An -criterion for testing a hypothesis about the covariance function of a random sequence

Author:
T. O. Yanevich

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **92** (2015).

Journal:
Theor. Probability and Math. Statist. **92** (2016), 163-173

MSC (2010):
Primary 60G15; Secondary 60G10

DOI:
https://doi.org/10.1090/tpms/990

Published electronically:
August 10, 2016

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An -criterion for testing a hypothesis about the covariance function for a centered stationary Gaussian sequence is constructed in this paper. The criterion is analyzed for some particular cases by using the simulated data.

**1.**T. W. Anderson,*The statistical analysis of time series*, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR**0283939****2.**George E. P. Box, Gwilym M. Jenkins, and Gregory C. Reinsel,*Time series analysis*, 4th ed., Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2008. Forecasting and control. MR**2419724****3.**G. E. P. Box and David A. Pierce,*Distribution of residual autocorrelations in autoregressive-integrated moving average time series models*, J. Amer. Statist. Assoc.**65**(1970), 1509–1526. MR**0273762****4.**Peter J. Brockwell and Richard A. Davis,*Time series: theory and methods*, 2nd ed., Springer Series in Statistics, Springer-Verlag, New York, 1991. MR**1093459****5.**V. V. Buldygin and Yu. V. Kozachenko,*Metric characterization of random variables and random processes*, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR**1743716****6.**Willa W. Chen and Rohit S. Deo,*A generalized Portmanteau goodness-of-fit test for time series models*, Econometric Theory**20**(2004), no. 2, 382–416. MR**2044276**, https://doi.org/10.1017/S0266466604202067**7.**O. Ē. Kamenshchikova and T. O. Yanevich,*Approximation of 𝐿_{𝑝}(Ω)-processes*, Teor. Ĭmovīr. Mat. Stat.**83**(2010), 59–68 (Ukrainian, with English and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**83**(2011), 71–82. MR**2768849**, https://doi.org/10.1090/S0094-9000-2012-00842-9**8.**Yu. V. Kozachenko and T. O. Ianevych,*Some goodness of fit tests for random sequences*, Lith. J. Statist.**52**(2013), no. 1, 5-13.**9.**Yurij V. Kozachenko and Olexander V. Stus,*Square-Gaussian random processes and estimators of covariance functions*, Math. Commun.**3**(1998), no. 1, 83–94 (English, with English and Croatian summaries). MR**1648867****10.**G. M. Ljung and G. E. P. Box,*On a measure on lack of fit in time series models*, Biometrica**65**(1978), no. 2, 297-303.**11.**A. I. McLeod and W. K. Li,*Diagnostic checking ARMA time series models using squared-residual autocorrelations*, J. Time Ser. Anal.**4**(1983), no. 4, 269–273. MR**738587**, https://doi.org/10.1111/j.1467-9892.1983.tb00373.x**12.**Carl Edward Rasmussen and Christopher K. I. Williams,*Gaussian processes for machine learning*, Adaptive Computation and Machine Learning, MIT Press, Cambridge, MA, 2006. MR**2514435****13.**O. O. Vasylyk, Yu. V. Kozachenko, and T. O. Yakovenko,*Simulation of stationary random sequences*, Visnyk Kyiv Univ. Ser. Fiz. Mat. Nauk (2009), no. 1, 7-10. (Ukrainian)**14.**Yu. V. Kozachenko and T. O. Yakovenko,*A criterion for testing hypothesis about the covariance function of a stationary Gaussian random sequence*, Visnyk Uzhgorod Univ. Ser. Mat. Inform. (2010), no. 20, 39-43. (Ukrainian)

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Additional Information

**T. O. Yanevich**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, 01601, Kyiv, Ukraine

Email:
yata452@univ.kiev.ua

DOI:
https://doi.org/10.1090/tpms/990

Keywords:
Square Gaussian random variables,
random sequences,
time series,
covariance functions

Received by editor(s):
May 5, 2015

Published electronically:
August 10, 2016

Article copyright:
© Copyright 2016
American Mathematical Society