longmemo (version 1.1-3)

Qeta: 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)

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.

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.

Author

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

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.

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

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)