Nonparametric Density Estimation From Covariate Information

An increasing number of statistical problems arise in connection with functional calibration. In each case, inexpensive indirect data in a particular context are combined with direct expensive-to-acquire data from different but related settings to estimate quantities in the former case. Sometimes (e.g., in chemometrics problems where spectroscopic calibration is used) the indirect data are functional. But more commonly, they are scalar or vector-valued, and the functional component is the quantity that we wish to estimate. The problem treated here is of the latter type. We observe data that give us access to the distribution of U given V, and from these and data on U, we wish to estimate the density of V. The motivating real datasets are of age and covariate information in fish populations. We suggest two methodologies, each of which is based on transforming the problem to one involving inversion of a symmetric linear operator. Our techniques have connections to methods for functional data analysis and for a variety of mixture and deconvolution problems, as well as to calibration techniques.