Beta negative log-likelihood

`nlogL = betalike(params,data)`

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

`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.

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

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