Orbits of 𝐿¹-functions under doubly stochastic transformations

Ryff, John V.

doi:10.1090/s0002-9947-1965-0209866-5

Cited by 57 publications

(56 citation statements)

References 10 publications

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

Section: As Tf(x) = \P(x Dy)f(y)fel Oo and ^P(x A)/{dx) = /{A) Formentioning

confidence: 79%

“…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].…”

Section: As Tf(x) = \P(x Dy)f(y)fel Oo and ^P(x A)/{dx) = /{A) Formentioning

confidence: 99%

“…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.…”

Section: It Is Easily Shown That T E L~umentioning

confidence: 99%

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Uniform approximation of doubly stochastic operators

Kim¹

1968

Pacific J. Math.

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

Section: As Tf(x) = \P(x Dy)f(y)fel Oo and ^P(x A)/{dx) = /{A) Formentioning

confidence: 79%

Section: As Tf(x) = \P(x Dy)f(y)fel Oo and ^P(x A)/{dx) = /{A) Formentioning

confidence: 99%

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Uniform approximation of doubly stochastic operators

Kim¹

1968

Pacific J. Math.

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“…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…”

Section: A Strictly Monotone Regression Estimatementioning

confidence: 99%

Strictly monotone and smooth nonparametric regression for two or more variables

Dette

Scheder

2006

Can J Statistics

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In this article a new monotone nonparametric estimate for a regression function of two or more variables is proposed. The method starts with an unconstrained nonparametric regression estimate and uses successively one-dimensional isotonization procedures. In the case of a strictly monotone regression function, it is shown that the new estimate is first order asymptotic equivalent to the unconstrained estimate, and asymptotic normality of an appropriate standardization of the estimate is established. Moreover, if the regression function is not monotone in one of its arguments, the constructed estimate has approximately the same L p -norm as the initial unconstrained estimate. The methodology is also illustrated by means of a simulation study, and two data examples are analyzed.

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“…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…”

Section: Introductionmentioning

confidence: 99%

A note on uniform consistency of monotone function estimators

Neumeyer

2007

Statistics & Probability Letters

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Recently, Dette, Neumeyer and Pilz (2005a) proposed a new monotone estimator for strictly increasing nonparametric regression functions and proved asymptotic normality. We explain two modifications of their method that can be used to obtain monotone versions of any nonparametric function estimators, for instance estimators of densities, variance functions or hazard rates. The method is appealing to practitioners because they can use their favorite method of function estimation (kernel smoothing, wavelets, orthogonal series,. . . ) and obtain a monotone estimator that inherits desirable properties of the original estimator. In particular, we show that both monotone estimators share the same rates of uniform convergence (almost sure or in probability) as the original estimator.MSC 2000: 62G05

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Orbits of 𝐿¹-functions under doubly stochastic transformations

Cited by 57 publications

References 10 publications

Uniform approximation of doubly stochastic operators

Uniform approximation of doubly stochastic operators

Strictly monotone and smooth nonparametric regression for two or more variables

A note on uniform consistency of monotone function estimators

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