On Uses of Mean Absolute Deviation: Shape Exploring and Distribution Function Estimation

TL;DR

This paper explores the use of mean absolute deviation for data visualization and proposes new nonparametric methods for estimating the cumulative distribution function, demonstrating improved accuracy over classical estimators through simulations and real data applications.

Contribution

It introduces a general nonparametric class of distribution function estimators based on mean absolute deviation, including novel Richardson extrapolation methods.

Findings

Richardson extrapolation improves mean squared error over classical estimators

Proposed methods perform comparably to spline-based approaches for small samples

Application to real data demonstrates practical utility in hazard concentration estimation

Abstract

Mean absolute deviation function is used to explore the pattern and the distribution of the data graphically to enable analysts gaining greater understanding of raw data and to foster quick and a deep understanding of the data as an important fundament for successful data analytic. Furthermore, new nonparametric approaches for estimating the cumulative distribution function based on the mean absolute deviation function are proposed. These new approaches are meant to be a general nonparametric class that includes the empirical distribution function as a special case. Simulation study reveals that the Richardson extrapolation approach has a major improvement in terms of average squared errors over the classical empirical estimators and has comparable results with smooth approaches such as cubic spline and constrained linear spline for practically small samples. The properties of the proposed estimators are studied. Moreover, the Richardson approach applied for real data application and used to estimate the hazardous concentration five percent.