Qeta {longmemo}R Documentation

Function to be minimized for approx. MLE of frARIMA or frGn

Description

Qmeta() is up to scaling the log likelihood function of the two models indicated.

Usage

Qeta(eta, model = c("fGn","fARIMA"), n, yper, pq.ARIMA,
     verbose = getOption("verbose"), 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".
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.

Details

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

Value

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

Note

yper[1] must be the periodogram I(λ_1) at the frequency 2π/n (i.e., not the frequency zero !).

Author(s)

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

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)

[Package longmemo version 0.9-6 Index]