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Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence

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  • Heyde, C. C.
  • Gay, R.

Abstract

In this paper we establish central limit theorems for the smoothed unbiased periodogram [integral operator][pi]-[pi]...[integral operator][pi]-[pi]g([omega],[theta]){I*T,X([omega])-EI*T,X([omega])}d[omega]1...d[omega]r, where {Xt} is a stationary r-dimensional random process or random field, possibly with long-range dependence, which is not necessarily Gaussian. Here I*T,X([omega]) is the unbiased periodogram and g([omega],[theta]) is a smoothing function satisfying modest regularity conditions. This result implies asymptotic normality of the asymptotic quasi-likelihood estimator of a distributional characteristic [theta] of the process {Xt} under very general conditions. In particular, these results show the asymptotic optimality of the Whittle estimation procedure for both short and long-range dependence in the absence of the Gaussian assumption, and extend those of Giraitis and Surgailis (1990) for the case r = 1.

Suggested Citation

  • Heyde, C. C. & Gay, R., 1993. "Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 45(1), pages 169-182, March.
  • Handle: RePEc:eee:spapps:v:45:y:1993:i:1:p:169-182
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    Cited by:

    1. Casas, Isabel & Gao, Jiti, 2008. "Econometric estimation in long-range dependent volatility models: Theory and practice," Journal of Econometrics, Elsevier, vol. 147(1), pages 72-83, November.
    2. Hualde, Javier, 2013. "A simple test for the equality of integration orders," Economics Letters, Elsevier, vol. 119(3), pages 233-237.
    3. Gao, Jiti, 2002. "Modeling long-range dependent Gaussian processes with application in continuous-time financial models," MPRA Paper 11973, University Library of Munich, Germany, revised 18 Sep 2003.
    4. Pai, Jeffrey & Ravishanker, Nalini, 2009. "A multivariate preconditioned conjugate gradient approach for maximum likelihood estimation in vector long memory processes," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1282-1289, May.
    5. Arteche, Josu, 2004. "Gaussian semiparametric estimation in long memory in stochastic volatility and signal plus noise models," Journal of Econometrics, Elsevier, vol. 119(1), pages 131-154, March.
    6. Ravishanker, Nalini & Ray, Bonnie K., 2002. "Bayesian prediction for vector ARFIMA processes," International Journal of Forecasting, Elsevier, vol. 18(2), pages 207-214.
    7. Zaffaroni, Paolo & d'Italia, Banca, 2003. "Gaussian inference on certain long-range dependent volatility models," Journal of Econometrics, Elsevier, vol. 115(2), pages 199-258, August.
    8. Robinson, P.M. & Vidal Sanz, J., 2006. "Modified Whittle estimation of multilateral models on a lattice," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1090-1120, May.
    9. Gao, Jiti & Anh, Vo & Heyde, Chris, 2002. "Statistical estimation of nonstationary Gaussian processes with long-range dependence and intermittency," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 295-321, June.
    10. Leonenko, N.N. & Sakhno, L.M., 2006. "On the Whittle estimators for some classes of continuous-parameter random processes and fields," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 781-795, April.
    11. Anh, V.V. & Leonenko, N.N. & Sakhno, L.M., 2007. "Statistical inference using higher-order information," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 706-742, April.
    12. Pai, Jeffrey & Ravishanker, Nalini, 2009. "Maximum likelihood estimation in vector long memory processes via EM algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4133-4142, October.
    13. Jaroslav Mohapl, 1998. "On Maximum Likelihood Estimation for Gaussian Spatial Autoregression Models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(1), pages 165-186, March.
    14. Hosoya, Yuzo, 1996. "The quasi-likelihood approach to statistical inference on multiple time-series with long-range dependence," Journal of Econometrics, Elsevier, vol. 73(1), pages 217-236, July.
    15. Pai, Jeffrey & Ravishanker, Nalini, 2015. "Fast approximate likelihood evaluation for stable VARFIMA processes," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 160-168.
    16. Ayache, Antoine & Lévy Véhel, Jacques, 2004. "On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 119-156, May.
    17. Rosa Espejo & Nikolai Leonenko & Andriy Olenko & María Ruiz-Medina, 2015. "On a class of minimum contrast estimators for Gegenbauer random fields," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 657-680, December.
    18. repec:bla:jtsera:v:38:y:2017:i:2:p:204-224 is not listed on IDEAS
    19. Beran, Jan & Ghosh, Sucharita & Schell, Dieter, 2009. "On least squares estimation for long-memory lattice processes," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2178-2194, November.

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