Asymptotic equivalence for nonparametric generalized linear models

Abstract

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's de®ciency distance D; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f t i of a regression function f at a grid point t i (nonparametric GLM). When f is in a HoÈ lder ball with exponent b b 1 2 Y we establish global asymptotic equivalence to observations of a signal Cf t in Gaussian white noise, where C is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

“…'s, when we compare the almost sure invariance principle (rate N − δ 2+2δ ) and the (weak) invariance principle (rate N − δ 3+2δ ). As a potential application of the obtained results we point out the asymptotic equivalence of statistical experiments as developed in [12], [13], [14], whose scope can be extended to various models under weak dependence constraints.…”

Asymptotic statistical equivalence for scalar ergodic diffusions

“…Main result. A common way of globalising a local equivalence result makes use of the variance stabilising transformation (see Grama and Nussbaum (1998) for the exact definition). In our case this amounts to seeking a functional T whose differential DT (b)[h] at the point b = b 0 is equal to √ µ b 0 h. Indeed, for such a functional the Kullback-Leibler divergence between the laws of the Gaussian random measures dZ…”

“…The asymptotic equivalence for nonparametric experiments is conceptually more demanding and by now the class of models that are provably asymptotically equivalent to the three core models of signal detection, regression and density estimation is still limited. Grama and Nussbaum (1998) have proved asymptotic equivalence for generalised linear models, which has recently been extended to a wider nonparametric class in Grama and Nussbaum (2002), Jähnisch and Nussbaum (2003). Brown, Cai, Low, and Zhang (2002) consider specifically nonparametric regression with random design and provide a constructive asymptotic equivalence result.…”

Слежение За Гладкой Функцией Регрессии

“…из вестно, что регрессионная оценка по наблюдениям (6) асимптотически эквивалентна, в смысле Ле Кама [6], оценке / для модели (2) (при гауссовском шуме в [1] и при про извольном шуме в [3]). В свете результатов [1] и [3] естественно предположить, что подходящий фильтр для дискретного времени может быть взят по образу и подобию фильтра (3) (здесь и далее ради простоты обозначений пишем U вместо U n ):…”

“…В свете результатов [1] и [3] естественно предположить, что подходящий фильтр для дискретного времени может быть взят по образу и подобию фильтра (3) (здесь и далее ради простоты обозначений пишем U вместо U n ):…”