Recursive and non-recursive kernel estimation of negative cumulative residual extropy under α-mixing dependence condition

The Shannon’s entropy function has a complementary dual function namely extropy and it facilitates the comparison of uncertainties of two random variables (see Lad et al. Stat Sci 30:40–58, 2015). Following the work of Lad et al. (Stat Sci 30:40–58, 2015), various generalizations/extensions of extropy measure are discussed in the literature analogous to that of Shannon’s entropy. Accordingly, a negative cumulative residual extropy is introduced by Tahmasebi and Toomaj (Commun Stat Theor Methods, 2020. https://doi.org/10.1080/03610926.2020.1831541). In the present work, we provide nonparametric kernel type estimators for the negative cumulative residual extropy based on the observations under study are dependent. Various properties including asymptotic properties of the proposed estimators are derived under suitable regularity conditions. A Monte-Carlo simulation study is carried out to find out the bias and mean squared error of the estimators.