Question

What is the log- likelihood function?

Answer 1

To solve this question we need to know the concept of statistics and probability log likelihood function is a function $F\left( x \right)$ which is defined to be the natural logarithmic of the likelihood function $L\left( x \right)$ . In mathematical form it is written as$F\left( x \right)\text{=ln L}\left( x \right)$. Here “$\ln$” is the natural logarithmic function having base as $e$, which is an exponent.

Complete step-by-step answer:
The question asks us the explanation for the log-likelihood function. Before we take the discussion to the log- likelihood function, let us know something about likelihood function. So likelihood function in the field of statistics is measure of how well the function fits a set of observations of statistical models of a sample of data for unknown parameters. Now taking our discussion to the log- likelihood function. So log- likelihood function is that kind of function which is the natural logarithm of likelihood function. Mathematically presented as: $F\left( x \right)\text{=ln L}\left( x \right)$, where $L\left( x \right)$ is a likelihood function and $F\left( x \right)$ is a log- likelihood function.
Likelihood function in the expanded notation form is:
$L\left( x \right)=\prod\limits_{i}^{n}{f\left( \dfrac{{{y}_{i}}}{x} \right)}$
So, the log- likelihood function will be written as:
$F\left( x \right)=\prod\limits_{i}^{n}{\ln f\left( \dfrac{{{y}_{i}}}{x} \right)}$
In the above expansion “$i$” vary till “n” which is stated in the problem.

Note: The log likelihood function is typically used to derive the maximum likelihood estimator of the parameter. The log likelihood function is important as it ensures that the maximum value of the log of the probability occurs at the same point as the original probability function. So we can work with the simpler log-likelihood rather than that of the original likelihood.