Generalized inverse transformation method via representative points in statistical simulation

Discrete approximations of continuous random variables play a crucial role in various areas of research and application, offering advantages in computational efficiency, interpretability, and modeling flexibility. This paper investigates discrete representations of continuous random variables using the mean-squared error criterion (MSE-RPs) and the inverse transformation method. We introduce a novel discrete approximation to the normal distribution that surpasses conventional MSE-RPs obtained from the normal density, particularly in matching lower-order moments. Furthermore, we propose a two-step generalized inverse transformation method to generate approximate MSE-RPs of random variables, inspired by the remarkable performance of the inverse transformation method in statistical simulation. Overall, the generalized inverse transformation method offers a more efficient and reliable alternative for obtaining discrete approximations to target continuous distributions, especially in scenarios where explicit computation and derivation of density functions are challenging or computationally expensive. Moreover, we extend our investigation to the case where the target distribution is a convolution of two random variables, thereby expanding the applicability of our proposed method. Furthermore, our findings hold potential applications in Monte Carlo simulation and resampling techniques.