What distributions are in the exponential family?

What distributions are in the exponential family?

The normal, exponential, log-normal, gamma, chi-squared, beta, Dirichlet, Bernoulli, categorical, Poisson, geometric, inverse Gaussian, von Mises and von Mises-Fisher distributions are all exponential families. Some distributions are exponential families only if some of their parameters are held fixed.

Is maximum likelihood estimator of exponential distribution biased?

In this case, the MLE estimate of the rate parameter λ of an exponential distribution Exp(λ) is biased, however, the MLE estimate for the mean parameter µ = 1/λ is unbiased. Thus, the exponential distribution makes a good case study for understanding the MLE bias.

Is exponential distribution exponential family?

The exponential distribution is a one-parameter exponential family (appropriately enough), in the rate parameter r ∈ ( 0 , ∞ ) . The gamma distribution is a two-parameter exponential family in the shape parameter k ∈ ( 0 , ∞ ) and the scale parameter b ∈ ( 0 , ∞ ) .

What is maximum likelihood distribution?

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.

Is exponential family convex?

The cumulant function of an exponential family is a lower semicontinuous proper convex function.

What is the maximum likelihood estimator of λ?

STEP 1 Calculate the likelihood function L(λ). log(xi!) STEP 3 Differentiate logL(λ) with respect to λ, and equate the derivative to zero to find the m.l.e.. Thus the maximum likelihood estimate of λ is ̂λ = ¯x STEP 4 Check that the second derivative of log L(λ) with respect to λ is negative at λ = ̂λ.

Is beta distribution exponential family?

The family of beta(α,β) distributions is an exponential family. η is called the natural parameter.