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

Abstract

Let $X$ and $Y$ be two random variables with continuous distribution functions $F$ and $G$ and means $\mu$ and $\xi$. In a linear model, the crucial property of the contrast $\Delta = \xi - \mu$ is that $X + \Delta =_\mathscr{L} Y$, where $= _\mathscr{L}$ denotes equality in law. When the linear model does not hold, there is no real number $\Delta$ such that $X + \Delta = _\mathscr{L} Y$. However, it is shown that if parameters are allowed to be function valued, there is essentially only one function $\Delta(\bullet)$ such that $X + \Delta(X) = _\mathscr{L} Y$, and this function can be defined by $\Delta(x) = G^{-1}(F(x)) - x$. The estimate $\hat{\Delta}_N(x) = G_n^{-1}(F_m(x)) - x$ of $\Delta(x)$ is considered, where $G_n$ and $F_m$ are the empirical distribution functions. Confidence bands based on this estimate are given and the asymptotic distribution of $\hat{\Delta}_N(\bullet)$ 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.