Abstract
Suppose that we observe independent, identically distributed random pairs (X1; Y1), (X2; Y2), . . . , (Xn; Yn). Our goal is to estimate regression functions such as the conditional mean or nquantile of Y given X, where 0 0 is some tuning parameter. This framework is extended further in order to include binary or Poisson regression, and to include local variation penalties. The latter are needed in order to construct estimators adapting to inhomogenous smoothness of f . For the general framework we develop noniterative algorithms for the solution of the minimization problems which are closely related to the taut string algorithm (cf. Davies and Kovac 2001).
Highlights
Suppose that we observe pairs (x1, Y1), (x2, Y2), . . . , with fixed numbers x1 ≤ x2 ≤ · · · ≤ xn and independent random variables Y1, Y2, . . . , Yn
We assume that the distribution function of Yi depends on xi, i.e
In order to estimate the mean function μ, one can take ρ(z) := z2/2. This particular case has been treated in detail by Mammen and van de Geer (1997) and Davies and Kovac (2001); see the remark following Lemma 2.2
Summary
Suppose that we observe pairs (x1, Y1), (x2, Y2), . . . , (xn, Yn) with fixed numbers x1 ≤ x2 ≤ · · · ≤ xn and independent random variables Y1, Y2, . . . , Yn. In order to estimate the mean function μ, one can take ρ(z) := z2/2 This particular case has been treated in detail by Mammen and van de Geer (1997) and Davies and Kovac (2001); see the remark following Lemma 2.2. The latter authors describe an algorithm with running time O(n), the taut string method, to minimize the functional T above. On each constant interval the function value is equal to the mean of the corresponding observations, except for local extrema of the fit In their discussion of Davies and Kovac (2001), Mammen and van de Geer mention the possibility to replace sample means just by sample quantiles, in order to treat Example II.
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