Joint Behaviour of Semirecursive Kernel Estimators of the Location and of the Size of the Mode of a Probability Density Function

Let θ and μ denote the location and the size of the mode of a probability density. We study the joint convergence rates of semirecursive kernel estimators of θ and μ. We show how the estimation of the size of the mode allows measuring the relevance of the estimation of its location. We also enlighten that, beyond their computational advantage on nonrecursive estimators, the semirecursive estimators are preferable to use for the construction of confidence regions.