### Function Reference: `tlsfit`

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

Estimate parameters and confidence intervals for the Location-scale Student’s T distribution.

`muhat = tlsfit (x)` returns the maximum likelihood estimates of the parameters of the location-scale T distribution given the data in x. `paramhat(1)` is the location parameter, mu, `paramhat(2)` is the scale parameter, sigma, and `paramhat(3)` is the degrees of freedom, nu.

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

`[…] = tlsfit (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.

`[…] = tlsfit (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))`.

`[…] = tlsfit (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))`.

`[…] = tlsfit (…, 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.TolX = 1e-6`

Further information about the location-scale Student’s T distribution can be found at https://en.wikipedia.org/wiki/Student%27s_t-distribution#Location-scale_t_distribution

See also: tlscdf, tlsinv, tlspdf, tlsrnd, tlslike, tlsstat

Source Code: tlsfit

### Example: 1

     ## Sample 3 populations from 3 different location-scale T distibutions
     randn ("seed", 1);    # for reproducibility
     randg ("seed", 2);    # for reproducibility
     r1 = tlsrnd (-4, 3, 1, 2000, 1);
     randn ("seed", 3);    # for reproducibility
     randg ("seed", 4);    # for reproducibility
     r2 = tlsrnd (0, 3, 1, 2000, 1);
     randn ("seed", 5);    # for reproducibility
     randg ("seed", 6);    # for reproducibility
     r3 = tlsrnd (5, 5, 4, 2000, 1);
     r = [r1, r2, r3];

     ## Plot them normalized and fix their colors
     hist (r, [-21:21], [1, 1, 1]);
     h = findobj (gca, "Type", "patch");
     set (h(1), "facecolor", "c");
     set (h(2), "facecolor", "g");
     set (h(3), "facecolor", "r");
     ylim ([0, 0.25]);
     xlim ([-20, 20]);
     hold on

     ## Estimate their lambda parameter
     mu_sigma_nuA = tlsfit (r(:,1));
     mu_sigma_nuB = tlsfit (r(:,2));
     mu_sigma_nuC = tlsfit (r(:,3));

     ## Plot their estimated PDFs
     x = [-20:0.1:20];
     y = tlspdf (x, mu_sigma_nuA(1), mu_sigma_nuA(2), mu_sigma_nuA(3));
     plot (x, y, "-pr");
     y = tlspdf (x, mu_sigma_nuB(1), mu_sigma_nuB(2), mu_sigma_nuB(3));
     plot (x, y, "-sg");
     y = tlspdf (x, mu_sigma_nuC(1), mu_sigma_nuC(2), mu_sigma_nuC(3));
     plot (x, y, "-^c");
     hold off
     legend ({"Normalized HIST of sample 1 with μ=0, σ=2 and nu=1", ...
              "Normalized HIST of sample 2 with μ=5, σ=2 and nu=1", ...
              "Normalized HIST of sample 3 with μ=3, σ=4 and nu=3", ...
              sprintf("PDF for sample 1 with estimated μ=%0.2f, σ=%0.2f, and ν=%0.2f", ...
                      mu_sigma_nuA(1), mu_sigma_nuA(2), mu_sigma_nuA(3)), ...
              sprintf("PDF for sample 2 with estimated μ=%0.2f, σ=%0.2f, and ν=%0.2f", ...
                      mu_sigma_nuB(1), mu_sigma_nuB(2), mu_sigma_nuB(3)), ...
              sprintf("PDF for sample 3 with estimated μ=%0.2f, σ=%0.2f, and ν=%0.2f", ...
                      mu_sigma_nuC(1), mu_sigma_nuC(2), mu_sigma_nuC(3))})
     title ("Three population samples from different location-scale T distibutions")
     hold off