Orbits of 𝐿¹-functions under doubly stochastic transformations

Uniform approximation of doubly stochastic operators

“…In view of Proposition 1.1, our Strong Approximation Theorem 2.2 is sharper than that of [2,8]. We now prove…”

“…For each TG^ there is a unique T^e^ such that By the weak (strong) topology in & we mean the weak (strong) operator topology in £^ [2,8]. We note that the relative uniform topology on Φ 1 coincides with the uniform topology on Φ x introduced by Halmos [5].…”

“…2* Approximation theorems* The problem of approximating doubly stochastic operators is discussed by several authors [2,7,8]. In this section we discuss various approximation theorems.…”

Strictly monotone and smooth nonparametric regression for two or more variables

“…To be precise, let g : [0, 1] → R denote an arbitrary function on a given interval, say [0, 1], we define for t ∈Im(g) Note that this function is always increasing and coincides with the inverse of the function g , say g −1 , in the case where g is strictly increasing. Some properties of a related function have been discussed in the context of nondecreasing/nonincreasing rearrangements [see, e.g., Ryff (1965), Ryff (1970), and Bennett and Sharpley (1988) among others]. The function g −1 I is not necessarily smooth, but smoothing can easily be accomplished by considering the function…”

A note on uniform consistency of monotone function estimators

“…Note that Dette et al's (2005a) proof for the asymptotic distribution ofĥ defined in (1.1) and its inverse is not easily generalized to obtain asymptotic results about the estimator based on (1.2). The approach to use the inverseĥ −1 as an estimator for g, whereĥ is defined in (1.2) is related to nondecreasing rearrangements of data considered by Ryff (1965Ryff ( ,1970, and is in principle similar to Polonik's (1995Polonik's ( ,1998 work, who constructs estimators for a density f from the identity…”