Evaluation of bias in higher-order spectral estimation
Authors:
V. V. Anh, N. N. Leonenko and L. M. Sakhno
Journal:
Theor. Probability and Math. Statist. 80 (2010), 1-14
MSC (2000):
Primary 62M15, 62M30
DOI:
https://doi.org/10.1090/S0094-9000-2010-00790-3
Published electronically:
August 18, 2010
MathSciNet review:
2541947
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper is concerned with the estimation of integral functionals of higher-order spectral densities for stationary random fields. It is shown that in some cases the problem of bias due to edge effects can be resolved by tapering.
References
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence, J. Appl. Probab. 41A (2004), 35–53. Stochastic methods and their applications. MR 2057564, DOI https://doi.org/10.1017/s0021900200112197
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Minimum contrast estimation of random processes based on information of second and third orders, J. Statist. Plann. Inference 137 (2007), no. 4, 1302–1331. MR 2301481, DOI https://doi.org/10.1016/j.jspi.2006.03.001
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Statistical inference using higher-order information, J. Multivariate Anal. 98 (2007), no. 4, 706–742. MR 2322125, DOI https://doi.org/10.1016/j.jmva.2006.09.009
- R. Bentkus, Cumulants of estimates of the spectrum of a stationary sequence, Litovsk. Mat. Sb. 16 (1976), no. 4, 37–61, 252–253 (Russian, with Lithuanian and English summaries). MR 0431571
- R. Bentkus and V. Rutkauskas, The asymptotic behavior of the first two moments of second order spectral estimates, Litovsk. Mat. Sb. 13 (1973), no. 1, 29–45, 230 (Russian, with Lithuanian and English summaries). MR 0319334
- R. Bentkus, R. Rudzkis, and Ju. Sušinskas, The average of the estimates of the spectrum of a homogeneous field, Litovsk. Mat. Sb. 14 (1974), no. 3, 67–74, 235 (Russian, with Lithuanian and English summaries). MR 0386193
- David R. Brillinger, The frequency analysis of relations between stationary spatial series., Proc. Twelfth Biennial Sem. Canad. Math. Congr. on Time Series and Stochastic Processes; Convexity and Combinatorics (Vancouver, B.C., 1969) Canad. Math. Congr., Montreal, Que., 1970, pp. 39–81. MR 0273763
- David R. Brillinger and Murray Rosenblatt, Asymptotic theory of estimates of $k$-th order spectra, Spectral Analysis Time Series (Proc. Advanced Sem., Madison, Wis., 1966), John Wiley, New York, 1967, pp. 153–188. MR 0211566
- Rainer Dahlhaus, Spectral analysis with tapered data, J. Time Ser. Anal. 4 (1983), no. 3, 163–175. MR 732895, DOI https://doi.org/10.1111/j.1467-9892.1983.tb00366.x
- Rainer Dahlhaus, A functional limit theorem for tapered empirical spectral functions, Stochastic Process. Appl. 19 (1985), no. 1, 135–149. MR 780726, DOI https://doi.org/10.1016/0304-4149%2885%2990045-6
- R. Dahlhaus and H. Künsch, Edge effects and efficient parameter estimation for stationary random fields, Biometrika 74 (1987), no. 4, 877–882. MR 919857, DOI https://doi.org/10.1093/biomet/74.4.877
- Michael Eichler, Empirical spectral processes and their applications to stationary point processes, Ann. Appl. Probab. 5 (1995), no. 4, 1161–1176. MR 1384370
- Xavier Guyon, Parameter estimation for a stationary process on a $d$-dimensional lattice, Biometrika 69 (1982), no. 1, 95–105. MR 655674, DOI https://doi.org/10.1093/biomet/69.1.95
- Xavier Guyon, Random fields on a network, Probability and its Applications (New York), Springer-Verlag, New York, 1995. Modeling, statistics, and applications; Translated from the 1992 French original by Carenne Ludeña. MR 1344683
- Daniel MacRae Keenan, Limiting behavior of functionals of higher-order sample cumulant spectra, Ann. Statist. 15 (1987), no. 1, 134–151. MR 885728, DOI https://doi.org/10.1214/aos/1176350257
- V. P. Leonov and A. N. Sirjaev, On a method of semi-invariants, Theor. Probability Appl. 4 (1959), 319–329. MR 123345, DOI https://doi.org/10.1137/1104031
- P. M. Robinson, Nonparametric spectrum estimation for spatial data, J. Statist. Plann. Inference 137 (2007), no. 3, 1024–1034. MR 2301732, DOI https://doi.org/10.1016/j.jspi.2006.06.021
- Ludmila Sakhno, Bias control in the estimation of spectral functionals, Theory Stoch. Process. 13 (2007), no. 1-2, 225–233. MR 2343825
- Masanobu Taniguchi, On estimation of the integrals of the fourth order cumulant spectral density, Biometrika 69 (1982), no. 1, 117–122. MR 655676, DOI https://doi.org/10.1093/biomet/69.1.117
- Masanobu Taniguchi, Minimum contrast estimation for spectral densities of stationary processes, J. Roy. Statist. Soc. Ser. B 49 (1987), no. 3, 315–325. MR 928940
- Rama Chellappa and Anil Jain (eds.), Markov random fields, Academic Press, Inc., Boston, MA, 1993. Theory and application. MR 1214376
