### Function Reference: `betafit`

statistics: paramhat = betafit (x)
statistics: [paramhat, paramci] = betafit (x)
statistics: [paramhat, paramci] = betafit (x, alpha)
statistics: [paramhat, paramci] = betafit (x, alpha, freq)
statistics: [paramhat, paramci] = betafit (x, alpha, options)
statistics: [paramhat, paramci] = betafit (x, alpha, freq, options)

Estimate parameters and confidence intervals for the Beta distribution.

`paramhat = betafit (x)` returns the maximum likelihood estimates of the parameters of the Beta distribution given the data in vector x. `paramhat([1, 2])` corresponds to the α and β shape parameters, respectively. Missing values, `NaNs`, are ignored.

`[paramhat, paramci] = betafit (x)` returns the 95% confidence intervals for the parameter estimates.

`[…] = betafit (x, alpha)` also returns the `100 * (1 - alpha)` percent confidence intervals of the estimated parameter. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals.

`[…] = betafit (params, x, freq)` accepts a frequency vector, freq, of the same size as x. freq must contain non-negative integer frequencies for the corresponding elements in x. By default, or if left empty, `freq = ones (size (x))`.

`[paramhat, paramci] = nbinfit (x, alpha, options)` specifies control parameters for the iterative algorithm used to compute ML estimates with the `fminsearch` function. options is a structure with the following fields and their default values:

- `options.Display = "off"`
- `options.MaxFunEvals = 400`
- `options.MaxIter = 200`
- `options.TolX = 1e-6`

The Beta distribution is defined on the open interval (0,1). However, `betafit` can also compute the unbounded beta likelihood function for data that include exact zeros or ones. In such cases, zeros and ones are treated as if they were values that have been left-censored at `sqrt (realmin)` or right-censored at `1 - eps/2`, respectively.

Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution

See also: betacdf, betainv, betapdf, betarnd, betalike, betastat

Source Code: betafit

### Example: 1

     ## Sample 2 populations from different Beta distibutions
     randg ("seed", 1);   # for reproducibility
     r1 = betarnd (2, 5, 500, 1);
     randg ("seed", 2);   # for reproducibility
     r2 = betarnd (2, 2, 500, 1);
     r = [r1, r2];

     ## Plot them normalized and fix their colors
     hist (r, 12, 15);
     h = findobj (gca, "Type", "patch");
     set (h(1), "facecolor", "c");
     set (h(2), "facecolor", "g");
     hold on

     ## Estimate their shape parameters
     a_b_A = betafit (r(:,1));
     a_b_B = betafit (r(:,2));

     ## Plot their estimated PDFs
     x = [min(r(:)):0.01:max(r(:))];
     y = betapdf (x, a_b_A(1), a_b_A(2));
     plot (x, y, "-pr");
     y = betapdf (x, a_b_B(1), a_b_B(2));
     plot (x, y, "-sg");
     ylim ([0, 4])
     legend ({"Normalized HIST of sample 1 with α=2 and β=5", ...
              "Normalized HIST of sample 2 with α=2 and β=2", ...
              sprintf("PDF for sample 1 with estimated α=%0.2f and β=%0.2f", ...
                      a_b_A(1), a_b_A(2)), ...
              sprintf("PDF for sample 2 with estimated α=%0.2f and β=%0.2f", ...
                      a_b_B(1), a_b_B(2))})
     title ("Two population samples from different Beta distibutions")
     hold off