ltsa (version 1.4.6)

TrenchLoglikelihood: Loglikelihood function of stationary time series using Trench algorithm

Description

The Trench matrix inversion algorithm is used to compute the exact concentrated loglikelihood function.

Usage

TrenchLoglikelihood(r, z)

Arguments

r

autocovariance or autocorrelation at lags 0,...,n-1, where n is length(z)

z

time series data

Value

The loglikelihood concentrated over the parameter for the innovation variance is returned.

Details

The concentrated loglikelihood function may be written Lm(beta) = -(n/2)*log(S/n)-0.5*g, where beta is the parameter vector, n is the length of the time series, S=z'M z, z is the mean-corrected time series, M is the inverse of the covariance matrix setting the innovation variance to one and g=-log(det(M)).

References

McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software.

See Also

DLLoglikelihood

Examples

#compute loglikelihood for white noise
z<-rnorm(100)
TrenchLoglikelihood(c(1,rep(0,length(z)-1)), z)


#simulate a time series and compute the concentrated loglikelihood using DLLoglikelihood and
#compare this with the value given by TrenchLoglikelihood.
phi<-0.8
n<-200
r<-phi^(0:(n-1))
z<-arima.sim(model=list(ar=phi), n=n)
LD<-DLLoglikelihood(r,z)
LT<-TrenchLoglikelihood(r,z)
ans<-c(LD,LT)
names(ans)<-c("DLLoglikelihood","TrenchLoglikelihood")