The Annals of Statistics

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

Kjell Doksum

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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.

Article information

Source
Ann. Statist., Volume 2, Number 2 (1974), 267-277.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342662

Digital Object Identifier
doi:10.1214/aos/1176342662

Mathematical Reviews number (MathSciNet)
MR356350

Zentralblatt MATH identifier
0277.62034

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62G15: Tolerance and confidence regions 62G10: Hypothesis testing 62P10: Applications to biology and medical sciences

Keywords
Nonlinear models two-sample problem shift function empirical probability plot

Citation

Doksum, Kjell. Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case. Ann. Statist. 2 (1974), no. 2, 267--277. doi:10.1214/aos/1176342662. https://projecteuclid.org/euclid.aos/1176342662


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