Estimating Densities of Functions of Observations

Density estimates, such as histograms and more sophisticated versions, are important in applied and theoretical statistics. In applied statistics, a density estimate provides the data analyst witha a graphical overview of the shape of the distribution. In theoretical statistics, yje shape pf the density allows the researcher to link the data to families of curves, perhaps indexed parametrically. By estimating a density nonparametrically, certain aspects of the data can be viewed without the limitation a priori imposing limitations of a class of paramtric curves. In this paper, we introduce density for functions of observations. To motivate the study, one type of function that is used in the interpoint distance between observtions arising in spatial statistics from the fields of biometry and regional science. A second type of function consider are the sums of observations as might occur in claims models in insurance. The nonparametric density estimates are introduced and certain computational issues are discussed. A central limit theorem for the estimator is provided. What is interesting about this asymptotic result is that, under certain mild conditions, the density estimate enjoys a rate of convergence similar to parametric estimates. This rate of convergence is much faster than the usual rate of convergence in nonparametric density estimation