Statistics: bisafit

### Function Reference: `bisafit`

statistics: paramhat = bisafit (x)
statistics: [paramhat, paramci] = bisafit (x)
statistics: [paramhat, paramci] = bisafit (x, alpha)
statistics: […] = bisafit (x, alpha, censor)
statistics: […] = bisafit (x, alpha, censor, freq)
statistics: […] = bisafit (x, alpha, censor, freq, options)

Estimate mean and confidence intervals for the Birnbaum-Saunders distribution.

`muhat = bisafit (x)` returns the maximum likelihood estimates of the parameters of the Birnbaum-Saunders distribution given the data in x. `paramhat(1)` is the scale parameter, beta, and `paramhat(2)` is the shape parameter, gamma.

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

`[…] = bisafit (x, alpha)` also returns the `100 * (1 - alpha)` percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in `[]` for alpha to use the default values.

`[…] = bisafit (x, alpha, censor)` accepts a boolean vector, censor, of the same size as x with `1`s for observations that are right-censored and `0`s for observations that are observed exactly. By default, or if left empty, `censor = zeros (size (x))`.

`[…] = bisafit (x, alpha, censor, freq)` accepts a frequency vector, freq, of the same size as x. freq typically contains integer frequencies for the corresponding elements in x, but it can contain any non-integer non-negative values. By default, or if left empty, `freq = ones (size (x))`.

`[…] = bisafit (…, 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`

Further information about the Birnbaum-Saunders distribution can be found at https://en.wikipedia.org/wiki/Birnbaum%E2%80%93Saunders_distribution

See also: bisacdf, bisainv, bisapdf, bisarnd, bisalike, bisastat

Source Code: bisafit

### Example: 1

     ## Sample 3 populations from different Birnbaum-Saunders distibutions
     rand ("seed", 5);    # for reproducibility
     r1 = bisarnd (1, 0.5, 2000, 1);
     rand ("seed", 2);    # for reproducibility
     r2 = bisarnd (2, 0.3, 2000, 1);
     rand ("seed", 7);    # for reproducibility
     r3 = bisarnd (4, 0.5, 2000, 1);
     r = [r1, r2, r3];

     ## Plot them normalized and fix their colors
     hist (r, 80, 4.2);
     h = findobj (gca, "Type", "patch");
     set (h(1), "facecolor", "c");
     set (h(2), "facecolor", "g");
     set (h(3), "facecolor", "r");
     ylim ([0, 1.1]);
     xlim ([0, 8]);
     hold on

     ## Estimate their α and β parameters
     beta_gammaA = bisafit (r(:,1));
     beta_gammaB = bisafit (r(:,2));
     beta_gammaC = bisafit (r(:,3));

     ## Plot their estimated PDFs
     x = [0:0.1:8];
     y = bisapdf (x, beta_gammaA(1), beta_gammaA(2));
     plot (x, y, "-pr");
     y = bisapdf (x, beta_gammaB(1), beta_gammaB(2));
     plot (x, y, "-sg");
     y = bisapdf (x, beta_gammaC(1), beta_gammaC(2));
     plot (x, y, "-^c");
     hold off
     legend ({"Normalized HIST of sample 1 with β=1 and γ=0.5", ...
              "Normalized HIST of sample 2 with β=2 and γ=0.3", ...
              "Normalized HIST of sample 3 with β=4 and γ=0.5", ...
              sprintf("PDF for sample 1 with estimated β=%0.2f and γ=%0.2f", ...
                      beta_gammaA(1), beta_gammaA(2)), ...
              sprintf("PDF for sample 2 with estimated β=%0.2f and γ=%0.2f", ...
                      beta_gammaB(1), beta_gammaB(2)), ...
              sprintf("PDF for sample 3 with estimated β=%0.2f and γ=%0.2f", ...
                      beta_gammaC(1), beta_gammaC(2))})
     title ("Three population samples from different Birnbaum-Saunders distibutions")
     hold off