Identifying the closest symmetric distribution or density function (Q1099525)

This paper addresses the problem of estimating the ``closest'' symmetric distribution or density function to the underlying distribution or density of a data sample, where closeness is measured by any of several (weighted) norms. The main results of the paper say that the ``closest'' symmetric distribution to the empirical distribution \(F_ n\) in various norms is of the form \[ \{F_ n(x)+1-F_ n((2\theta_ n-x)-)\}/2 \] for a suitable estimator \(\theta_ n\) of the ``best location of symmetry''. In the special cases of unweighted sup-norm and integrated square error norm, the explicit forms of the estimators \(\theta_ n\) are given. Analogous results are presented for closest symmetric densities.