## The Annals of Statistics

### Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations

Michael H. Neumann

#### Abstract

We derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors. We construct, on a sufficiently rich probability space, such a pairing of the random variables of both experiments that the set of observations $X_1,\ldots,X_n}$ from the time series model is nearly the same as the set of observations $Y_1,\ldots,Y_n}$ from the i.i.d. model. With a high probability, all sets of the form $({X_1,\ldots,X_n}\\Delta{Y_1,\ldots,Y_n})\bigcap([a_1,b_1]\times\ldots\times[a_d,b_d])$ contain no more than $O({[n^1/2 \prod(b_i-a_i)]+ 1} \log(n))$ elements, respectively. Although this does not imply very much for parametric problems, it has important implications in nonparametric statistics. It yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i.i.d. model. Moreover, it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap. Using these results we derive simultaneous confidence bands as well as supremum­type nonparametric tests based on reasoning for the i.i.d. model.

#### Article information

Source
Ann. Statist., Volume 26, Number 5 (1998), 2014-2048.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1024691367

Digital Object Identifier
doi:10.1214/aos/1024691367

Mathematical Reviews number (MathSciNet)
MR1673288

Zentralblatt MATH identifier
0930.62038

Subjects
Primary: 62G07: Density estimation
Secondary: 62G09: Resampling methods 62M07: Non-Markovian processes: hypothesis testing

#### Citation

Neumann, Michael H. Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations. Ann. Statist. 26 (1998), no. 5, 2014--2048. doi:10.1214/aos/1024691367. https://projecteuclid.org/euclid.aos/1024691367

