Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case

Let X and Y be two random variables with continuous distribution functions F and G and means and. In a linear model, the crucial property of the contrast = - is that X + =L Y, where = L denotes equality in law. When the linear model does not hold, there is no real number such that X + = L Y. However, it is shown that if parameters are allowed to be function valued, there is essentially only one function () such that X + (X) = L Y, and this function can be defined by (x) = G^-1 (F (x) ) - x. The estimate N (x) = Gₙ^-1 (Fₘ (x) ) - x of (x) is considered, where Gₙ and Fₘ are the empirical distribution functions. Confidence bands based on this estimate are given and the asymptotic distribution of N () is derived. For general models in analysis of variance, contrasts that can be expressed as sums of differences of means can be replaced by sums of functions of the above kind.