The selection of the number of terms in an orthogonal series cumulative function estimator

Let h(.) be a continuous, strictly positive probability density function over an interval [a, b] and H(.) its associated cumulative distribution function (cdf). Given a sample set \(X_{1},\ldots ,X_{n}\) of independent identically distributed variables, we want to estimate H(.) from this sample set. The present work has two goals. The first one is to propose an estimator of a cdf based on an orthogonal trigonometric series and to give its statistical and asymptotic proprieties (bias, variance, mean square error, mean integrated squared error, convergence of the bias, convergence of variance, convergence of the mean squared error, convergence of the mean integrated squared error, uniform convergence in probability and the rate of convergence of the mean integrated squared error). The second is to introduce a new method for the selection of a “smoothing parameter”. The comparison by simulation between this method and Kronmal–Tarter’s method, shows that the new method is more performant in the sense of the mean integrated square error.