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March, 1974 Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case
Kjell Doksum
Ann. Statist. 2(2): 267-277 (March, 1974). DOI: 10.1214/aos/1176342662

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.

Citation

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Kjell Doksum. "Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case." Ann. Statist. 2 (2) 267 - 277, March, 1974. https://doi.org/10.1214/aos/1176342662

Information

Published: March, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0277.62034
MathSciNet: MR356350
Digital Object Identifier: 10.1214/aos/1176342662

Subjects:
Primary: 62G05
Secondary: 62G10, 62G15, 62P10

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 2 • March, 1974
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