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

The Annals of Statistics · 1974 · Volume 2 · Issue 2

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.

Keywords

62G05, 62G10, 62G15, 62P10, empirical probability plot, nonlinear models, shift function, two-sample problem

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