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- W3100994739 abstract "We consider bivariate observations $(X_1,Y_1), ldots, (X_n,Y_n)$ such that, conditional on the $X_i$, the $Y_i$ are independent random variables with distribution functions $F_{X_i}$, where $(F_x)_x$ is an unknown family of distribution functions. Under the sole assumption that $x mapsto F_x$ is isotonic with respect to stochastic order, one can estimate $(F_x)_x$ in two ways: (i) For any fixed $y$ one estimates the antitonic function $x mapsto F_x(y)$ via nonparametric monotone least squares, replacing the responses $Y_i$ with the indicators $1_{[Y_i le y]}$. (ii) For any fixed $beta in (0,1)$ one estimates the isotonic quantile function $x mapsto F_x^{-1}(beta)$ via a nonparametric version of regression quantiles. We show that these two approaches are closely related, with (i) being more flexible than (ii). Then, under mild regularity conditions, we establish rates of convergence for the resulting estimators $hat{F}_x(y)$ and $hat{F}_x^{-1}(beta)$, uniformly over $(x,y)$ and $(x,beta)$ in certain rectangles as well as uniformly in $y$ or $beta$ for a fixed $x$." @default.
- W3100994739 created "2020-11-23" @default.
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- W3100994739 date "2020-01-01" @default.
- W3100994739 modified "2023-10-02" @default.
- W3100994739 title "Monotone least squares and isotonic quantiles" @default.
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- W3100994739 doi "https://doi.org/10.1214/19-ejs1659" @default.
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