Statistics: invgfit

### Function Reference: `invgfit`

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

Estimate mean and confidence intervals for the inverse Gaussian distribution.

`mu0 = invgfit (x)` returns the maximum likelihood estimates of the parameters of the inverse Gaussian distribution given the data in x. `paramhat(1)` is the scale parameter, mu, and `paramhat(2)` is the shape parameter, lambda.

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

`[…] = invgfit (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.

`[…] = invgfit (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))`.

`[…] = invgfit (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))`.

`[…] = invgfit (…, 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 inverse Gaussian distribution can be found at https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

See also: invgcdf, invginv, invgpdf, invgrnd, invglike

Source Code: invgfit

### Example: 1

     ## Sample 3 populations from different inverse Gaussian distibutions
     rand ("seed", 5); randn ("seed", 5);   # for reproducibility
     r1 = invgrnd (1, 0.2, 2000, 1);
     rand ("seed", 2); randn ("seed", 2);   # for reproducibility
     r2 = invgrnd (1, 3, 2000, 1);
     rand ("seed", 7); randn ("seed", 7);   # for reproducibility
     r3 = invgrnd (3, 1, 2000, 1);
     r = [r1, r2, r3];

     ## Plot them normalized and fix their colors
     hist (r, [0.1:0.1:3.2], 9);
     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, 3]);
     hold on

     ## Estimate their MU and LAMBDA parameters
     mu_lambdaA = invgfit (r(:,1));
     mu_lambdaB = invgfit (r(:,2));
     mu_lambdaC = invgfit (r(:,3));

     ## Plot their estimated PDFs
     x = [0:0.1:3];
     y = invgpdf (x, mu_lambdaA(1), mu_lambdaA(2));
     plot (x, y, "-pr");
     y = invgpdf (x, mu_lambdaB(1), mu_lambdaB(2));
     plot (x, y, "-sg");
     y = invgpdf (x, mu_lambdaC(1), mu_lambdaC(2));
     plot (x, y, "-^c");
     hold off
     legend ({"Normalized HIST of sample 1 with μ=1 and λ=0.5", ...
              "Normalized HIST of sample 2 with μ=2 and λ=0.3", ...
              "Normalized HIST of sample 3 with μ=4 and λ=0.5", ...
              sprintf("PDF for sample 1 with estimated μ=%0.2f and λ=%0.2f", ...
                      mu_lambdaA(1), mu_lambdaA(2)), ...
              sprintf("PDF for sample 2 with estimated μ=%0.2f and λ=%0.2f", ...
                      mu_lambdaB(1), mu_lambdaB(2)), ...
              sprintf("PDF for sample 3 with estimated μ=%0.2f and λ=%0.2f", ...
                      mu_lambdaC(1), mu_lambdaC(2))})
     title ("Three population samples from different inverse Gaussian distibutions")
     hold off