Abstract
It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we prove that the slopes of the greatest convex minorant are distributed as order statistics of the running averages. This result implies an exact non-asymptotic formula for the squared error risk of least squares in homoscedastic isotonic regression when the true sequence is constant that holds for every exchangeable error distribution.
Highlights
Isotonic regression refers to a noisy observation vector Y the problem of estimating a which is assumed to be an amddonitoivtoenpeersteuqrubeantcieonθ1∗of≤θ⋯∗ ≤ =(θθn∗1∗,b.a.s.e,dθn∗o)n,Y = θ∗ + σZ, where the components Z1, . . . , Zn of Z are assumed to have zero mean and unit variance
As Mn is a closed convex cone, θas defined above exists uniquely; it can be computed in O(n) time by the pool adjacent violators algorithm [4, 11]
Combined with (1.1), our result provides a sharp non-asymptotic bound on the risk of isotonic regression for any exchangeable noise vector
Summary
Isotonic regression refers to a noisy observation vector Y the problem of estimating a which is assumed to be an amddonitoivtoenpeersteuqrubeantcieonθ1∗of≤θ⋯∗. This bound gives the right behavior as the right hand side of equation (1.3) but only as n → ∞ We improve this result by proving for every n ≥ 1 that δn(⊗ni=1η) is always equal to the nth harmonic number Hn for every probability measure η having mean 0 and variance 1. As long as Z is exchangeable, we show in Theorem 2.2 that (ΠMn (Z))k has the same distribution as Z(k), the kth order statistic of the running averages Zj. We prove Theorem 2.2, using a characterization of the components of the isotonic LSE as the left-hand slopes of the greatest convex minorant of the random walk with increments Z1, . Zn. We prove Theorem 2.2, using a characterization of the components of the isotonic LSE as the left-hand slopes of the greatest convex minorant of the random walk with increments Z1, .
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