Polygonal smoothing of the empirical distribution function

Delphine Blanke (1) , Denis Bosq (2)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We present two families of polygonal estimators of the distribution function: the first family is based on the knowledge of the support while the second addresses the case of an unknown support. Polygonal smoothing is a simple and natural method for regularizing the empirical distribution function Fn but its properties have not been studied deeply. First, consistency and exponential type inequalities are derived from well-known convergence properties of Fn. Then, we study their mean integrated squared error (MISE) and we establish that polygonal estimators may improve the MISE of Fn. We conclude by some numerical results to compare these estimators globally, and also together with the integrated kernel distribution estimator.

Keywords: Distribution function estimation; Polygonal estimator; Cumulative frequency polygon; Order statistics; Mean integrated squared error; Exponential inequality; Smoothed processes

Domains: Mathematics [math]; Statistics [math.ST]

Delphine Blanke, Denis Bosq. Polygonal smoothing of the empirical distribution function. Statistical Inference for Stochastic Processes, 2018, 21 (2), pp.263-287. ⟨10.1007/s11203-018-9183-y⟩. ⟨hal-02062903⟩