Abstract
We consider a, discrete time, weakly stationary bidimensional process, for which the spectral measure is the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines. In this paper we are interested in estimating the spectral density of the absolutely continuous measure and of the density on the lines. For this aim, by using the double kernel method, we construct consistent estimators of these densities and we study their asymptotic behaviors in term of the mean squared error with rate.
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References
Bartlett MS (1955) An introduction to stochastic processes with special reference to methods and applications, 2nd edn. Cambridge University Press, London
Benhenni K, Rachdi M (2007) Bispectrum estimation for a continuous-time stationary processes from random sampling. In: Christos H. Skiadas (eds) Recent advances in stochastic modelling and data analysis. World Scientific Publishing Co Pte Ltd, Singapore, pp 442–453
Charlot F, Rachdi M (2008) On the statistical properties of a stationary process sampled by a stationary point process. Stat Probab Lett 78(4): 456–462
Bosq D (1998) Nonparametric statistics for stochastic processes: estimation and prediction, 2nd edn. Lecture notes in statistics, vol 110. Springer, Berlin
Brillinger DR (1981) Time series. Data analysis and theory. Expanded edition (English), holden-day series in time series analysis. San Francisco etc.: Holden- Day, Inc. XII, p 540
Heine V (1955) Models for two dimensional stationary stochastic processes. Biometrika 42: 170–178
Lii KS, Masry E (1994) Spectral estimation of time stationary processes from random sampling. Stoch Process Appl 52: 39–64
Longuett-Higgins MS (1957) The statistical analysis of a randomly moving surface. Philos Trans Roy Soc Lond Ser A(249): 287–321
Pierson WJ, Tick LJ (1957) Stationary random processes in meteorology and oceanography. Bull Inst Int Stat 35(2): 271–281
Porcu E, Crujeiras RM, Mateu J, González-Manteiga W (2007) On the second order properties of the multidimensional periodogram for regularly spaced data. Theory Prob Appl (SIAM) (in press)
Priestley M (1981) Spectral analysis and time series, probability and mathematical statistics. Academic Press, London
Rachdi M (1998) Estimation de la densité de la mesure spectrale mixte pour un processus p-adique stationnaire. Ann. de l’I.S.U.P. 42(2–3): 75–91
Rachdi M, Sabre R (1998) Le choix optimal de la largeur de fenêtre spectrale pour un champ aléatoire. Trait Signal 15(6): 569–575
Sabre R (1995) Spectral density estimation for stationary random fields. Appl Math 2: 107–133
Sabre R (2002) Aliasing free for symmetric stable random fields. Egypt Stat J 46: 53–75
Walker AM, Young A (1955) The analysis of observations on the variations of the latitude. Mon Not R Astron Soc 115(443)
Whittle P (1954) On the stationary processes on the plane. Biometrika 41: 434–449
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Rachdi, M., Sabre, R. Mixed-spectra analysis for stationary random fields. Stat Methods Appl 18, 333–358 (2009). https://doi.org/10.1007/s10260-008-0107-7
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DOI: https://doi.org/10.1007/s10260-008-0107-7