Likelihood Function/Examples

Examples of Likelihood Functions

Arbitrary Independent Observations

Let S$ be a sample of $n$ independent observations of a random variable with a given probability distribution.

The likelihood function $\map {\mathrm L} \theta$ is:

$\map {\mathrm L} \theta := \map f {x_1, x_2, \ldots, x_n}$

Because of independence:

$\map {\mathrm L} \theta = \map f {x_1, \theta} \map f {x_2, \theta} \cdots,\map f {x_n, \theta}$

Suppose $\theta_1$ and $\theta_2$ are values of $\theta$.

Suppose that:

$\map {\mathrm L} {\theta_2} < \map {\mathrm L} {\theta_1}$

This implies that the sample has a smaller value of the joint frequency function if the unknown parameter is $\theta_2$ rather than $\theta_1$.

This in turn means that the sample is less likely to have come from a population where $\theta = \theta_2$ rather than where $\theta = \theta_1$.

This line of reasoning leads to the concept of maximum likelihood estimation.