bisafit
              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 1s for
 observations that are right-censored and 0s 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
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 ## 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
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