Qeta: Approximate Log Likelihood for Fractional Gaussian Noise /...

In longmemo: Statistics for Long-Memory Processes (Book Jan Beran), and Related Functionality

Approximate Log Likelihood for Fractional Gaussian Noise / Fractional ARIMA

Description

Qeta() (= \tilde{Q}(\eta) of Beran(1994), p.117) is up to scaling the negative log likelihood function of the specified model, i.e., fractional Gaussian noise or fractional ARIMA.

Usage

Qeta(eta, model = c("fGn","fARIMA"), n, yper, pq.ARIMA, give.B.only = FALSE)

Arguments

eta	parameter vector = (H, phi[1:p], psi[1:q]).
model	character specifying the kind model class.
n	data length
yper	numeric vector of length (n-1)%/% 2, the periodogram of the (scaled) data, see per.
pq.ARIMA	integer, = c(p,q) specifying models orders of AR and MA parts — only used when model = "fARIMA".
give.B.only	logical, indicating if only the B component (of the Values list below) should be returned. Is set to TRUE for the Whittle estimator minimization.

Details

Calculation of A, B and T_n = A/B^2 where A = 2\pi/n \sum_j 2*[I(\lambda_j)/f(\lambda_j)], B = 2\pi/n \sum_j 2*[I(\lambda_j)/f(\lambda_j)]^2 and the sum is taken over all Fourier frequencies \lambda_j = 2\pi*j/n, (j=1,\dots,(n-1)/2).

f is the spectral density of fractional Gaussian noise or fractional ARIMA(p,d,q) with self-similarity parameter H (and p AR and q MA parameters in the latter case), and is computed either by specFGN or specARIMA.

cov(X(t),X(t+k)) = \int \exp(iuk) f(u) du

Value

a list with components

n	= input
H	(input) Hurst parameter, = eta[1].
eta	= input
A, B	defined as above.
Tn	the goodness of fit test statistic Tn= A/B^2 defined in Beran (1992)
z	the standardized test statistic
pval	the corresponding p-value P(W > z)
theta1	the scale parameter \hat{\theta_1} = \frac{{\hat\sigma_\epsilon}^2}{2\pi} such that f()= \theta_1 f_1() and integral(\log[f_1(.)]) = 0.
spec	scaled spectral density f_1 at the Fourier frequencies \omega_j, see FEXPest; a numeric vector.

Note

yper[1] must be the periodogram I(\lambda_1) at the frequency 2\pi/n, i.e., not the frequency zero !

Author(s)

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

References

Jan Beran (1992). A Goodness-of-Fit Test for Time Series with Long Range Dependence. JRSS B 54, 749–760.

Beran, Jan (1994). Statistics for Long-Memory Processes; Chapman & Hall. (Section 6.1, p.116–119; 12.1.3, p.223 ff)

See Also

WhittleEst computes an approximate MLE for fractional Gaussian noise / fractional ARIMA, by minimizing Qeta.

Examples

data(NileMin)
y <- NileMin
n <- length(y)
yper <- per(scale(y))[2:(1+ (n-1) %/% 2)]
eta <- c(H = 0.3)
q.res <- Qeta(eta, n=n, yper=yper)
str(q.res)

longmemo documentation built on Aug. 8, 2025, 6:15 p.m.