We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.
Polytomous discrimination index is a novel and important diagnostic accuracy measure for multi-category classification. After reconstructing its probabilistic definition, we propose a nonparametric approach to the estimation of polytomous discrimination index based on an empirical sample of biomarker values. In this paper, we provide the finite-sample and asymptotic properties of the proposed estimators and such analytic results may facilitate the statistical inference. Simulation studies are performed to examine the performance of the nonparametric estimators. Two real data examples are analysed to illustrate our methodology.
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