Nonparametric estimation of unrestricted distributions and their jumps
Mynbaev K. Martins-Filho C. Henderson D.J.
June 2022Statistical Society of Canada
Canadian Journal of Statistics
2022#50Issue 2638 - 662 pp.
We consider-135 nonparametric estimation of a distribution function F associated with a random variable X based on a random sample. First, for x, a point of continuity of F, we define a class of estimators for F(x) and obtain their rates of convergence. Contrary to the existing literature, we impose no restriction on the existence or smoothness of the derivatives of F. The traditional kernel estimator for F(x) is a member of the class. Second, for x that is either the location of a jump discontinuity or an isolated point of the support, we define a class of estimators for the jump px = F(x) − F(x−) and obtain their rates of convergence. Again, no additional restriction is imposed on F beyond right-continuity. Our results are of significant practical use, as there are numerous examples in economics, finance and biomedicine of distributions that have point masses. Our main insight is also applied to obtain new inversion theorems for characteristic functions and explicit estimates for convergence rates. A small simulation study provides some evidence on the finite sample properties of our proposed estimators and contrasts their performance with some existing alternatives. An empirical section illustrates the use of our estimators using data on global elevation and data associated with “P-hacking” in economics journals.
Discontinuous distribution functions , estimation of distribution function jumps , inversion theorems , nonparametric distribution estimation
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New Economic School, Satbayev University, Almaty, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
Department of Economics, University of Colorado, Boulder, CO, United States
Department of Economics, Finance and Legal Studies, University of Alabama, Tuscaloosa, AL, United States
School of Mathematical Sciences, Nankai University, Tianjin, China
New Economic School
Institute of Mathematics and Mathematical Modeling
Department of Economics
Department of Economics
School of Mathematical Sciences
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