Fig. 3.

Quantile residual analysis of the empirically-estimated null CDF of the T_pb statistic versus the null CDF associated with the ${\chi }_4^2$ distribution, theoretically suggested by Fisher’s method. a The kernel density approximations of the distributions of the ${\chi }_4^2$ (left) and the empirically-estimated (right) quantile residuals demonstrate closer agreement of the latter with standard normality than the former. b The Q-Q plot (green marks) of the ${\chi }_4^2$ (left) and the empirically-estimated (right) quantile residuals against the corresponding true quantile residuals corroborate the result in (a)