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Functional CLT for nonparametric estimates of the spectrum and change- point problem for a spectral function. (English. Russian original) Zbl 0761.62123

Lith. Math. J. 30, No. 4, 302-322 (1990); translation from Lit. Mat. Sb. 30, No. 4, 674-697 (1990).
Summary: We prove the c.l.t. (on convergence in the Skorokhod space of two- parameter functions) for spectral statistics analogous to statistics of Kolmogorov-Smirnov type connected with the change-point of the spectrum of a stationary moving average sequence. The paper generalizes and supplements the results of D. Picard [Adv. Appl. Probab. 17, 841- 867 (1985; Zbl 0585.62151)] relating to the case of a Gaussian sequence.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62G07 Density estimation
60F17 Functional limit theorems; invariance principles

Citations:

Zbl 0585.62151
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References:

[1] Z. Antoshevskii and R. Bentkus, ?Asymptotic estimate of the spectral function of a stationary Gaussian sequence,? Liet. Mat. Rinkinys,16, No. 2, 5-19 (1976). · Zbl 0334.62038
[2] T. V. Arak, ?Rate of convergence of the distribution of the maximal deviation of an estimate of the spectrum of a Gaussian random sequence,? Dokl. Akad. Nauk SSSR,201, No. 5, 1019-1021 (1971). · Zbl 0268.62031
[3] R. Bentkus, ?Asymptotic behavior of an estimate of the spectral function of a multidimensional stationary Gaussian sequence,? Liet. Mat. Rinkinys,11, No. 4, 745-760 (1971). · Zbl 0229.62049
[4] R. Bentkus, ?Asymptotic normality of an estimate of the spectral function,? Liet. Mat. Rinkinys,12, No. 3, 5-18 (1972).
[5] R. Bentkus, ?Cumulants of estimates of the spectrum of a stationary sequence,? Liet. Mat. Rinkinys,16, No. 4, 37-61 (1976). · Zbl 0373.62056
[6] P. Billingsley, Convergence of Probability Measures [Russian translation], Nauka, Moscow (1977).
[7] L. Giraitis, ?Central limit theorem for polynomial forms. I,? Liet. Mat. Rinkinys,29, No. 2, 266-289 (1989). · Zbl 0687.60028
[8] I. A. Ibragimov, ?Estimate of the spectral function of a stationary Gaussian process,? Teor. Veroyatn. Primen.,8, No. 4, 391-430 (1963).
[9] I. A. Ibragimov and T. M. Tovstik, ?Estimate of spectral functions of a class of, stationary random sequences,? Vestn. Leningr. Gos. Univ., No. 1, 42-57 (1964).
[10] N. Kligene and L. Tel’ksnis, ?Methods of detection of moments of change of properties of stochastic processes,? Avtomat. Telemekh., No. 10, 5-56 (1983).
[11] R. Leipus, ?Weak convergence of two-parameter empirical fields in change-point problems,? Liet. Mat. Rinkinys,28, No. 4, 716-723 (1988). · Zbl 0673.62039
[12] T. L. Malevich, ?Asymptotic behavior of an estimate of the spectral function of a stationary Gaussian process,? Teor. Veroyatn. Primen.,9, No. 2, 386-390 (1964).
[13] A. N. Shiryaev, ?Optimal methods in problems of fastest discovery,? Teor. Veroyatn. Primen.,8, No. 1, 26-51 (1963).
[14] F. Avram, ?On bilinear forms in Gaussian random variables and Teoplitz metrices,? Prob. Th. Rel. Fiels,79, No. 1, 37-45 (1988). · Zbl 0648.60043
[15] P. J. Bickel and M. J. Wichura ?Convergance criteria for multiparameter stochastic processes and some applications,?. Ann. Math. Statist.,42, No. 5, 1656-1670 (1971). · Zbl 0265.60011
[16] D. R. Brillinger, ?Asymptotic properties of spectral estimates of second order,? Biometrika,56, No. 2, 375-390 (1969). · Zbl 0179.23902
[17] M. Csorgo and L. Horvath, ?nonparametric methods for changepoint problems,? in: Handbook of Statistics, Vol. 7 (P. R. Krishnaiah and C. R. Rao, eds.), North-Holland, Elsevier (1988), pp. 403-425.
[18] R. Dahlhaus, ?Empirical spectral processes and their applications to time series analysis,? Stoch. Proc. Appl.,30, No. 1, 69-83 (1988). · Zbl 0655.60033
[19] R. Dahlhaus, ?A functional limit theorem for tapered empirical spectral functions,? Stoch. Proc. Appl.,19, No. 1, 135-149 (1985). · Zbl 0568.60038
[20] J. Deshayes and D. Picard, ?Off-line statistical analysis of changepoint models using nonparametric and likelihood methods,? Lect. Notes Control Inform. Sci.,77, 103-168 (1986).
[21] T. W. Epps, ?Testing that a Gaussian process is stationary,? Ann. Stat.,16, No. 4, 1667-1683 (1988). · Zbl 0653.62063
[22] R. Fox and M. S. Taqqu, ?Central limit theorems for quadratic forms in random variables having long-range dpendence,? Proc. Th. Rel. Fields,74, No. 2, 213-240 (1987). · Zbl 0586.60019
[23] L. Giraitis and D. Surgailis, ?CLT for quadratic forms in strongly dependent linear variables and application to asymptotical normality of Whittle’s estimate,? (to appear). · Zbl 0717.62015
[24] U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series, Almqvist and Wiksell, Stockholm (1956). · Zbl 0072.36401
[25] D. L. Hawkins, ?Retrospective and sequential tests for a change in distribution based on Kolmogorov-Smirnoy type statistics,? Sequential Analysis,7, No. 1, 23-51 (1988). · Zbl 0657.62090
[26] I. B. MacNeill, ?Limit processes for co-spectral and quadrature spectral distribution function,? Ann. Math. Statist.,42, No. 1, 81-96 (1971). · Zbl 0218.62112
[27] G. Neuhaus, ?On weak convergence of stochastic processes with multidimensional time parameter,? Ann. Math. Stat.,42, No. 4, 1285-1295 (1971). · Zbl 0222.60013
[28] D. Picard, ?Testing and estimating change-points in time series.,? Adv. Appl. Probab.,17, No. 4, 841-867 (1985). · Zbl 0585.62151
[29] D. A. Wolfe and E. Schechtman, ?Nonparametric statistical procedures for the changepoint problem,? J. Stat. Plan. Inf.,9, No. 3, 389-396 (1984). · Zbl 0561.62039
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