probplot for Chi2 and noncentral chi2 distribution

I am looking at GLR tests and I need to verify that my test statistic is chi2/noncentrally chi2 distributed. As far as I can see there isn't a way to make a chi2 probplot, am I mistaken or is there a workaround?

Answer 1

[Sai Pavan]

Hello Morten,

I understand that you are looking for ways to verify whether the test statistic is chi-square or noncentral chi-square distributed.

The "probplot" function automatically generates a probability plot comparing the empirical distribution of a dataset to a specified theoretical distribution. As "probplot" doesn't natively support chi-square or noncentral chi-square distributions out of the box, we can create a workaround by using the theoretical quantiles from the chi-square or noncentral chi-square distribution as a custom theoretical distribution in "probplot" to compare against the data.

Please refer to the below code snippet that illustrates this work around:

% Sample data
data1 = randn(100,1).^2; % Example dataset

% Chi-square distribution parameters
df1 = 2; % Degrees of freedom

% Generate theoretical quantiles
p1 = ((1:length(data1)) - 0.5) / length(data1); % Empirical CDF values
theoreticalQuantiles1 = chi2inv(p1, df1);

% Generate probability plot
probplot(data1);
hold on;
plot(sort(data1), theoreticalQuantiles1, 'r-', 'LineWidth', 2);
hold off;
legend('Empirical Data', 'Chi-square Theoretical Quantiles', 'Location', 'best');
title('Probability Plot against Chi-square Distribution');

% Sample data
data2 = ncx2rnd(2, 5, 100, 1); % Example dataset with df=2, noncentrality=5

% Noncentral chi-square distribution parameters
df2 = 2; % Degrees of freedom
noncentrality = 5; % Noncentrality parameter

% Generate theoretical quantiles
p2 = ((1:length(data2)) - 0.5) / length(data2); % Empirical CDF values
theoreticalQuantiles2 = ncx2inv(p2, df2, noncentrality);

% Generate probability plot
figure
probplot(data2);
hold on;
plot(sort(data2), theoreticalQuantiles2, 'r-', 'LineWidth', 2);
hold off;
legend('Empirical Data', 'Noncentral Chi-square Theoretical Quantiles', 'Location', 'best');
title('Probability Plot against Noncentral Chi-square Distribution');

Please refer to the below documentation to learn more about:

• "chi2inv" function: https://www.mathworks.com/help/stats/chi2inv.html
• "ncx2inv" function: https://www.mathworks.com/help/stats/ncx2inv.html

Hope it helps!