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Nonparametric mixture models are commonly used to estimate the number of unobserved species for their robustness of modelling heterogeneity. In particular, Poisson mixtures are popular in this regard, but they are also known to have the boundary problem that causes unstable estimation. A family of shape-restricted distributions known as discrete k -monotone distributions is proposed for species richness estimation. These distributions are in fact also nonparametric mixtures and can thus be fitted rapidly via some algorithms that are made available recently. As shown by empirical studies, as compared with Poisson mixtures, their use avoids the boundary problem and gives more stable and more accurate estimates. [ABSTRACT FROM AUTHOR]

Highlights • A new nabla Taylor formula expanded at b ≥ n is derived for a discrete function. • Two kinds of series representations of are derived for nabla discrete fractional sum/difference. • Some essential properties of nabla discrete fractional calculus are investigated. • Besides fixed initial instant case, nabla fractional calculus with fixed memory step are introduced. Abstract This paper addresses the description and analysis problems of nabla discrete fractional calculus. The series representation framework is developed first, including two Taylor series expanded at the initial instant and the current time, respectively. Under this framework, several essential properties of fractional sum/difference are presented and investigated. Notably, the short memory principle is introduced for nabla discrete fractional calculus, along with which two corresponding series are studied. [ABSTRACT FROM AUTHOR]