Empirical regression as a conditional expected value of a special distribution-mixture for a modelfree quantitative recording of stochastical relations

An “empirical” distribution function F̂(x, y) is estimated from measured points (xi, yi), i =1(1)n, of a continuous two‐dimensional random variable (X, Y) with unknown continuous density function f(x, y). The density function F̂(x, y) of F̂(x, y) is a mixture of n two‐dimensional normal densities. The first order moments of F̂(x, y) are the sample means x and y, whilst the second order moments are only proportional to the sample variances and the sample covariance. This “empirical” distribution F̂(x, y) is used for evaluation of an empirical regression curve where a free parameter has to be fixed by an optimality criterion. The procedure is demonstrated by an example from morphometrical research.