Statistics: loglfit

### 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 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 `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))`.

`[…] = 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 μ, s, 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

### Example: 1

     ## 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