#### References

• [1] Amosova, N. N. (1972). On limit theorems for probabilities of moderate deviations. Vestnik Leningrad. Univ. 13 5-14 (in Russian).
• [2] Ango Nze, P. (1992). Crit eres d'ergodicit´e de quelques mod eles a repr´esentation markovienne. C. R. Acad. Sci. Paris S er. I Math. 315 1301-1304.
• [3] Ango Nze, P. and Doukhan, P. (1993). Estimation fonctionnelle de s´eries temporelles m´elangeantes. C. R. Acad. Sci. Paris S er. I Math. 317 405-408.
• [4] Ango Nze, P. and Rios, R. (1995). Estimation L de la fonction de densit´e d`un processus faiblement d´ependant: les cas absolument r´egulier et fortement m´elangeant. C. R. Acad. Sci. Paris S er. I Math. 320 1259-1262.
• [5] Bickel, P. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095.
• [6] Br´emaud, P. and Massouli´e, L. (1996). Stability of nonlinear Hawkes processes. Ann. Probab. 24 1563-1588.
• [7] B ¨uhlmann, P. (1994). Blockwise bootstrapped empirical process for stationary sequences. Ann. Statist. 22 995-1012.
• [8] Dhompongsa, S. (1984). A note on the almost sure approximation of the empirical process of weakly dependent random vectors. Yokohama Math. J. 32 113-121.
• [9] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
• [10] Doukhan, P., Massart, P. and Rio, E. (1995). Invariance principles for absolutely regular empirical processes. Ann. Inst. H. Poincar´e Probab. Statist. 31 393-427.
• [11] Doukhan, P. and Portal, F. (1987). Principe d'invariance faible pour la fonction de r´epartition empirique dans un cadre multidimensionnel et m´elangeant. Probab. Math. Statist. 8 117-132.
• [12] Efron, B. (1979). Bootstrap methods: another look at the jackknife. Ann. Statist. 7 1-26.
• [13] Efron, B. (1982). The Jackknife, the Bootstrap and Other Resampling Plans. SIAM, Philadelphia.
• [14] Falk, M. and Reiss, R.-D. (1989a). Weak convergence of smoothed and nonsmoothed bootstrap quantile estimates. Ann. Probab. 17 362-371.
• [15] Falk, M. and Reiss, R.-D. (1989b). Bootstrapping the distance between smooth bootstrap and sample quantile distribution. Probab. Theory Related Fields 82 177-186.
• [16] Faraway, J. J. and Jhun, M. (1990). Bootstrap choice of bandwidth for density estimation. J. Amer. Statist. Assoc. 85 1119-1122.
• [17] Franke, J., Kreiss, J.-P., Mammen, E. and Neumann, M. H. (1998). Properties of the nonparametric autoregressive bootstrap. Discussion Paper 54/98, SFB 373, Humboldt Univ., Berlin.
• [18] Friedman, J., Stuetzle, W. and Schroeder, A. (1984). Projection pursuit density estimation. J. Amer. Statist. Assoc. 79 599-608.
• [19] Gy ¨orfi, L., H¨ardle, W., Sarda, P. and Vieu, P. (1989). Nonparametric Curve Estimation from Time Series. Springer, Berlin.
• [20] Hall, P. (1991). On convergence rates of suprema. Probab. Theory Related Fields 89 447- 455.
• [21] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
• [22] Hall, P. (1993). On Edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. J. R. Statist. Soc. Ser. B 55 291-304.
• [23] Hall, P., DiCiccio, T. J. and Romano, J. P. (1989). On smoothing and the bootstrap. Ann. Statist. 17 692-704.
• [24] Hall, P. and Hart, J. D. (1990). Convergence rates in density estimation for data from infinite-order moving average processes. Probab. Theory Related Fields 87 253-274.
• [25] H¨ardle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926-1947.
• [26] Hart, J. D. (1996). Some automated methods of smoothing time-dependent data. J. Nonparametr. Statist. 6 115-142.
• [27] Koml ´os, J., Major, P. and Tusn´ady, G. (1975). An approximation of partial sums of independent rv's and the sample distribution function. Z. Wahrsch. Verw. Gebiete 32 111-131.
• [28] Konakov, V., L¨auter, H. and Liero, H. (1995). Nonparametric versus parametric goodness of fit. Statistics. To appear.
• [29] K ¨unsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann. Statist. 17 1217-1241.
• [30] Masry, E. (1994). Probability density estimation from dependent observations using wavelet orthonormal bases. Statist. Probab. Lett. 21 181-194.
• [31] Mokkadem, A. (1988). Mixing properties of ARMA processes. Stochastic Process. Appl. 29 309-315.
• [32] Mokkadem, A. (1990). Propri´et´es de m´elange des processus autor´egessifs poly nomiaux. Ann. Inst. H. Poincar´e Probab. Statist. 26 219-260.
• [33] Neumann, M. H. (1995). Automatic bandwidth choice and confidence intervals in nonparametric regression. Ann. Statist. 23 1937-1959.
• [34] Neumann, M. H. and Kreiss, J.-P. (1998). Regression-ty pe inference in nonparametric autoregression. Ann. Statist. 26 1570-1613.
• [35] Neumann, M. H. and Paparoditis, E. (1998). A nonparametric test for the stationary density. Discussion Paper 58/98, SFB 373, Humboldt Univ., Berlin.
• [36] Pham, T. D. (1986). The mixing property of bilinear and generalised random coefficient autoregressive models. Stochastic Process. Appl. 23 291-300.
• [37] Pham, T. D. and Tran, L. T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297-303.
• [38] Reiss, R.-D. (1993). A Course on Point Processes. Springer, New York.
• [39] Robinson, P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185-207.
• [40] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
• [41] Silverman, B. W. and Young, G. A. (1987). The bootstrap: to smooth or not to smooth? Biometrika 74 469-479.
• [42] Spokoiny, V. G. (1996). Adaptive and spatially adaptive testing of a nonparametric hy pothesis. Math. Methods Statist. To appear.
• [43] Takahata, H. and Yoshihara, K. (1987). Central limit theorems for integrated square error of nonparametric density estimators based on absolutely regular random sequences. Yokohama Math. J. 35 95-111.
• [44] Tran, L. T. (1990). Density estimation for time series by histograms. J. Statist. Plann. Inference 40 61-79.
• [45] Yu, B. (1993). Density estimation in the L norm for dependent data with applications to the Gibbs sampler. Ann. Statist. 21 711-735.