Maximum Likelihood Distribution

Question

I know that in Maximum Likelihood Distribution we have data points. From those data points, we calculate parameters for maximum likelihood.

In this we assume that the data points has been generated from some distribution (Binomial, Normal, Gaussian, Exponential, etc.)

I want to know on what basis the distribution is assumed? Means we can have multiple type of distribution generating same data.

Answer 1

For $X_1, X_2, \dots ,X_n$ independently distributed $$f(x_1, x_2, \dots ,x_n) = f_{X_1}(x_1) f_{X_2}(x_2) \cdots f_{X_n}(x_n) $$ where $f(x_1, x_2, \dots ,x_n)$ is joint density function and $f_{X_n}(\cdot)$ are the marginal density functions.

For $X_1, X_2, \dots ,X_n$ independent and identically distributed $$f(x_1, x_2, \dots ,x_n) = f_{X}(x_1) f_{X}(x_2) \cdots f_{X}(x_n) $$ where $f(x_1, x_2, \dots ,x_n)$ is joint density function and $f_X(\cdot)$ is the marginal density function.

This is the Likelihood function usually expressed as $L=\prod_{i=1}^{n} f_X(x_i)$. The log of $L$ is the log-likelihood function which is used to estimate the parameters of the distribution. That would be the basis for the Likelihood function.

The vector $(x_1, x_2, \dots ,x_n)$ is generated from some function $f_X(\cdot)$ (Binomial, Normal, Gaussian, Exponential, etc.). The choice of $f_X(\cdot)$ is based upon the nature of the data. For instance, if we knew that $x_i$ can only be positive, we may use an exponential distribution, etc. That would be the basis for the assumed distribution.