The asymptotic distributions of kernel estimators of the mode

Summary In a decreasing sequence of intervals centered on the true mode the normalized kernel estimate of the density converges weakly to a nonstationary Gaussian random process. The expected value of this process is a parabola through the origin. The covariance function of this process depends on the smoothness of the kernel. When the kernel is mean-square differentiable the location of the maximum of this process has a normal distribution. When the kernel is discontinuous the location of the maximum has a distribution related to a solution of the heat equation.Research supported in part by the National Science Foundation under grant MCS-78-02422 and MCS-80-05115 to Carnegie-Mellon University