Nonparametric deconvolution problem for dependent sequences. (English) Zbl 1320.62073
Summary: We consider the nonparametric estimation of the density function of weakly and strongly dependent processes with noisy observations. We show that in the ordinary smooth case the optimal bandwidth choice can be influenced by long range dependence, as opposite to the standard case, when no noise is present. In particular, if the dependence is moderate the bandwidth, the rates of mean-square convergence and, additionally, central limit theorem are the same as in the i.i.d. case. If the dependence is strong enough, then the bandwidth choice is influenced by the strength of dependence, which is different when compared to the non-noisy case. Also, central limit theorem are influenced by the strength of dependence. On the other hand, if the density is supersmooth, then long range dependence has no effect at all on the optimal bandwidth choice.
MSC:
| 62G05 | Nonparametric estimation |
| 62G07 | Density estimation |
| 60F05 | Central limit and other weak theorems |
References:
| [1] | Beran, J. and Feng, Y. (2001). Local polynomial estimation with a FARIMA-GARCH error process., Bernoulli , 7 , 733-750. · Zbl 0985.62033 · doi:10.2307/3318539 |
| [2] | Beran, J. and Feng, Y. (2001). Local polynomial fitting with long-memory, short-memory and antipersistent errors., Ann. Inst. Statist. Math. , 54 , 291-311. · Zbl 1012.62033 · doi:10.1023/A:1022469818068 |
| [3] | Bosq, D. (1996)., Nonparametric Statistics for Stochastic Processes . Lecture Notes in Statistics 110 . Springer, New York. · Zbl 0857.62081 |
| [4] | Carroll, R.J. and Hall, P. (1988). Optimal Rates of Convergence for Deconvoling a Density., J. Amer. Statist. Assoc. 83 , 1184-1186. · Zbl 0673.62033 · doi:10.2307/2290153 |
| [5] | Claeskens, G. and Hall, P. (2002). Effect of dependence on stochastic measures of accuracy of density estimators., Ann. Statist. 30 , 431-454. · Zbl 1012.62031 · doi:10.1214/aos/1021379860 |
| [6] | Csőrgö, S. and Mielniczuk, J. (1995). Density estimation under long-range dependence., Ann. Statist. 23 , 990-999. · Zbl 0843.62037 · doi:10.1214/aos/1176324632 |
| [7] | Doukhan, P. (1984)., Mixing: Properties and Examples . Lecture Notes in Statisitcs. Springer. |
| [8] | Estevez, G. and Vieu, P. (2003). Nonparametric estimation under long memory dependence., Nonparametirc Statistics 15 , 535-551. · Zbl 1054.62032 · doi:10.1080/10485250310001604668 |
| [9] | Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems., Ann. Statist. 19 , 1257-1272. · Zbl 0729.62033 · doi:10.1214/aos/1176348248 |
| [10] | Fan, J. (1991). Asymptotic normality for deconvolving kernel density estimators., Sankya Ser. A 53 , 97-110. · Zbl 0729.62034 |
| [11] | Fan, J. and Masry, E. (1992). Multivariate Regression Estimation with Errors-in-Variables: Asymptotic Normality for Mixing Processes., J. Mult. Anal. 43 , 237-271. · Zbl 0769.62028 · doi:10.1016/0047-259X(92)90036-F |
| [12] | Giraitis, L. and Surgailis, D. (1999). Central limit theorem for the empirical process of a linear sequence with long memory., J. Statist. Plann. Inference 80 , 81-93. · Zbl 0943.60035 · doi:10.1016/S0378-3758(98)00243-2 |
| [13] | Hall, P. and Hart, J.D. (1990). Convergence rates in density estimation for data from infinite-order moving average processes., Probab. Th. Rel. Fields 87 , 253-274. · Zbl 0695.60043 · doi:10.1007/BF01198432 |
| [14] | Hall, P., Lahiri, S. N. and Truong, Y. K. (1995). On bandwidth choice for density estimation with dependent data., Ann. Statist. 23 , 2241-2263. · Zbl 0854.62039 · doi:10.1214/aos/1034713655 |
| [15] | 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. · Zbl 0862.60026 · doi:10.1214/aos/1032526953 |
| [16] | Ho, H.-C. and Hsing, T. (1997). Limit theorems for functionals of moving averages., Ann. Probab. 25 , 1636-1669. · Zbl 0903.60018 · doi:10.1214/aop/1023481106 |
| [17] | Ioannides, D.A. and Papanastassiou, D.P. (2001). Estimating the Distribution Function of a Stationary Process Involving Measurement Errors., Statist. Inf. for Stoch. Proc. 4 , 181-198. · Zbl 0984.62061 · doi:10.1023/A:1017996326631 |
| [18] | Masry, E. (1991). Multivariate Probability Density Deconvolution for Stationary Random Processes., IEEE Trans. Inf. Th. 37 , 1105-1115. · Zbl 0732.60045 · doi:10.1109/18.87002 |
| [19] | Masry, E. (1993). Strong consistency and rates for deconvolution of multivariate densities of stationary processes., Stochastic Process. Appl. 47 , 53-74. · Zbl 0797.62071 · doi:10.1016/0304-4149(93)90094-K |
| [20] | Masry, E. (1993). Asymptotic normality for deconvolving estimators of multivariate densities for stationary processes., J. Mult. Anal. 44 , 47-68. · Zbl 0783.62065 · doi:10.1006/jmva.1993.1003 |
| [21] | Masry, E. (2003). Deconvolving Multivariate Kernel Density Estimates From Contaminated Associated Observations., IEEE Trans. Inf. Th. 49 , 2941-2952. · Zbl 1302.62085 · doi:10.1109/TIT.2003.818415 |
| [22] | Masry, E. and Mielniczuk, J. (1999). Local linear regression estimation for time series with long-range dependence., Stoch. Proc. Appl. 82 , 173-193. · Zbl 0991.62024 · doi:10.1016/S0304-4149(99)00015-0 |
| [23] | Mielniczuk, J. (1997). On the asymptotic mean integrated squared error of a kernel density estimator for dependent data., Statist. Probab. Letters 34 , 53-58. · Zbl 0902.62047 · doi:10.1016/S0167-7152(96)00165-4 |
| [24] | Mielniczuk, J. and Wu, W. B. (2004). On random-design model with dependent errors., Statistica Sinica , 1105-1126. · Zbl 1060.62046 |
| [25] | Stefanski, L.A. and Carroll, R.J. (1990). Deconvoltuing kernel density estimators., Statistics 21 , 169-184. · Zbl 0697.62035 · doi:10.1080/02331889008802238 |
| [26] | Wu, W.B. (2003). Empirical processes of long-memory sequences., Bernoulli 9 , 809-831. · Zbl 1188.62288 · doi:10.3150/bj/1066418879 |
| [27] | Wu, W.B. (2005). On the Bahadur representation of sample quantiles for dependent sequences., Ann. Statist. 33 , 1934-1963. · Zbl 1080.62024 · doi:10.1214/009053605000000291 |
| [28] | Wu, W. B. and Mielniczuk, J. (2002) Kernel density estimation for linear processes., Ann. Statist. 30 , 1441-1459. · Zbl 1015.62034 · doi:10.1214/aos/1035844982 |
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