Beta parameter estimates

`phat = betafit(data)`

[phat,pci] = betafit(data,alpha)

`phat = betafit(data)`

computes
the maximum likelihood estimates of the beta distribution parameters *a* and *b* from
the data in the vector `data`

and returns a column
vector containing the *a* and *b* estimates,
where the beta cdf is given by

$$F(x|a,b)=\frac{1}{B(a,b)}{\displaystyle \underset{0}{\overset{x}{\int}}{t}^{a-1}}{(1-t)}^{b-1}dt$$

and *B*( · ) is the Beta function. 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, `betafit`

maximizes 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.

`[phat,pci] = betafit(data,alpha)`

returns
confidence intervals on the *a* and *b* parameters
in the 2-by-2 matrix `pci`

. The first column of the
matrix contains the lower and upper confidence bounds for parameter *a*,
and the second column contains the confidence bounds for parameter *b*.
The optional input argument `alpha`

is a value in
the range [0, 1] specifying the width of the confidence intervals.
By default, `alpha`

is `0.05`

, which
corresponds to 95% confidence intervals. The confidence intervals
are based on a normal approximation for the distribution of the logs
of the parameter estimates.

This example generates 100 beta distributed observations. The
true *a* and *b* parameters
are 4 and 3, respectively. Compare these to the values returned in `p`

by
the beta fit. Note that the columns of `ci`

both
bracket the true parameters.

data = betarnd(4,3,100,1); [p,ci] = betafit(data,0.01) p = 5.5328 3.8097 ci = 3.6538 2.6197 8.3781 5.5402

[1] Hahn, Gerald J., and S. S. Shapiro. *Statistical
Models in Engineering*. Hoboken, NJ: John Wiley &
Sons, Inc., 1994, p. 95.