  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.
