Description Usage Arguments Details Value Note Author(s) References See Also Examples
Qeta()
(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.
1 |
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 |
pq.ARIMA |
integer, = c(p,q) specifying models orders of AR and
MA parts — only used when |
give.B.only |
logical, indicating if only the |
Calculation of A, B and T_n = A/B^2 where A = 2π/n ∑_j 2*[I(λ_j)/f(λ_j)], B = 2π/n ∑_j 2*[I(λ_j)/f(λ_j)]^2 and the sum is taken over all Fourier frequencies λ_j = 2π*j/n, (j=1,…,(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
a list with components
n |
= input |
H |
(input) Hurst parameter, = |
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 theta1^ = sigma_e^2 / (2 pi) such that f()= θ_1 f_1() and integral(\log[f_1(.)]) = 0. |
spec |
scaled spectral density f_1 at the Fourier frequencies
ω_j, see |
yper[1] must be the periodogram I(λ_1) at the frequency 2π/n, i.e., not the frequency zero !
Jan Beran (principal) and Martin Maechler (fine tuning)
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)
WhittleEst
computes an approximate MLE for fractional
Gaussian noise / fractional ARIMA, by minimizing Qeta
.
1 2 3 4 5 6 7 |
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