Nonparametric Estimation and Testing on Discontinuity of Positive   Supported Densities: A Kernel Truncation Approach
  Discontinuity in density functions is of economic importance and interest. For instance, in studies on regression discontinuity designs, discontinuity in the density of a running variable suggests violation of the no-manipulation assumption. In this paper we develop estimation and testing procedures on discontinuity in densities with positive support. Our approach is built on splitting the gamma kernel (Chen, 2000) into two parts at a given (dis)continuity point and constructing two truncated kernels. The jump-size magnitude of the density at the point can be estimated nonparametrically by two kernels and a multiplicative bias correction method. The estimator is easy to implement, and its convergence properties are delivered by various approximation techniques on incomplete gamma functions. Based on the jump-size estimator, two versions of test statistics for the null of continuity at a given point are also proposed. Moreover, estimation theory of the entire density in the presence of a discontinuity point is explored. Monte Carlo simulations confirm nice finite-sample properties of the jump-size estimator and the test statistics. 
Benedikt Funke and Masayuki Hirukawa
math.ST stat.TH