Statistical properties of detrended fluctuation analysis

Abstract

The main goal of this work is to consider the detrended fluctuation analysis (DFA), proposed by Peng et al. [Mosaic organization of DNA nucleotides, Phys. Rev. E. 49(5) (1994), 1685–1689]. This is a well-known method for analysing the long-range dependence in non-stationary time series. Here we describe the DFA method and we prove its consistency and its exact distribution, based on the usual i.i.d. assumption, as an estimator for the fractional parameter d. In the literature it is well established that the nucleotide sequences present long-range dependence property. In this work, we analyse the long dependence property in view of the autoregressive moving average fractionally integrated ARFIMA(p, d, q) processes through the analysis of four nucleotide sequences. For estimating the fractional parameter d we consider the semiparametric regression method based on the periodogram function, in both classical and robust versions; the semiparametric R/S(n) method, proposed by Hurst [Long term storage in reservoirs, Trans. Am. Soc. Civil Eng. 116 (1986), 770–779] and the maximum likelihood method (see [R. Fox and M.S. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann. Statist. 14 (1986), 517–532]), by considering the approximation suggested by Whittle [Hypothesis Testing in Time Series Analysis (1953), Hafner, New York].

Keywords:

• long memory
• detrended fluctuation analysis
• semiparametric estimation
• robustness

AMS Classification: :

• Primary: 91B70
• Secondary: 00A05

Acknowledgements

N. Crato acknowledges the partial support by FCT-Fundação para a Ciência e Tecnologia (Programme FEDER/POCI 2010), Portugal. The work of R.R. Linhares was supported by CNPq-Brazil. S.R.C. Lopes acknowledges the partial supported by CNPq-Brazil, by CAPES-Brazil, by Millennium Institute in Probability and also by Pronex Probabilidade e Processos Estocásticos - E-26/170.008/2008 -APQ1.

The authors thank two anonymous referees and both the editor and the associate editor for their valuable comments and suggestions that improved the final version of the manuscript.