Asymptotic equivalence for nonparametric regression

By Ion Grama, Michael Nussbaum. 2024

64 citations recorded by the source corpus.

We consider a nonparametric model $\mathcal{E}^{n},$ generated by independent observations $X_{i},$ $i=1,...,n,$ with densities $p(x,\theta_{i}),$ $i=1,...,n,$ the parameters of which $\theta _{i}=f(i/n)\in \Theta $ are driven by the values of an unknown function $f:[0,1]\rightarrow \Theta $ 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 Gaussian shift model $Y_{i}=\Gamma (f(i/n))+\varepsilon _{i},$ where $\varepsilon_{1},...,\varepsilon _{n}$ are i.i.d. standard normal r.v.'s, the function $\Gamma (\theta ):\Theta \rightarrow \mathrm{R}$ satisfies $\Gamma ^{\prime}(\theta )=\sqrt{I(\theta )}$ and $I(\theta )$ is the Fisher information corresponding to the density $p(x,\theta ).$

Subfields: Artificial Intelligence, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/74e2f882defda174723591d57ce04929ce0f257e

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Citations in the corpus

Advanced Distribution Theory for SiZer (2006, background): “(For some interesting mathematics that makes these connections precise, see Nussbaum 1996; Brown and Low 1996; Brown, Cai, and Low 2002; Grama and Nussbaum 1998, 2002.)”
Simple and honest confidence intervals in nonparametric regression (2016, methodology): “Brown and Low (1996) establish a formal sense in which this white noise model, with λ replaced by the function λn(t) = [σ (2)(t)/(nd(t))], is asymptotically equivalent to the fixed design regression model.”
Asymptotic equivalence for inference on the volatility from noisy observations (2011, methodology): “Indeed, this follows readily from results by Grama and Nussbaum (2002) who in their Section 4.”
Asymptotic statistical equivalence for scalar ergodic diffusions (2006, methodology): “The next theorem is an adaptation of Theorem 7.7 in Liptser and Shiryaev (2001) to our purposes, see also Theorem IV.4.23 in Jacod and Shiryaev (2003).”
Bayesian aspects of some nonparametric problems (2000, unknown): “Cox (1993), Diaconis and Freedman (1986, 1998) and Freedman (1999) discuss aspects of the asymptotic behavior of Bayes estimates for a special subfamily of conjugate priors.”
Nonparametric inference with generalized likelihood ratio tests (2007, methodology): “The model keeps all important features of nonparametric regression model, as demonstrated by Brown and Low (1996) and Grama and Nussbaum (2002) .”