loglfit
              Estimate mean and confidence intervals for the log-logistic distribution.
 mu0 = loglfit (x) returns the maximum likelihood
 estimates of the parameters of the log-logistic distribution given the data
 in x.  paramhat(1) is the scale parameter, a, and
 paramhat(2) is the shape parameter, b.
 [paramhat, paramci] = loglfit (x) returns the 95%
 confidence intervals for the parameter estimates.
 […] = loglfit (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.
 […] = loglfit (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)).
 […] = loglfit (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)).
 […] = loglfit (…, 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 log-logistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution
MATLAB compatibility: MATLAB uses an alternative parameterization given by the pair , i.e. mu and s, in analogy with the logistic distribution. Their relation to the a and b parameters is given below:
a = exp (mu)
 b = 1 / s
 See also: loglcdf, loglinv, loglpdf, loglrnd, logllike
Source Code: loglfit
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 ## Sample 3 populations from different log-logistic distibutions
 rand ("seed", 5)  # for reproducibility
 r1 = loglrnd (1, 1, 2000, 1);
 rand ("seed", 2)   # for reproducibility
 r2 = loglrnd (1, 2, 2000, 1);
 rand ("seed", 7)   # for reproducibility
 r3 = loglrnd (1, 8, 2000, 1);
 r = [r1, r2, r3];
 ## Plot them normalized and fix their colors
 hist (r, [0.05:0.1:2.5], 10);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 3.5]);
 xlim ([0, 2.0]);
 hold on
 ## Estimate their MU and LAMBDA parameters
 a_bA = loglfit (r(:,1));
 a_bB = loglfit (r(:,2));
 a_bC = loglfit (r(:,3));
 ## Plot their estimated PDFs
 x = [0.01:0.1:2.01];
 y = loglpdf (x, a_bA(1), a_bA(2));
 plot (x, y, "-pr");
 y = loglpdf (x, a_bB(1), a_bB(2));
 plot (x, y, "-sg");
 y = loglpdf (x, a_bC(1), a_bC(2));
 plot (x, y, "-^c");
 legend ({"Normalized HIST of sample 1 with α=1 and β=1", ...
          "Normalized HIST of sample 2 with α=1 and β=2", ...
          "Normalized HIST of sample 3 with α=1 and β=8", ...
          sprintf("PDF for sample 1 with estimated α=%0.2f and β=%0.2f", ...
                  a_bA(1), a_bA(2)), ...
          sprintf("PDF for sample 2 with estimated α=%0.2f and β=%0.2f", ...
                  a_bB(1), a_bB(2)), ...
          sprintf("PDF for sample 3 with estimated α=%0.2f and β=%0.2f", ...
                  a_bC(1), a_bC(2))})
 title ("Three population samples from different log-logistic distibutions")
 hold off
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