Parametric versus nonparametric: The fitness coefficient

Olkin and Spiegelman introduced a semiparametric estimator of the density defined as a mixture between the maximum likelihood estimator and the kernel density estimator. Due to the absence of any leave‐one‐out strategy and the hardness of estimating the Kullback–Leibler loss of kernel density estimate, their approach produces unsatisfactory results. This article investigates an alternative approach in which only the kernel density estimate is modified. From a theoretical perspective, the estimated mixture parameter is shown to converge in probability to one if the parametric model is true and to zero otherwise. From a practical perspective, the utility of the approach is illustrated on real and simulated data sets.