Estimating Densities of Functions of Observations

Abstract

Density estimates, such as histograms and more sophisticated versions, are important in applied and theoretical statistics. In applied statistics, a density estimate gives the data analyst a graphical overview of the shape of the distribution. This overview allows the data analyst to arrive immediately at a qualitative impression of the location, scale, and various aspects of the extremes of the distribution. In theoretical statistics, the shape of 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 restriction of a priori imposing limitations of a class of parametric curves. In this article we introduce density estimation for functions of observations. To motivate the study, one type of function used is the interpoint distance between observations arising in spatial statistics from the fields of biometry and regional science. A second type of function considered is the sum 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. This asymptotic result is interesting because, 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.

Key Words:

• Insurance statistics
• Nonparametric estimation
• Spatial statistics
• U statistics