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The probability distribution function (and thus likelihood function) for exponential families contain products of factors involving exponentiation. The logarithm of such a function is a sum of products, again easier to differentiate than the original function.
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
The terms "distribution" and "family" are often used loosely: Specifically, an exponential family is a set of distributions, where the specific distribution varies with the parameter; [a] however, a parametric family of distributions is often referred to as "a distribution" (like "the normal distribution", meaning "the family of normal distributions"), and the set of all exponential families ...
In statistics, the method of estimating equations is a way of specifying how the parameters of a statistical model should be estimated. This can be thought of as a generalisation of many classical methods—the method of moments , least squares , and maximum likelihood —as well as some recent methods like M-estimators .
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.
In the four survival function graphs shown above, the shape of the survival function is defined by a particular probability distribution: survival function 1 is defined by an exponential distribution, 2 is defined by a Weibull distribution, 3 is defined by a log-logistic distribution, and 4 is defined by another Weibull distribution.
It is a generalization of the idea of using the sum of squares of residuals (SSR) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood. It plays an important role in exponential dispersion models and generalized linear models. Deviance can be related to Kullback-Leibler divergence. [1]
The Bernoulli distributions for form an exponential family. The maximum likelihood estimator of based on a random sample is the sample mean. The probability mass distribution function of a Bernoulli experiment along with its corresponding cumulative distribution function.