Abstract
We estimate the marginal density function of a long-range dependent linear process by the kernel estimator. We assume the innovations are i.i.d. Then it is known that the term of the sample mean is dominant in the MISE of the kernel density estimator when the dependence is beyond some level which depends on the bandwidth and that the MISE has asymptotically the same form as for i.i.d. observations when the dependence is below the level. We call the latter the case where the dependence is not very strong and focus on it in this paper. We show that the asymptotic distribution of the kernel density estimator is the same as for i.i.d. observations and the effect of long-range dependence does not appear. In addition we describe some results for weakly dependent linear processes.
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References
Beran, J. (1994). Statistics for Long-memory Processes, Chapman & Hall, New York.
Castellana, J. V. and Leadbetter, M. R. (1986). On smoothed probability density estimation for stationary processes, Stochastic Process. Appl., 21, 179–193.
Cheng, B. and Robinson, P. M. (1991). Density estimation in strongly dependent nonlinear time series, Statist. Sinica, 1, 335–359.
Chow, Y. S. and Teicher, H. (1988). Probability Theory, 2nd ed., Springer, New York.
Doukhan, P. (1994). Mixing: Properties and Examples, Springer, New York.
Giraitis, L., Koul, H. L., and Surgailis, D. (1996). Asymptotic normality of regression estimators with long memory errors, Statist. Probab. Lett., 29, 317–335.
Györfi, L., Härdle, W., Sarda, P. and Vieu, P. (1989). Nonparametric Curve Estimation from Time Series, Springer, New York.
Hall, P. and Hart, J. D. (1990). Convergence rates for density estimation for data from infinite order moving average sequences, Probab. Theory Related Fields, 87, 253–274.
Hall, P., Lahiri, S. N. and Truong, Y. K. (1995). On bandwidth choice for density estimation with dependent data, Ann. Statist., 23, 2241–2263.
Hallin, M. and Tran, L. T. (1996). Kernel density estimation for linear processes: asymptotic normality and optimal bandwidth derivation, Ann. Inst. Statist. Math., 48, 429–449.
Hidalgo, J. (1997). Non-parametric estimation with strongly dependent multivariate time series, J. Time Ser. Anal., 18, 95–122.
Ho, H.-C. (1996). On central and non-central limit theorems in density estimation for sequences of long-range dependence, Stochastic Processes Appl., 63, 153–174.
Ho, H.-C. and Hsing, T. (1996). On the asymptotic expansion of the empirical process of long-memory moving averages, Ann. Statist., 24, 992–1024.
Ho, H.-C. and Hsing, T. (1997). Limit theorems for functionals of moving averages, Ann. Probab., 25, 1636–1669.
Liebscher, E. (1996). Strong convergence of sums of α-mixing random variables with applications to density estimation, Stochastic Process. Appl., 65, 69–80.
Mielniczuk, J. (1997). Long and short-range dependent sums of infinite-order moving averages and regression estimation, Acta Sci. Math., 63, 301–316.
Robinson, P. M. (1991). Nonparametric functional estimation for long-memory time series, Nonparametric and Semiparametric Methods in Econometrics and Statistics (eds. W. A. Barnett, J. Powell and G. Tauchen), 437–457, Cambridge University Press, Cambridge.
Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman & Hall, London.
Tran, L., Roussas, G., Yakowitz, S., and Van, B. T. (1996). Fixed-design regression for linear time series, Ann. Statist., 24, 975–991.
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Honda, T. Nonparametric Density Estimation for a Long-Range Dependent Linear Process. Annals of the Institute of Statistical Mathematics 52, 599–611 (2000). https://doi.org/10.1023/A:1017504723799
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DOI: https://doi.org/10.1023/A:1017504723799