Abstract

This paper focuses mainly on the problem of computing the (Formula presented.), (Formula presented.), moment of a random variable (Formula presented.) in which the (Formula presented.) ’s are positive real numbers and the (Formula presented.) ’s are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of (Formula presented.), especially when not all (Formula presented.) ’s are equal. We analytically solve this problem by showing that the (Formula presented.) moment of (Formula presented.) can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the (Formula presented.) moment of (Formula presented.) when (Formula presented.) is a combination of statistically independent (Formula presented.) and (Formula presented.) in which the (Formula presented.) ’s are distributed according to normal or Maxwell–Boltzmann distributions and the (Formula presented.) ’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding (Formula presented.) conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.