Monotone Least Squares and Isotonic Quantiles
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We consider bivariate observations (X_1,Y_1),...,(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 F_x is isotonic in x 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 F_x(y) via nonparametric monotone least squares, replacing the responses Y_i with the indicators 1_[Y_i < y]. (ii) For any fixed β∈ (0,1) one estimates the isotonic quantile function x F_x^-1(β) via a nonparametric version of regression quantiles. We show that these two approaches are closely related, with (i) being a bit more flexible than (ii). Then, under mild regularity conditions, we establish rates of convergence for the resulting estimators F̂_x(y) and F̂_x^-1(β), uniformly over (x,y) and (x,β) in certain rectangles.
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