Polygonal smoothing of the empirical distribution function
Let \( X_{1},\ldots,X_{n} \) be i.i.d. random variables with cumulative distribution function (cdf) \(F\). The most classical and simple estimate of the cdf \(F\) is the empirical distribution function (edf) \( F_{n}\left(x \right) = \frac{1}{n} \sum_{i=1}^{n} \mathbb{I}_{\left( -\infty,x\right] } \left(X_{i} \right) \), \( x\in \mathbb{R} \), where \( \mathbb{I}_{A} \) denotes the indicator function of the set \(A\). In the case, when \(F\) is absolutely continuous, the edf \(F_{n}\) is not continuous. There are several smoothing techniques to correct this drawback. For estimating \(F\), the authors consider a polygonal smoothing of the edf \(F_{n}\). From the authors' abstract: 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 \(F_n\) but its properties have not been studied deeply. First, consistency and exponential type inequalities are derived from well-known convergence properties of \(F_n\). Then, we study their mean integrated squared error (MISE) and we establish that polygonal estimators may improve the MISE of \(F_n\). We conclude by some numerical results to compare these estimators globally, and also together with the integrated kernel distribution estimator.
scientific article; zbMATH DE number 6916511
scientific article