×

zbMATH — the first resource for mathematics

On invariant distribution function estimation for continuous-time stationary processes. (English) Zbl 1084.62084
Summary: This paper is concerned with the asymptotic behaviour of the empirical distribution function for a large class of continuous-time weakly dependent stationary processes. Under mild mixing conditions the empirical distribution function is an unbiased consistent estimator of the marginal distribution function of the process. For strongly mixing processes this estimator is asymptotically normal. We propose a consistent estimator of the asymptotic variance, and then study the functional central limit theorem for the empirical distribution function.

MSC:
62M09 Non-Markovian processes: estimation
62G30 Order statistics; empirical distribution functions
60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley. · Zbl 0172.21201
[2] Billingsley, P. (1995) Probability and Measure, 3rd edition. New York: Wiley. · Zbl 0822.60002
[3] Castellana, J.V. and Leadbetter, M.R. (1986) On smoothed probability density estimation for stationary process. Stochastic Process Appl., 21, 179-193. · Zbl 0588.62156 · doi:10.1016/0304-4149(86)90095-5
[4] Davydov, Yu. (2001) On convergence of empirical measures. Statist. Inference Stochastic Process., 4(1), 1-15.
[5] Dehay, D. and Kutoyants, Yu.A. (2004) On confidence intervals for distribution function and density of ergodic diffusion process. J. Statist. Plann. Inference, 124(1), 63-73. · Zbl 1094.62106 · doi:10.1016/S0378-3758(03)00195-2
[6] Dehling, H., Mikosch, T. and Sørensen, M (2002) Empirical Process Techniques for Dependent Data. Boston: Birkhäuser. · Zbl 1021.62036
[7] Doukhan, P. (1984) Mixing: Properties and Examples, Lecture Notes in Statist. 85. Berlin: Springer- Verlag.
[8] Dudley, R.M. (1984) A course on empirical processes. In P.L. Hennequin (ed.), Ecole d\'É té de Probabilités de Saint Flour XII-1982, Lecture Notes in Math. 1097, pp. 1-142. Berlin: Springer- Verlag. · Zbl 0554.60029
[9] Ibragimov, I.A. and Linnik, Yu.V. (1971) Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff. · Zbl 0219.60027
[10] Kutoyants, Yu.A. (2003) Statistical Inference for Ergodic Diffusion Processes. London: Springer- Verlag. · Zbl 1012.62091
[11] Meyn, S.P. and Tweedie, R.L. (1993a) Stability of Markovian processes II: Continuous-time processes and sampled chains. Adv. Appl. Probab., 25, 487-517. JSTOR: · Zbl 0781.60052 · doi:10.2307/1427521 · links.jstor.org
[12] Meyn, S.P. and Tweedie, R.L. (1993b) Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab., 25, 518-548. JSTOR: · Zbl 0781.60053 · doi:10.2307/1427522 · links.jstor.org
[13] Negri, I. (1998) Stationary distribution function estimation for ergodic diffusion process. Statist. Inference Stochastic Process., 1(1), 61-84. · Zbl 1061.62553 · doi:10.1023/A:1009997126882
[14] Pollard, D. (1990) Empirical Processes: Theory and Applications, NSF-CBMS Reg. Conf. Ser. Probab. Statist. 2. Hayward, CA, and Alexandrias, VA: Institute of Mathematical Statistics and American Statistical Association. · Zbl 0741.60001
[15] Rio, E. (2000) Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants, Math. Appl. 31. Berlin: Springer-Verlag. · Zbl 0944.60008
[16] Shorack, G.R. and Wellner, J.A. (1986) Empirical Processes with Applications to Statistics. New York: Wiley. · Zbl 1170.62365
[17] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. New York: Springer-Verlag. · Zbl 0862.60002
[18] Veretennikov, A.Yu. (1988) Bounds for the mixing rate in the theory of stochastic equations. Theory Probab. Appl., 32(2), 273-281. · Zbl 0663.60046 · doi:10.1137/1132036
[19] Veretennikov, A.Yu. (1999) On Castellana-Leadbetterś condition for diffusion density estimation. Statist. Inference Stochastic Process., 2(1), 1-9. · Zbl 0958.62078 · doi:10.1023/A:1009996608986
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.