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Function Reference: burrfit

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

Estimate mean and confidence intervals for the Burr type XII distribution.

muhat = burrfit (x) returns the maximum likelihood estimates of the parameters of the Burr type XII distribution given the data in x. paramhat(1) is the scale parameter, lambda, paramhat(2) is the first shape parameter, c, and paramhat(3) is the second shape parameter, k

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

[…] = burrfit (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.

[…] = burrfit (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)).

[…] = burrfit (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)).

[…] = burrfit (…, options) specifies control parameters for the iterative algorithm used to compute the maximum likelihood estimates. options is a structure with the following field and its default value:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

Further information about the Burr type XII distribution can be found at https://en.wikipedia.org/wiki/Burr_distribution

See also: burrcdf, burrinv, burrpdf, burrrnd, burrlike, burrstat

Source Code: burrfit

Example: 1

  

 ## Sample 3 populations from different Burr type XII distibutions
 rand ("seed", 4);    # for reproducibility
 r1 = burrrnd (3.5, 2, 2.5, 10000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = burrrnd (1, 3, 1, 10000, 1);
 rand ("seed", 9);    # for reproducibility
 r3 = burrrnd (0.5, 2, 3, 10000, 1);
 r = [r1, r2, r3];

 ## Plot them normalized and fix their colors
 hist (r, [0.1:0.2:20], [18, 5, 3]);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "c");
 set (h(2), "facecolor", "g");
 set (h(3), "facecolor", "r");
 ylim ([0, 3]);
 xlim ([0, 5]);
 hold on

 ## Estimate their α and β parameters
 lambda_c_kA = burrfit (r(:,1));
 lambda_c_kB = burrfit (r(:,2));
 lambda_c_kC = burrfit (r(:,3));

 ## Plot their estimated PDFs
 x = [0.01:0.15:15];
 y = burrpdf (x, lambda_c_kA(1), lambda_c_kA(2), lambda_c_kA(3));
 plot (x, y, "-pr");
 y = burrpdf (x, lambda_c_kB(1), lambda_c_kB(2), lambda_c_kB(3));
 plot (x, y, "-sg");
 y = burrpdf (x, lambda_c_kC(1), lambda_c_kC(2), lambda_c_kC(3));
 plot (x, y, "-^c");
 hold off
 legend ({"Normalized HIST of sample 1 with λ=3.5, c=2, and k=2.5", ...
          "Normalized HIST of sample 2 with λ=1, c=3, and k=1", ...
          "Normalized HIST of sample 3 with λ=0.5, c=2, and k=3", ...
  sprintf("PDF for sample 1 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ...
          lambda_c_kA(1), lambda_c_kA(2), lambda_c_kA(3)), ...
  sprintf("PDF for sample 2 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ...
          lambda_c_kB(1), lambda_c_kB(2), lambda_c_kB(3)), ...
  sprintf("PDF for sample 3 with estimated λ=%0.2f, c=%0.2f, and k=%0.2f", ...
          lambda_c_kC(1), lambda_c_kC(2), lambda_c_kC(3))})
 title ("Three population samples from different Burr type XII distibutions")
 hold off
plotted figure