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## Asymptotic equivalence for nonparametric regression

Venue: | Math. Methods Statist |

Citations: | 22 - 0 self |

### Citations

419 |
Asymptotic methods in statistical decision theory
- Cam
- 1986
(Show Context)
Citation Context ...means of a functional Hungarian construction for partial sums, established in Grama and Nussbaum [8]. The abovementioned results are part of an effort to extend Le Cam’s asymptotic theory (see Le Cam =-=[12]-=- and Le Cam and Yang [13]) to a class of general models with infinite dimensional parameters which cannot be estimated at the usual ”root-n” rate n−1/2. The case of the infinite dimensional parameters... |

340 | Optimal global rates of convergence for nonparametric regression - Stone - 1982 |

256 |
Fundamentals of statistical exponential families: with applications in statistical decision theory
- Brown
- 1986
(Show Context)
Citation Context ... Section 3 arises when the parametric experiment E = (X,X , {Pθ : θ ∈ Θ}) is an onedimensional linearly indexed exponential family, where Θ is a possibly infinite interval on the real line (see Brown =-=[2]-=-). This means that the measures Pθ are absolutely continuous w.r.t. a σ-finite measure µ(dx) with densities (in the canonical form) p(x, θ) = Pθ(dx) µ(dx) = exp (θU(x)− V (θ)) , x ∈ X, θ ∈ Θ,(4.2) whe... |

207 |
Asymptotic equivalence of nonparametric regression and white
- Brown, Low
- 1996
(Show Context)
Citation Context ...be asymptotically equivalent if the Le Cam pseudodistance between them tends to 0 as n→∞. Such a relation between the model (1.2) and its continuous time analog was first established by Brown and Low =-=[3]-=-. In a paper by Nussbaum [15] the accompanying Gaussian model for the density estimation from an i.i.d. sample was found to be the white noise model (1.2) with the root of the density as a signal. The... |

126 |
Asymptotics in statistics: some basic concepts
- Cam, Yang
- 1990
(Show Context)
Citation Context ...garian construction for partial sums, established in Grama and Nussbaum [8]. The abovementioned results are part of an effort to extend Le Cam’s asymptotic theory (see Le Cam [12] and Le Cam and Yang =-=[13]-=-) to a class of general models with infinite dimensional parameters which cannot be estimated at the usual ”root-n” rate n−1/2. The case of the infinite dimensional parameters which are estimable at t... |

52 |
Approximation theorems for independent and weakly dependent random variables, The Annals of Probability 7
- Philipp
- 1978
(Show Context)
Citation Context ... model En, generated by independent observations Xi, i = 1, ..., n, with densities p(x, θi), i = 1, ..., n, the parameters of which θi = f(i/n) ∈ Θ are driven by the values of an unknown function f : =-=[0, 1]-=-→ Θ in a smoothness class. The main result of the paper is that, under regularity assumptions, this model can be approximated, in the sense of the Le Cam deficiency pseudodistance, by a nonparametric ... |

38 | Asymptotic equivalence theory for nonparametric regression with random design
- Brown, Cai, et al.
- 2002
(Show Context)
Citation Context ...). An overview of the technique of proof can be found at the end of the Section 3. A nonparametric regression model with random design, but Gaussian noise was recently treated by Brown, Low and Zhang =-=[5]-=-. We focus here on the nongaussian case, assuming a regular nonrandom design i/n, i = 1, . . . , n: the model is generated by a parametric family of densities p(x, θ), θ ∈ Θ, where θ is assumed to tak... |

22 |
Asymptotic equivalence of density estimation and white
- Nussbaum
- 1996
(Show Context)
Citation Context ... if the Le Cam pseudodistance between them tends to 0 as n→∞. Such a relation between the model (1.2) and its continuous time analog was first established by Brown and Low [3]. In a paper by Nussbaum =-=[15]-=- the accompanying Gaussian model for the density estimation from an i.i.d. sample was found to be the white noise model (1.2) with the root of the density as a signal. The case of generalized linear m... |

21 |
Komlos—Major–Tusnady approximation for the general empirical process and haar expansions of classes of functions
- Koltchinskii
- 1994
(Show Context)
Citation Context ...=1 f(i/n) ( X̃i −Ni )∣∣∣∣∣ > x(log n)2 ) ≤ c0 exp (−c1λnx) , x ≥ 0, where c0 and c1 are absolute constants. This result is an analog of the functional strong approximation established by Koltchinskii =-=[9]-=- for the empirical processes. Remark 9.1. Note that the r.v.’s X1, ..., Xn are not assumed to be identically distributed. We also do not assume any additional richness of the probability space (Ω,F , ... |

