Documentation

betalike

Beta negative log-likelihood

Syntax

nlogL = betalike(params,data)
[nlogL,AVAR] = betalike(params,data)

Description

nlogL = betalike(params,data) returns the negative of the beta log-likelihood function for the beta parameters a and b specified in vector params and the observations specified in the column vector data.

The elements of data must lie in the open interval (0, 1), where the beta distribution is defined. However, it is sometimes also necessary to fit a beta distribution to data that include exact zeros or ones. For such data, the beta likelihood function is unbounded, and standard maximum likelihood estimation is not possible. In that case, betalike computes a modified likelihood that incorporates the zeros or ones by treating them as if they were values that have been left-censored at sqrt(realmin) or right-censored at 1-eps/2, respectively.

[nlogL,AVAR] = betalike(params,data) also returns AVAR, which is the asymptotic variance-covariance matrix of the parameter estimates if the values in params are the maximum likelihood estimates. AVAR is the inverse of Fisher's information matrix. The diagonal elements of AVAR are the asymptotic variances of their respective parameters.

betalike is a utility function for maximum likelihood estimation of the beta distribution. The likelihood assumes that all the elements in the data sample are mutually independent. Since betalike returns the negative beta log-likelihood function, minimizing betalike using fminsearch is the same as maximizing the likelihood.

Examples

This example continues the betafit example, which calculates estimates of the beta parameters for some randomly generated beta distributed data.

r = betarnd(4,3,100,1);
[nlogl,AVAR] = betalike(betafit(r),r)
nlogl =

-27.5996

AVAR =

0.2783    0.1316
0.1316    0.0867