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Function Reference: loglfit

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

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 mean parameter, mu, and paramhat(2) is the scale parameter, sigma.

[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 loglogistic distribution can be found at https://en.wikipedia.org/wiki/Log-logistic_distribution

OCTAVE/MATLAB use an alternative parameterization given by the pair μ, σ, i.e. mu and sigma, in analogy with the logistic distribution. Their relation to the α and b parameters used in Wikipedia are given below:

  • mu = log (a)
  • sigma = 1 / a

See also: loglcdf, loglinv, loglpdf, loglrnd, logllike, loglstat

Source Code: loglfit

Example: 1

Sample 3 populations from different log-logistic distributions

 rand ('seed', 5)  # for reproducibility
 r1 = loglrnd (0, 1, 2000, 1);
 rand ('seed', 2)   # for reproducibility
 r2 = loglrnd (0, 0.5, 2000, 1);
 rand ('seed', 7)   # for reproducibility
 r3 = loglrnd (0, 0.125, 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 distributions')
 hold off
plotted figure