Qeta {longmemo} | R Documentation |
Qmeta() is up to scaling the log likelihood function of the two models indicated.
Qeta(eta, model = c("fGn","fARIMA"), n, yper, pq.ARIMA, verbose = getOption("verbose"), give.B.only = FALSE)
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" . |
verbose |
logical indicating if diagnostic output should be produced during fitting. |
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. |
Calculation of A, B and T_n = A/B^2 where A = 2π/n sum_j 2*[I(λ_j)/f(λ_j)], B = 2π/n sum_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.
cov(X(t),X(t+k)) = int exp(iuk) f(u) du
a list with components
n |
= input |
H |
Hurst parameter, = eta[1] . |
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 such that f=theta1*spec and
integral(log[spec])=0. |
spec |
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)
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)