20 |
Exact asymptotic minimax estimate of a nonparametric regression in the uniform norm, Theory Probab
- Korostelev
- 1993
(Show Context)
Citation Context ...ould also require c(β) → ∞ for convergence to 0 in (G1)). In the case of the Gaussian location-type regression ((1.1) for normal ξi) this is a consequence of the optimal constant result of Korostelev =-=[10]-=-. The extension to our nongaussian regression models would be of technical nature; for the density estimation model it has been verified in Korostelev and Nussbaum [11] and applied in a similar contex... |

20 | The asymptotic minimax constant for sup-norm loss in nonparametric density estimation
- Korostelev, Nussbaum
- 1996
(Show Context)
Citation Context ...l constant result of Korostelev [10]. The extension to our nongaussian regression models would be of technical nature; for the density estimation model it has been verified in Korostelev and Nussbaum =-=[11]-=- and applied in a similar context to here in Lemma 9.3 of [15] . Remark 3.3. The function Γ (θ) can be related to so called variance-stabilizing transformation, which we proceed to introduce. Let X1, ... |

19 | C.: Asymptotic nonequivalence of nonparametric experiments when the smoothness index is 1/2
- BROWN, ZHANG
- 1998
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Citation Context ...ts En, n = 1, 2, ... is asymptotically equivalent to the sequence of Gaussian experiments Gn, n = 1, 2, ... : ∆ (En,Gn)→ 0, as n→∞. Remark 3.1. Examples in Efromovich and Samarov [6], Brown and Zhang =-=[4]-=- [see also Brown and Low [3]] show that asymptotic equivalence, in general, fails to hold true when β ≤ 1/2. Remark 3.2. Assumption (G1) is related to attainable rates of convergence in the sup-norm ‖... |

18 |
Mathematical Theory of Statistics. Walter de Gruyter
- Strasser
- 1985
(Show Context)
Citation Context ...such that sup (θ,u) 1 |u− θ|1+δ (∫ X ( s (x, u)− s (x, θ)− (u− θ) •s (x, θ) )2 µ (dx) )1/2 <∞, where sup is taken over all pairs (θ, u) satisfying θ, u ∈ Θ, |u− θ| ≤ ε. It is well-known (see Strasser =-=[17]-=-) that there is a map • l (θ) ∈ L2 (X,X , µ) such that the function • s (θ) in condition (R1) can be written as • s (θ) = 1 2 • l (θ) √ p (θ), µ-a.s. on X.(3.4) Moreover, • l (θ) ∈ L2 (X,X , Pθ) and E... |

14 |
The minimax principle in asymptotic statistical theory. École d’Été de Probabilités de St
- MILLAR
- 1983
(Show Context)
Citation Context ...mensional parameters which cannot be estimated at the usual ”root-n” rate n−1/2. The case of the infinite dimensional parameters which are estimable at this rate was considered for instance in Millar =-=[14]-=-. The approach used in the proofs of the present results is quite different from that in the ”root-n” case and was suggested by the papers of Brown and Low [3] and Nussbaum [15] (see also Grama and Nu... |

11 |
Asymptotic equivalence of nonparametric regression and white noise model has its limits
- Efromovich, Samarov
- 1996
(Show Context)
Citation Context ...sequence of experiments En, n = 1, 2, ... is asymptotically equivalent to the sequence of Gaussian experiments Gn, n = 1, 2, ... : ∆ (En,Gn)→ 0, as n→∞. Remark 3.1. Examples in Efromovich and Samarov =-=[6]-=-, Brown and Zhang [4] [see also Brown and Low [3]] show that asymptotic equivalence, in general, fails to hold true when β ≤ 1/2. Remark 3.2. Assumption (G1) is related to attainable rates of converge... |

4 |
A functional Hungarian construction for sums of independent random variables. Unpublished manuscript
- Grama, Nussbaum
- 2001
(Show Context)
Citation Context ...≤ L, x, y ∈ [0, 1], where L is an absolute constant. In the sequel, the notation L(X) = L(Y ) for r.v.’s means equality of their distributions. The following assertion is proved in Grama and Nussbaum =-=[8]-=-. Theorem 9.1. A sequence of independent r.v.’s X̃1, ..., X̃n can be constructed on the probability space (Ω,F , P ) such that L(X̃i) = L(Xi), i = 1, ..., n, and sup f∈H(1/2,L) P (∣∣∣∣∣ n∑ i=1 f(i/n) ... |

1 |
Asymptotic equivalence for generalized linear models, Probability Theory and Related
- Grama, Nussbaum
- 1998
(Show Context)
Citation Context ...model (1.2) with the root of the density as a signal. The case of generalized linear models (i.e. a class of nonparametric regression models with non-additive noise) was treated in Grama and Nussbaum =-=[7]-=-. However none of the above results covers observations of the form (1.1). It is the aim of the present paper to develop an asymptotic equivalence theory for a more general class Date: October, 2001. ... |