Imprecise Functional Estimation: The Cumulative Distribution Case

Abstract

In this paper, we propose an adaptation of the Parzen Rosenblatt cumulative distribution function estimator that uses maxitive kernels. The result of this estimator, on every point of the domain of F, the cumulative distribution to be estimated, is interval valued instead of punctual valued. We prove the consistency of our approach with the classical Parzen Rosenblatt estimator, since, according to consistency conditions between the maxitive kernel involved in the imprecise estimator and the summative kernel involved in the precise estimator, our imprecise estimate contains the precise Parzen Rosenblatt estimate.

Pseudo-likelihood Estimation

“…the necessity measure N π x ∆ ). As shown in [6] when κ ∈ M(π), then ∀∆ ∈ IR + , ∀x ∈ Ω, F n κ∆ (x) ∈ F n π∆ (x).…”

“…It defines two dual confidence measures on Ω: a possibility measure Π π and a necessity measure N π by: ∀A ∈ P(Ω), Π π (A) = sup x∈A π(x) and N π (A) = 1 − sup x ∈A π(x). Based on [2], a maxitive kernel π defines a convex set M(π) of summative kernels [6]:…”

“…A maxitive kernel based imprecise estimate of the cumulative distribution function has been proposed in [6]. It is defined for all x ∈ Ω by:…”

“…The computation of the lower and the upper bounds of the imprecise cumulative distribution estimator, defined by (9), is given in [6] by:…”

Interval Estimation of Value-at-Risk Based on Nonparametric Models

“…However, as in the above mentioned methods, there is a systematic bias induced by the choice of the kernel used to estimate the CDF. Our proposal is to focus on the new nonparametric approach developed by Loquin and Strauss (2008a) to estimate the CDF. This approach makes use of the ability of a new kind of interpolating kernel, called maxitive kernel, to represent a convex set of conventional kernels-that we call summative kernels Loquin and Strauss (2008b).…”