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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 ...
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
Consider the problem of estimating the rate parameter, λ of the exponential distribution which has the probability density function: (;) = {,,, <Suppose that a sample of data is available from which either the sample mean, ¯, or the sample median, m, can be calculated.
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 ...
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.
A likelihood function arises from a probability density function considered as a function of its distributional parameterization argument. For example, consider a model which gives the probability density function f X ( x ∣ θ ) {\displaystyle \;f_{X}(x\mid \theta )\;} of observable random variable X {\displaystyle \,X\,} as a function of a ...
For example, given a problem as ... , is from a natural exponential family. Moreover, the likelihood ratio is ... of an exponential distribution is a straight line ...