Apart from the theoretical contributions, non-central chi-squared distribution has been applied in mathematical physics, social and behavioral research, communication theory, management science and many other diversified research areas. Non-central chi-squared distribution has been derived mostly by considering the non-centrality parameter  and sometimes by taking the non-centrality parameter  Here we also consider an alternative non-centrality parameter  We calculated the parametric characteristics for each of the three densities using cumulant generating function and recurrence relation. Finally, we made a head-to-head comparison among the characteristics obtained from these three unique densities. It is found that only second non-centrality parameter gives us exact correspondence in parametric characteristics calculated using both the methods of calculation.

cumulants, kurtosis, moment generating function, non-central beta distribution, non-central F distribution, non-central family of densities, non-central t distribution, skewness.