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The bias of isotonic regression. (English) Zbl 1436.62137

Summary: We study the bias of the isotonic regression estimator. While there is extensive work characterizing the mean squared error of the isotonic regression estimator, relatively little is known about the bias. In this paper, we provide a sharp characterization, proving that the bias scales as \(O(n^{-\beta /3})\) up to log factors, where \(1\leq \beta \leq 2\) is the exponent corresponding to Hölder smoothness of the underlying mean. Importantly, this result only requires a strictly monotone mean and that the noise distribution has subexponential tails, without relying on symmetric noise or other restrictive assumptions.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics

Software:

isotone
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References:

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