter values that maximize the likelihood function. The process can be viewed as finding the parameters that maximize the likelihood of getting the data we observed for a particular set of statistical models.

Suppose we have the data points (random samples) \(X_1, X_2, ..., X_n\) which belong to a distribution which depends on one or more unknown parameters \(\theta_1, \theta_2, ..., \theta_m\) with probability density (or mass) function \(f(x_i; \theta_1, \theta_2, ..., \theta_m)\). Here, \(x_i\)s are the observed values for \(X_i\)s. Our task is to find the value of parameters that maximize the probability or likelihood of getting the observed value of the data. i.e., We need to maximize the following quantity (which is called as the Likelihood function):

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Maximum Likelihood Estimation