Infinite density at the median and the typical shape of stock return distributions

Statistics are developed to test for the presence of an asymptotic discontinuity (or infinite density or peakedness) in a probability density at the median. The approach makes use of work by Knight (1998) on L₁ estimation asymptotics in conjunction with nonparametric kernel density estimation methods. The size and power of the tests are assessed, and conditions under which the tests have good performance are explored in simulations. The new methods are applied to stock returns of leading companies across major U.S. industry groups. The results confirm the presence of infinite density at the median as a new significant empirical evidence for stock return distributions.