### Function Reference: `nakafit`

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

Estimate mean and confidence intervals for the Nakagami distribution.

`mu0 = nakafit (x)` returns the maximum likelihood estimates of the parameters of the Nakagami distribution given the data in x. `paramhat(1)` is the shape parameter, mu, and `paramhat(2)` is the spread parameter, omega.

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

`[…] = nakafit (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.

`[…] = nakafit (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))`.

`[…] = nakafit (params, x, censor, freq)` accepts a frequency vector, freq, of the same size as x. freq must contain non-negative integer frequencies for the corresponding elements in x. By default, or if left empty, `freq = ones (size (x))`.

`[…] = nakafit (…, 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 Nakagami distribution can be found at https://en.wikipedia.org/wiki/Nakagami_distribution

See also: nakacdf, nakainv, nakapdf, nakarnd, nakalike, nakastat

Source Code: nakafit

### Example: 1

     ## Sample 3 populations from different Nakagami distibutions
     randg ("seed", 5)  # for reproducibility
     r1 = nakarnd (0.5, 1, 2000, 1);
     randg ("seed", 2)   # for reproducibility
     r2 = nakarnd (5, 1, 2000, 1);
     randg ("seed", 7)   # for reproducibility
     r3 = nakarnd (2, 2, 2000, 1);
     r = [r1, r2, r3];

     ## Plot them normalized and fix their colors
     hist (r, [0.05:0.1:3.5], 10);
     h = findobj (gca, "Type", "patch");
     set (h(1), "facecolor", "c");
     set (h(2), "facecolor", "g");
     set (h(3), "facecolor", "r");
     ylim ([0, 2.5]);
     xlim ([0, 3.0]);
     hold on

     ## Estimate their MU and LAMBDA parameters
     mu_omegaA = nakafit (r(:,1));
     mu_omegaB = nakafit (r(:,2));
     mu_omegaC = nakafit (r(:,3));

     ## Plot their estimated PDFs
     x = [0.01:0.1:3.01];
     y = nakapdf (x, mu_omegaA(1), mu_omegaA(2));
     plot (x, y, "-pr");
     y = nakapdf (x, mu_omegaB(1), mu_omegaB(2));
     plot (x, y, "-sg");
     y = nakapdf (x, mu_omegaC(1), mu_omegaC(2));
     plot (x, y, "-^c");
     legend ({"Normalized HIST of sample 1 with μ=0.5 and ω=1", ...
              "Normalized HIST of sample 2 with μ=5 and ω=1", ...
              "Normalized HIST of sample 3 with μ=2 and ω=2", ...
              sprintf("PDF for sample 1 with estimated μ=%0.2f and ω=%0.2f", ...
                      mu_omegaA(1), mu_omegaA(2)), ...
              sprintf("PDF for sample 2 with estimated μ=%0.2f and ω=%0.2f", ...
                      mu_omegaB(1), mu_omegaB(2)), ...
              sprintf("PDF for sample 3 with estimated μ=%0.2f and ω=%0.2f", ...
                      mu_omegaC(1), mu_omegaC(2))})
     title ("Three population samples from different Nakagami distibutions")
     hold off