Table of Contents

## 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.