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New robust confidence intervals for the mean under dependence. (English) Zbl 1455.60042

Summary: The goal of this paper is to indicate a new method for constructing normal confidence intervals for the mean, when the data is coming from stochastic structures with possibly long memory, especially when the dependence structure is not known or even the existence of the density function. More precisely we introduce a random smoothing suggested by the kernel estimators for the regression function. The normal confidence intervals are constructed under the sole condition that the sequence is ergodic and has finite second moments and a mild condition on the sample variance. Applications are presented for linear processes and reversible Markov chains with long memory.

MSC:

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
62G08 Nonparametric regression and quantile regression
62G15 Nonparametric tolerance and confidence regions

Software:

longmemo
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References:

[1] Beran, J., Statistics for Long-Memory Processes (1959), Chapman & Hall
[2] Beran, J., A test of location for data with slowly decaying serial correlations, Biometrika, 76, 261-269 (1989) · Zbl 0669.62080
[3] Billingsley, P., Probability and Measure (1995), John Willey & Sons · Zbl 0822.60002
[4] Billingsley, P., Convergence of Probability Measures (1999), John Willey & Sons · Zbl 0172.21201
[5] Bosq, D., Nonparametric Statistics for Stochastic Processes: Estimation and Prediction (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0857.62081
[6] Bosq, D.; Merlevède, F.; Peligrad, M., Asymptotic normality for density kernel estimators in discrete and continuous time, J. Multivariate Anal., 68, 79-95 (1999) · Zbl 0926.60024
[7] Bradley, R., Asymptotic normality of some kernel-type estimators of probability density, Statist. Probab. Lett., 1, 295-300 (1983) · Zbl 0521.62032
[8] Bradley, R. C., Introduction To Strong Mixing Conditions 1, 2, 3 (2007), Kendrick Press: Kendrick Press Heber City, UT · Zbl 1134.60004
[9] Collomb, G., Propriétés de convergence presque complète du prédicteur à noyau, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 66, 441-460 (1984) · Zbl 0525.62046
[10] Deligiannidis, G.; Utev, S., Variance of partial sums of stationary sequences, Ann. Probab., 41, 3606-3616 (2013) · Zbl 1291.60068
[11] Eisner, F.; Farkas, B.; Haase, M.; Nagel, R., Operator theoretic aspects of ergodic theory, (Graduate Texts in Mathematics, vol. 272 (2015), Springer) · Zbl 1353.37002
[12] Granger, C. W.; Joyeux, R., An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal., 1, 15-29 (1980) · Zbl 0503.62079
[13] Härdle, W., Smoothing Techniques with Implementation in S (1991), Springer-Veralg · Zbl 0716.62040
[14] Hong, S. Y.; Linton, O., Asymptotic properties of a nadaraya-watson type estimator for regression functions of infinite order (2016)
[15] Hosking, J. R.M., Fractional differencing, Biometrika, 68, 165-176 (1981) · Zbl 0464.62088
[16] Jones, M.; Marron, J.; Sheather, S., A brief survey of bandwidth selection for density estimation, J. Amer. Statist. Assoc., 91, 433, 401-407 (1996) · Zbl 0873.62040
[17] Krengel, U., Ergodic Theorems (1985), De Gruyter Studies in Mathematics: De Gruyter Studies in Mathematics Berlin · Zbl 0471.28011
[18] Laib, N.; Louani, D., Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties, J. Multivariate Anal., 101, 2266-2281 (2010) · Zbl 1198.62027
[19] Long, H.; Qian, L., Nadaraya-Watson estimator for stochastic processes driven by stable Lévy motions, Electron. J. Stat., 7, 1387-1418 (2013) · Zbl 1337.62204
[20] Longla, M., Remarks on the speed of convergence of mixing coefficients and applications, Statist. Probab. Lett., 83, 10, 2439-2445 (2013) · Zbl 1308.60086
[21] Longla, M., On mixtures of copulas and mixing coefficients, J. Multivariate Anal., 139, 259-265 (2015) · Zbl 1333.60161
[22] Longla, M.; Peligrad, M., Some aspects of modeling dependence in copula-based Markov chains, J. Multivariate Anal., 111, 234-240 (2012) · Zbl 1301.60089
[23] McElroy, T. S.; Politis, D. N., Spectral density and spectral distribution inference for long memory time series via fixed-b asymptotics, J. Econometrics, 182, 211-225 (2014) · Zbl 1311.62151
[24] Nadaraya, E. A., On estimating regression, Theory Probab. Appl., 9, 141-142 (1964) · Zbl 0136.40902
[25] Nelsen, R. B., An Introduction To Copulas (2006), Springer: Springer New York · Zbl 1152.62030
[26] Parzen, E., On estimation of a probability density function and mode, Ann. Math. Statist., 33, 3, 1065-1076 (1962) · Zbl 0116.11302
[27] Peligrad, M., Invariance principles for mixing sequences of random variables, Ann. Probab., 10, 968-981 (1982) · Zbl 0503.60044
[28] Peligrad, M., Properties of uniform consistency of the kernel estimators of density and of regression functions under dependence assumptions, Stoch. Stoch. Rep., 40, 147-168 (1992) · Zbl 0770.62032
[29] Peligrad, M., On the blockwise bootstrap for empirical processes for stationary sequences, Ann. Probab., 26, 877-901 (1998) · Zbl 0932.62055
[30] Peligrad, M.; Shao, Q.-M., Estimation of the variance for partial sums for \(r h o\)-mixing random variables, J. Multivariate Anal., 52, 140-157 (1995) · Zbl 0816.62027
[31] Thode, H., Testing for Normality (2002), Marcel Dekker: Marcel Dekker New York · Zbl 1032.62040
[32] Wang, Q.; Lin, X.-Y.; Gulati, C. M., Asymptotics for moving average processes with dependent innovations, Statist. Probab. Lett., 54, 347-356 (2001) · Zbl 0996.60041
[33] Watson, G. S., Smooth regression analysis, Sankhya A, 26, 359-372 (1964) · Zbl 0137.13002
[34] Yoshihara, K., Weakly Dependent Stochastic Sequences and their Applications: Volume IV: Curve Estimation Based on Weakly Dependent Data (1994), Sanseido: Sanseido Tokyo, Japan · Zbl 0876.62033
[35] Zhao, O.; Woodroofe, M.; Volný, D., A central limit theorem for reversible processes with nonlinear growth of variance, J. Appl. Probab., 47, 1195-1202 (2010) · Zbl 1208.60025
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