# betafit

Beta parameter estimates

## Syntax

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

## Description

`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

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

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.

## Examples

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```

## References

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