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In this paper we obtain root- consistency and functional central limit theorems in weighted -spaces for plug-in estimators of the two-step transition density in the classical stationary linear autoregressive model of order one, assuming essentially only that the innovation density has bounded variation. We also show that plugging in a properly weighted residual-based kernel estimator for the unknown innovation density improves on plugging in an unweighted residual-based kernel estimator. These weights are chosen to exploit the fact that the innovations have mean zero. If an efficient estimator for the autoregression parameter is used, then the weighted plug-in estimator for the two-step transition density is efficient. Our approach generalizes to invertible linear processes.
@article{PS_2009__13__135_0,
author = {Schick, Anton and Wefelmeyer, Wolfgang},
title = {Plug-in estimators for higher-order transition densities in autoregression},
journal = {ESAIM: Probability and Statistics},
pages = {135--151},
publisher = {EDP-Sciences},
volume = {13},
year = {2009},
doi = {10.1051/ps:2008001},
mrnumber = {2502027},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/ps:2008001/}
}
TY - JOUR AU - Schick, Anton AU - Wefelmeyer, Wolfgang TI - Plug-in estimators for higher-order transition densities in autoregression JO - ESAIM: Probability and Statistics PY - 2009 SP - 135 EP - 151 VL - 13 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/ps:2008001/ DO - 10.1051/ps:2008001 LA - en ID - PS_2009__13__135_0 ER -
%0 Journal Article %A Schick, Anton %A Wefelmeyer, Wolfgang %T Plug-in estimators for higher-order transition densities in autoregression %J ESAIM: Probability and Statistics %D 2009 %P 135-151 %V 13 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/ps:2008001/ %R 10.1051/ps:2008001 %G en %F PS_2009__13__135_0
Schick, Anton; Wefelmeyer, Wolfgang. Plug-in estimators for higher-order transition densities in autoregression. ESAIM: Probability and Statistics, Tome 13 (2009), pp. 135-151. doi : 10.1051/ps:2008001. http://geodesic.mathdoc.fr/articles/10.1051/ps:2008001/
[1] and , Kernel estimation for real-valued Markov chains. Sankhyā Ser. A 60 (1998) 1-17. | Zbl | MR
[2] , , and , Efficient and Adaptive Estimation for Semiparametric Models. Springer, New York (1998). | Zbl | MR
[3] , and , Adaptive estimation in time-series models. Ann. Statist. 25 (1997) 786-817. | Zbl | MR
[4] , Estimating densities of functions of observations. J. Amer. Statist. Assoc. 89 (1994) 517-525. | Zbl | MR
[5] and , On local U-statistic processes and the estimation of densities of functions of several variables. Ann. Statist. 35 (2007a) 1105-1145. | Zbl | MR
[6] and , Laws of the iterated logarithm for the local U-statistic process. J. Theoret. Probab. 20 (2007b) 457-485. | Zbl | MR
[7] , Some aspects of asymptotic theory with applications to time series models. Econometric Theory 11 (1995) 818-887. | MR
[8] and , Efficient estimation in nonlinear autoregressive time series models. Bernoulli 3 (1997) 247-277. | Zbl | MR
[9] , On adaptive estimation in stationary ARMA processes. Ann. Statist. 1 (1987a) 112-133. | Zbl | MR
[10] , On adaptive estimation in autoregressive models when there are nuisance functions. Statist. Decisions 5 (1987b) 59-76. | Zbl | MR
[11] and , Probability in Banach Spaces. Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 23, Springer, Berlin (1991). | Zbl | MR
[12] , and , Weighted residual-based density estimators for nonlinear autoregressive models. Statist. Sinica 15 (2005) 177-195. | Zbl | MR
[13] , Recursive nonparametric estimation in stationary Markov processes. Publ. Inst. Statist. Univ. Paris 29 (1984) 65-84. | Zbl | MR
[14] , Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 (1988) 237-249. | Zbl | MR
[15] , Empirical Likelihood. Monographs on Statistics and Applied Probability 92, Chapman & Hall / CRC, London (2001). | Zbl
[16] , Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math. 21 (1969) 73-87. | Zbl | MR
[17] , Nonparametric estimation in mixing sequences of random variables. J. Statist. Plann. Inference 18 (1988). 135-149. | Zbl | MR
[18] and , Rate of convergence of a convolution-type estimator of the marginal density of an MA(1) process. Stoch. Proc. Appl. 80 (1999) 129-155. | Zbl | MR
[19] and , On the estimation of the marginal density of a moving average process. Canad. J. Statist. 28 (2000) 799-815. | Zbl | MR
[20] and , Root- consistent and optimal density estimators for moving average processes. Scand. J. Statist. 31 (2004a) 63-78. | Zbl | MR
[21] and , Root- consistent density estimators for sums of independent random variables. J. Nonparametr. Statist. 16 (2004b) 925-935. | Zbl | MR
[22] and , Functional convergence and optimality of plug-in estimators for stationary densities of moving average processes. Bernoulli 10 (2004c) 889-917. | Zbl | MR
[23] and , Convergence rates in weighted -spaces of kernel density estimators for linear processes. Technical Report, Department of Mathematical Sciences, Binghamton University (2006). | Zbl
[24] and , Uniformly root- consistent density estimators for weakly dependent invertible linear processes. Ann. Statist. 35 (2007a) 815-843. | Zbl | MR
[25] and , Root- consistent density estimators of convolutions in weighted -norms. J. Statist. Plann. Inference 137 (2007b) 1765-1774. | Zbl | MR
[26] and , Root- consistency in weighted -spaces for density estimators of invertible linear processes. in: Stat. Inference Stoch. Process. 11 (2008) 281-310. | MR
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