Estimate the memory parameter

Estimates the memory parameter \(d\) from the leading score process, the centred curves projected onto the leading eigenfunction of a covariance-type operator. If the estimate is near the upper end of its search interval, the score process is integer differenced and \(d\) is estimated again, until the estimate falls below the upper end. The number of differences is then added back to the estimate.

Usage

estimate_memory(
  x,
  method = c("bc_lpwn2", "bc_lpwn1", "lpwn2", "lpwn1", "lw", "dfa"),
  cov_type = c("sample", "lrc_RS", "lrc_LRS"),
  m_const = 0.75,
  B = 100,
  burn_in = 200,
  R_noise = 0,
  tuning_const = 1,
  cov_band_const = 1,
  dfa_levels = 50,
  dfa_minnpts = 3,
  dfa_cutoff = 10^c(0.7, 2.5),
  seed = NULL,
  verbose = FALSE
)

Arguments

x: A matrix of curves (grid points by time).
method: Estimator: "lw" (local Whittle), "dfa" (detrended fluctuation analysis), "lpwn1" or "lpwn2" (local polynomial Whittle with noise, with one or two polynomial correction terms), or "bc_lpwn1" or "bc_lpwn2" (BC-LPWN1 or BC-LPWN2, the LPWN estimate bias-corrected by the prefiltered sieve bootstrap).
cov_type: Operator of the FPCA that gives the leading eigenfunction: "sample" (empirical covariance operator), "lrc_RS" (long-run covariance operator with an adaptive bandwidth) or "lrc_LRS" (long-run covariance operator in sandwich form).
m_const: Bandwidth exponent \(a\), so that \(\lfloor 1 + T^a \rfloor\) frequencies are used.
B, burn_in: Number of bootstrap samples and burn-in length (BC-LPWN).
R_noise: Number of polynomial terms for the noise in LPWN.
tuning_const: Constant \(c\). The score process is differenced while the estimate is at least \(1 - c / \log m\).
cov_band_const: Bandwidth constant for cov_type = "lrc_LRS".
dfa_levels, dfa_minnpts, dfa_cutoff: Number of block sizes, minimum number of points per block and range of block sizes in the DFA regression.
seed: Optional seed for the bootstrap.
verbose: Whether to show progress.

Value

An object of class farfima_memory with the estimate d and the leading score process scores. BC-LPWN also returns the uncorrected estimate d_initial, the bias and the bootstrap estimates.

Details

The local Whittle and LPWN code is adapted from the LongMemoryTS package (Leschinski, Voges and Wenger, 2019), and the DFA code from the fArma package (Wuertz, Setz and Chalabi, 2017).

References

Robinson, P. M. (1995). Gaussian semiparametric estimation of long range dependence. The Annals of Statistics.

Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., Stanley, H. E. and Goldberger, A. L. (1994). Mosaic organization of DNA nucleotides. Physical Review E.

Frederiksen, P., Nielsen, F. S. and Nielsen, M. O. (2012). Local polynomial Whittle estimation of perturbed fractional processes. Journal of Econometrics.

Paparoditis, E. (2018). Sieve bootstrap for functional time series. The Annals of Statistics.

Li, D., Robinson, P. M. and Shang, H. L. (2020). Long-range dependent curve time series. Journal of the American Statistical Association.

Rice, G. and Shang, H. L. (2017). A plug-in bandwidth selection procedure for long-run covariance estimation with stationary functional time series. Journal of Time Series Analysis.

Liu, C. and Shang, H. L. (2026). Bias correction of long-memory estimator of functional time series via the prefiltered sieve bootstrap. arXiv:2607.28177.

Leschinski, C., Voges, M. and Wenger, K. (2019). LongMemoryTS: Long Memory Time Series. R package version 0.1.0.

Wuertz, D., Setz, T. and Chalabi, Y. (2017). fArma: Rmetrics - Modelling ARMA Time Series Processes. R package version 3042.81.

Examples

x = sim_farfima(n = 300, d = 0.3, p = 1, q = 0, seed = 1)
estimate_memory(x, method = "lw")
#> Memory estimate d = 0.6358
#>   LW on the leading score process of the empirical covariance operator (Covar), bandwidth exponent 0.75
estimate_memory(x, method = "lpwn2")
#> Memory estimate d = 0.5871
#>   LPWN2 on the leading score process of the empirical covariance operator (Covar), bandwidth exponent 0.75