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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 ...
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.
The values for which both likelihood and spacing are maximized, the maximum likelihood and maximum spacing estimates, are identified. Suppose two values x (1) = 2, x (2) = 4 were sampled from the exponential distribution F(x;λ) = 1 − e −xλ, x ≥ 0 with unknown parameter λ > 0. In order to construct the MSE we have to first find the ...
A textbook example is parameter estimation of a probability distribution function.Consider the exponential distribution: =,, >The hypotheses are {: =: = >.Then the log-likelihood function (LLF) for one sample is
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.
For each θ, the likelihood function is a probability density function, and therefore =. By using the chain rule on the partial derivative of log f {\displaystyle \log f} and then dividing and multiplying by f ( x ; θ ) {\displaystyle f(x;\theta )} , one can verify that
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 ...