Frequency Polygons: Theory and Application

Abstract

In this article I investigate the theoretical properties and applications of the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. For estimating an unknown probability density function using a random sample, the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error, achieving the same rate of convergence to zero of the integrated mean squared error as non-negative kernel estimators. Data-based algorithms for constructing frequency polygons are discussed and illustrated. One is based on a histogram with bin width equal to 2.15sn-"', where s is an estimate of the standard deviation from a sample of size n. Another approach is based on the method of generalized crossvalidation. The bivariate frequency polygon is also investigated. Comparisons are made between the frequency polygon and other density estimators, Examples with data in one and two dimensions are presented.

Piecewise linear density estimation for sampled data

“…In both Theorems 4.2 and 4.3 we exhibit (in lim sup) the same n −4/5 -consistency obtained in Scott [27] with i.i.d. observations.…”

“…It is noteworthy that elementary estimators may also be efficient from a theoretical viewpoint. Thus and despite its high simplicity, the frequency polygon -defined in dimension one as the linear interpolant of the mid-points of an equally spaced histogram -is known to be as good as some more sophisticated density estimators in terms of MISE (see Scott [27]). …”

Calibrating histograms with application to economic data

“…In fact, it r~ay be shown that the first differences of a histogram are not consistent with the first derivative, and that an optimal histogram asymptotically has numerous bumps, many only a bin or two wide (recall the estimator of R(f') in equation (8) has a negative sign on the second term). It may be shown that the frequency polygon, a density estimator constructed by linear interpolation of the midpoints of a histogram, is consistent for both the density and its derivative when optimized with respect to MISE (Scott 1985a). The biased cross-validation estimate of the MISE of a frequency polygon is Of course, it is possible that a more statistically powerful density estimator such as the averaged shifted histogram (Scott 1985b) could in fact reveal the presence of a second mode, but at this point the work of the statisticians has to be completed by economic interpretation.…”

Vitesses optimale et suroptimale des polygones de fréquences pour les processus à temps continu

“…L'erreur quadratique intégrée du polygone de fréquences a déjà été étudiée dans les cas i.i.d. [8] et mélangeant [2]. Les résultats suivants apportent une majoration et des vitesses de convergence pour l'EQI en temps continu.…”