# What is maximum likelihood method in statistics?

## What is maximum likelihood method in statistics?

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable.

### How do you interpret gamma distribution?

The gamma and exponential distributions are equivalent when the gamma distribution has a shape value of 1. Remember that the shape value equals the number of events and the exponential distribution models times for one event. Therefore, a gamma distribution with a shape = 1 is the same as an exponential distribution.

What is gamma function in probability?

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution.

What is alpha and theta in gamma distribution?

The gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter. A random variable X that is gamma-distributed with shape α and rate β is denoted. The corresponding probability density function in the shape-rate parametrization is.

## How do you find the probability density of a gamma distribution?

If X follows a gamma distribution with shape α and scale β, then its probability density is Sometimes this is re-parameterized with β ⋆ = 1 / β, in which case you will need to change this accordingly. The likelihood function is just the density viewed as a function of the parameters.

### What are the parameters of gamma distribution?

Exponential family The gamma distribution is a two-parameter exponential family with natural parameters k − 1 and −1/ θ (equivalently, α − 1 and − β), and natural statistics X and ln (X). If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential family.

What is the exponential family of gamma distribution?

Exponential family. The gamma distribution is a two-parameter exponential family with natural parameters k − 1 and −1/ θ (equivalently, α − 1 and − β ), and natural statistics X and ln( X ). If the shape parameter k is held fixed, the resulting one-parameter family of distributions is a natural exponential family .

How do you find the likelihood function for maximum likelihood?

Therefore, the likelihood function L ( p) is, by definition: for 0 < p < 1. Simplifying, by summing up the exponents, we get : Now, in order to implement the method of maximum likelihood, we need to find the p that maximizes the likelihood L ( p).