The Annals of Statistics

Asymptotic equivalence of nonparametric autoregression and nonparametric regression

Ion G. Grama and Michael H. Neumann

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It is proved that nonparametric autoregression is asymptotically equivalent in the sense of Le Cam’s deficiency distance to nonparametric regression with random design as well as with regular nonrandom design.

Article information

Ann. Statist., Volume 34, Number 4 (2006), 1701-1732.

First available in Project Euclid: 3 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62B15: Theory of statistical experiments
Secondary: 62G07: Density estimation 62G20: Asymptotic properties

Asymptotic equivalence deficiency distance Gaussian approximation nonparametric autoregression nonparametric regression


Grama, Ion G.; Neumann, Michael H. Asymptotic equivalence of nonparametric autoregression and nonparametric regression. Ann. Statist. 34 (2006), no. 4, 1701--1732. doi:10.1214/009053606000000560.

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  • Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29--54.
  • Brown, L. D., Cai, T., Low, M. G. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688--707.
  • Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074--2097.
  • Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384--2398.
  • Carter, A. V. (2002). Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist. 30 708--730.
  • Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
  • Dalalyan, A. and Reiß, M. (2006). Asymptotic statistical equivalence for scalar ergodic diffusions. Probab. Theory Related Fields 134 248--282.
  • Delattre, S. and Hoffmann, M. (2002). Asymptotic equivalence for a null recurrent diffusion. Bernoulli 8 139--174.
  • Donoho, D. L. (1994). Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery. Probab. Theory Related Fields 99 145--170.
  • Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
  • Drees, H. (2001). Minimax risk bounds in extreme value theory. Ann. Statist. 29 266--294.
  • Genon-Catalot, V., Laredo, C. and Nussbaum, M. (2002). Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift. Ann. Statist. 30 731--753.
  • Golubev, G. K. and Nussbaum, M. (1990). A risk bound in Sobolev class regression. Ann. Statist. 18 758--778.
  • Grama, I. G. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167--214.
  • Grama, I. G. and Nussbaum, M. (2002). Asymptotic equivalence for nonparametric regression. Math. Methods Statist. 11 1--36.
  • Grama, I. G. and Nussbaum, M. (2002). A functional Hungarian construction for sums of independent random variables. Ann. Inst. H. Poincaré Probab. Statist. 38 923--957.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Korostelev, A. and Nussbaum, M. (1995). Density estimation in the uniform norm and white noise approximation. Preprint No. 154, Weierstrass Institute, Berlin.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
  • Le Cam, L. and Yang, G. (2000). Asymptotics in Statistics: Some Basic Concepts, 2nd ed. Springer, New York.
  • Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535--543.
  • Neumann, M. H. and Kreiss, J.-P. (1998). Regression-type inference in nonparametric autoregression. Ann. Statist. 26 1570--1613.
  • Nussbaum, M. (1996). Asymptotic equivalence of density estimation and white noise. Ann. Statist. 24 2399--2430.
  • Nussbaum, M. (1999). Minimax risk: Pinsker bound. Encyclopedia of Statistical Sciences Update 3 451--460. Wiley, New York.
  • Nussbaum, M. and Klemelä, J. (1998). Constructive asymptotic equivalence of density estimation and Gaussian white noise. Discussion Paper No. 53, Sonderforschungsbereich 373, Humboldt Univ., Berlin.
  • Robinson, P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185--207.
  • Strasser, H. (1985). Mathematical Theory of Statistics. de Gruyter, Berlin.