A comparative simulation study on the IFS distribution function estimator

Abstract

In this paper, we do a comparative simulation study of the standard empirical distribution function estimator versus a new class of nonparametric estimators of a distribution function F, called the iterated function system (IFS) estimator. The target distribution function F is supposed to have compact support. The IFS estimator of a distribution function F is considered as the fixed point of a contractive operator T defined in terms of a vector of parameters p and a family of affine maps W which can be both dependent on the sample ( X 1 , X 2 , … , X n ) . Given W , the problem consists in finding a vector p such that the fixed point of T is “sufficiently near” to F. It turns out that this is a quadratic constrained optimization problem that we propose to solve by penalization techniques. Analytical results prove that IFS estimators for F are asymptotically equivalent to the empirical distribution function (EDF) estimator. We will study the relative efficiency of the IFS estimators with respect to the empirical distribution function for small samples via the Monte Carlo approach. For well-behaved distribution functions F and for a particular family of the so-called wavelet maps the IFS estimators can be dramatically better than the empirical distribution function in the presence of missing data, i.e. when it is only possible to observe data on subsets of the whole support of F.