- J. Yuan and T. Subba Rao, Higher order spectral estimation for random fields, Multidimens. Systems Signal Process. 4 (1993), no. 1, 7–22. MR 1202189, DOI https://doi.org/10.1007/BF00986003
References
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Quasilikelihood-based higher-order spectral estimation of random processes and fields with possible long-range dependence, J. Appl. Probab. 41A (2004), 35–54. MR 2057564 (2005j:62169)
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Minimum contrast estimation of random processes based on information of second and third orders, J. Statist. Plann. Inference 137 (2007), 1302–1331. MR 2301481 (2008h:62218)
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Statistical inference using higher-order information, J. Multivariate Anal. 98 (2007), no. 4, 706–742. MR 2322125 (2009a:62345)
- R. Bentkus, Cumulants of estimates of the spectrum of a stationary sequence, Liet. Mat. Rink. 16 (1976), 37–61. (Russian) MR 0431571 (55:4568)
- R. Bentkus and R. Rutkauskas, On the asymptotics of the first two moments of second order spectral estimators, Litovsk. Mat. Sb. 13 (1973), 29–45. (Russian) MR 0319334 (47:7878)
- R. Bentkus, R. Rutkauskas, and Ju. Sušinskas, The average of the estimates of the spectrum of a homogeneous field, Liet. Mat. Rink. 14 (1974), 67–74. (Russian) MR 0386193 (52:7051)
- D. R. Brillinger, The frequency analysis of relations between stationary spatial series, Proceedings of the Twelfth Biennial Seminar of the Canadian Mathematical Congress, Montreal, 1970, pp. 39–81. MR 0273763 (42:8640)
- D. R. Brillinger and M. Rosenblatt, Asymptotic theory of estimates of $k$-th order spectra, Spectral Anal. Time Ser. (B. Harris, ed.), Wiley, New York, 1967. MR 0211566 (35:2444)
- R. Dahlhaus, Spectral analysis with tapered data, J. Time Ser. Anal. 4 (1983), 163–175. MR 732895 (85c:62246)
- R. Dahlhaus, A functional limit theorem for tapered empirical spectral functions, Stoch. Process. Appl. 19 (1985), 135–149. MR 780726 (86i:60089)
- R. Dahlhaus and H. Künsch, Edge effects and efficient parameter estimation for stationary random fields, Biometrika 74 (1987), 877–882, 39–81. MR 919857 (89g:62146)
- M. Eichler, Empirical spectral processes and their applications to stationary point processes, Ann. Appl. Probab. 5 (1995), no. 4, 1161–1176. MR 1384370 (97k:60062)
- X. Guyon, Parameter estimation for a stationary process on a $d$-dimensional lattice, Biometrika 69 (1982), 95–105. MR 655674 (83j:62128)
- X. Guyon, Random Fields on a Network: Modelling, Statistics and Applications, Springer, New York, 1995. MR 1344683 (96m:60119)
- D. M. Keenan, Limiting behavior of functionals of higher-order sample cumulant spectra, Ann. Stat. 15 (1987), no. 1, 134–151. MR 885728 (88k:62176)
- V. V. Leonov and A. N. Shiryaev, On a method of calculation of semi-invariants, Theor. Probab. Appl. 4 (1959), 319–329. MR 0123345 (23:A673)
- P. M. Robinson, Nonparametric spectrum estimation for spatial data, J. Statist. Plann. Inference 137 (2007), no. 3, 1024–1034. MR 2301732 (2008m:62157)
- L. Sakhno, Bias control in the estimation of spectral functionals, Theory Stoch. Process. 1329 (2007), no. 1–2, 225–233. MR 2343825 (2008m:62087)
- M. Taniguchi, On estimation of the integrals of the fourth order cumulant spectral density, Biometrika 69 (1982), 117–122. MR 655676 (83g:62144)
- M. Taniguchi, Minimum contrast estimation for spectral densities of stationary processes, J. R. Stat. Soc., Ser. B 49 (1987), 315–325. MR 928940 (89c:62163)
- J. Yuan and T. Subba Rao, Spectral estimation for random fields with applications to Markov modelling and texture classification, Markov Random Fields: Theory and Applications (R. Chellappa and A. K. Jain, eds.), Academic Press, Boston, 1993, pp. 179–209. MR 1214376 (93j:68224)
- J. Yuan and T. Subba Rao, Higher order spectral estimation for random fields, Multidimensional Syst. Signal Process. 4 (1993), no. 1, 7–22. MR 1202189 (93k:62198)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
62M15,
62M30
Retrieve articles in all journals
with MSC (2000):
62M15,
62M30
Additional Information
V. V. Anh
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australia
Email:
v.anh@qut.edu.au
N. N. Leonenko
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4YH, United Kingdom
Email:
LeonenkoN@Cardiff.ac.uk
L. M. Sakhno
Affiliation:
Department of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska 64, Kyiv 01033, Ukraine
Email:
lms@univ.kiev.ua
Keywords:
Spectral estimation,
bias,
higher-order spectral densities,
minimum contrast estimation
Received by editor(s):
March 12, 2009
Published electronically:
August 18, 2010
Additional Notes:
Partly supported by the Australian Research Council grant DP0559807 and the Commission of the European Communities grant PIRSES-GA-2008-230804 within the programme “Marie Curie Actions”
Article copyright:
© Copyright 2010
American Mathematical Society
