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).