Estimation of discontinuous probability density function of class size data^{†}

Jib Huh^{1}

Received December 28, 2022; Revised January 19, 2023; Accepted January 9, 2023.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Abstract
- When the probability density function has a discontinuity point, Huh (2002) proposed a kernel estimator of the location and the jump size of the discontinuity point, and showed their asymptotic properties. The hypothesis testing method for the existence of a discontinuity point was explained using the asymptotic distribution of the proposed jump size estimator. On the other hand, Cline and Hart (1999) proposed a kernel estimator of the discontinuous probability density function using the method of Schuster (1985) because the discontinuity point has the same problem as the boundary point in the kernel estimator. Huh (2002) separated samples based on discontinuity points and estimated the discontinuous probability density function with a boundary kernel function. In this study, we introduce an algorithm for estimating the number of discontinuity points in the probability density function using the hypothesis testing method for the existence of discontinuity point introduced by Huh (2002). By the algorithm, the number of discontinuity points are estimated in the probability density function of the 5th grade class size data in Angrist and Lavy (1999). The probability density function of the class size data is estimated and analyzed using the estimated the number and the locations of discontinuity points.
**Keywords**: Boundary kernel function, discontinuity point, jump size, one-sided kernel function, probability density function.