longmemo (version 1.1-3)

ckARMA0: Covariances of a Fractional ARIMA(0,d,0) Process

Description

Compute the Autocovariances of a fractional ARIMA(0,d,0) process (d = H - 1/2).

Usage

ckARMA0(n, H)

Value

numeric vector of length n of covariances

\(C(0) \ldots C(n-1)\).

Arguments

n

sample size (length of time series).

H

self-similarity (`Hurst') parameter.

Author

Jan Beran (principal) and Martin Maechler (speedup, fine tuning)

Details

The theoretical formula, $$C(k) = (-1)^k \Gamma(1-2d) / (\Gamma(k+1-d) \Gamma(1-k-d)) ,$$ where \(d = H - 1/2\), leads to over-/underflow for larger lags \(k\); hence use the asymptotical formula there.

References

Jan Beran (1994), p.63, (2.35) and (2.39).

See Also

ckFGN0 which does the same for fractional Gaussian noise.

Examples

str(C.8 <- ckARMA0(50, H = 0.8))
yl <- c(0,max(C.8))
plot(0:49, C.8, type = "h", ylim = yl)
plot(0:49, C.8, type = "h", log = "xy",
     main = "Log-Log  ACF for ARIMA(0,d,0)")