MINIMIZING-DISTANCE BETWEEN DISTRIBUTION FUNCTIONS
Eugene F. Schuster
Abstract: The problem addressed is that of finding the closest distribution function in a
class of distributions to a given theoretical or empirical distribution function
in the -norm. Applications considered are those of estimating the center
of symmetry in the one-sample problem and in estimating the shift in the
two-sample problem by minimizing the -distance between suitably chosen
empirical distribution functions. In both cases, the minimizing is shown to be
Galton’s estimator. The closest symmetric distribution function to the empirical in
-norm is identified as the average of the empirical distribution function and the
empirical distribution function of the data reflected about Galton’s estimator. The
minimizing techniques employed can be used to give new proofs of the corresponding
results for the -norm where the minimizing is the Hodges-Lehmann estimator.
in a
class of distributions 
to a given theoretical or empirical distribution function
in the 
-norm. Applications considered are those of estimating the center
of symmetry 
in the one-sample problem and in estimating the shift 
in the
two-sample problem by minimizing the 
-distance between suitably chosen
empirical distribution functions. In both cases, the minimizing 
is shown to be
Galton’s estimator. The closest symmetric distribution function to the empirical in
-norm is identified as the average of the empirical distribution function and the
empirical distribution function of the data reflected about Galton’s estimator. The
minimizing techniques employed can be used to give new proofs of the corresponding
results for the 
-norm where the minimizing 
is the Hodges-Lehmann estimator.