Monotone spectral density estimation. (English) Zbl 1209.62206

Summary: We propose two estimators of a monotone spectral density, that are based on the periodogram. These are the isotonic regression of the periodogram and the isotonic regression of the log-periodogram. We derive pointwise limit distribution results for the proposed estimators for short memory linear processes and long memory Gaussian processes and also that the estimators are rate optimal.


62M15 Inference from stochastic processes and spectral analysis
62E20 Asymptotic distribution theory in statistics
62G08 Nonparametric regression and quantile regression
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G07 Density estimation


sapa; ftnonpar
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[1] Anevski, D. and Hössjer, O. (2006). A general asymptotic scheme for inference under order restrictions. Ann. Statist. 34 1874-1930. · Zbl 1246.62019
[2] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[3] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods . Springer, New York. · Zbl 0709.62080
[4] Dahlhaus, R. (1989). Efficient parameter estimation for self-similar processes. Ann. Statist. 17 1749-1766. · Zbl 0703.62091
[5] Davies, P. L. and Kovac, A. (2004). Densities, spectral densities and modality. Ann. Statist. 32 1093-1136. · Zbl 1093.62042
[6] Dzhaparidze, K. (1986). Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series . Springer, New York. · Zbl 0584.62157
[7] Gill, R. D. and Levit, B. Y. (1995). Applications of the Van Trees inequality: A Bayesian Cramér-Rao bound. Bernoulli 1 59-79. · Zbl 0830.62035
[8] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388-400. JSTOR: · Zbl 04567029
[9] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Application . Academic Press, New York. · Zbl 0462.60045
[10] Holden, A. V. (1976). Models of the Stochastic Activity of Neurons . Springer, Berlin. · Zbl 0353.92001
[11] Hörmander, L. (2007). Notions of Convexity . Birkhäuser, Boston. · Zbl 1108.32001
[12] Hurvich, C. M., Deo, R. and Brodsky, J. (1998). The mean squared error of Geweke and Porter-Hudak’s estimator of the memory parameter of a long-memory time series. J. Time Ser. Anal. 19 19-46. · Zbl 0920.62108
[13] Kerner, C. and Harris, P. E. (1994). Scattering attenuation in sediments modeled by ARMA processes-validation of simple Q modeles. Geophysics 53 1813-1826.
[14] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Isoperimetry and Processes . Springer, Berlin. · Zbl 0748.60004
[15] Mammen, E. (1991). Estimating a smooth monotone regression function. Ann. Statist. 19 724-740. · Zbl 0737.62038
[16] McCoy, E. J., Walden, A. T. and Percival, D. B. (1998). Multitaper spectral estimation of power law processes. IEEE Trans. Signal Process. 46 655-668. · Zbl 0988.94505
[17] McHardy, I. and Czerny, B. (1987). Fractal X-ray time variablity and spectral invariance of the Seyfert galaxy NGC 5507. Nature 325 696-698.
[18] Mikosch, T. and Norvaiša, R. (1997). Uniform convergence of the empirical spectral distribution function. Stochastic Process. Appl. 70 85-114. · Zbl 0913.60032
[19] Moulines, E. and Soulier, P. (1999). Broadband log-periodogram regression of time series with long-range dependence. Ann. Statist. 27 1415-1439. · Zbl 0962.62085
[20] Munk, W. H. and Macdonald, G. J. F. (2009). The Rotation of the Earth, A Geophysical Discussion . Cambridge Univ. Press, Cambridge. · Zbl 1168.86001
[21] Percival, D. B. (1991). Characterization of frequency stability: Frequence-domain estimation of stability measures. IEEE Process. 79 961-972.
[22] Percival, D. B. and Walden, A. T. (1993). Spectral Analysis for Physical Applications . Cambridge Univ. Press, Cambridge. · Zbl 0796.62077
[23] Prakasa Rao, B. L. S. P. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23-36. · Zbl 0181.45901
[24] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference . Wiley, Chichester. · Zbl 0645.62028
[25] Robinson, P. M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048-1072. · Zbl 0838.62085
[26] Soulier, P. (2001). Moment bounds and central limit theorem for functions of Gaussian vectors. Statist. Probab. Lett. 54 193-203. · Zbl 0993.60019
[27] Wright, F. T. (1981). The asymptotic behavior of monotone regression estimates. Ann. Statist. 9 443-448. · Zbl 0471.62062
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