Moderate deviations for the kernel mode estimator and some applications

Let θ be the mode of a probability density and θn be its kernel estimator. In the case θn is known to fulfill a central limit theorem, we prove that it also satisfies a moderate deviations principle, and we apply this result to the analysis of confidence intervals for the mode. In the case the bandwidth is too small to be in the range where θn satisfies a central limit theorem, we obtain a moderate deviations upper bound for the kernel mode estimator; as an application of this result, we give an upper bound of the strong convergence rate of θn under assumptions under which only the strong consistency of the kernel mode estimator was known.