Kernel estimators for q-fractional diffusion processes with random effects using q-calculus

Abstract

. The main purpose of this article is to investigate the kernel estimators for a class of q-analog of fractional stochastic differential equations (q-FSDE) with random effects. Using q-calculus, we first present some properties of the kernel density estimators, such as Bias, variance and we need to introduce the q-analog of Lyapunov’s central limit theorem to prove the q-analog of asymptotic normality of kernel density estimators. Our intention is to use some basic concepts of q-calculus to study the asymptotic behavior of the kernel density estimators for the whole range H ∈ ( 1 2 , 1 ) . Eventually, we provide an illustrative example, namely q-fractional Langevin equation, to validate the efficacy of our outcomes.