The discrete q-Gaussian distribution [formula omitted]: Properties and parameters estimation

Abstract

We introduce a new discrete distribution, called the centered reduced discrete q-Gaussian Nq(0,1). This distribution connects classical Gaussian, discrete Uniform, and quantum q-Gaussian distributions. In this paper, we extend Nq(0,1) to Nq(μ,σ2), overcoming a limitation of some q-distributions like Diaz and Pariguan's q-Gaussian. Notably, Nq(0,1) has distinct shapes and parameters from the classical counterpart, providing additional flexible modeling approach. Results show the suggested discrete q-Gaussian as a useful alternative to the classical Gaussian for modeling data with hollow values or heavy-tailed tails. We explore properties of Nq(μ,σ2) and apply moments and maximum likelihood methods to estimate its parameters. Our analysis yields a key result on the concavity of the likelihood function, enabling efficient optimization algorithms for parameters estimation. Furthermore, we investigate a finite mixture of discrete q-Gaussians and apply the EM algorithm for parameters estimation. Finally, we conduct a simulation study to evaluate the model and estimation methods.