R_hl {rlfsm}

R high /low

Description

Defined as

R_{\textnormal{high}} (p,k)_n := \frac{\sum_{i=2k}^n \left| \Delta_{i,k}^{n,2} X \right|^p} {\sum_{i=k}^n \left| \Delta_{i,k}^{n,1} X \right|^p}, \qquad

R_{\textnormal{low}} (p,k)_n := \frac{\sum_{i=2k}^n \left| \Delta_{i,k}^{2} X \right|^p} {\sum_{i=k}^n \left| \Delta_{i,k}^{1} X \right|^p}

Usage

R_hl(p, k, path)

Arguments

p

power

k

increment order

path

sample path of lfsm on which the inference is to be performed

Details

The computation procedure for high- and low frequency cases is the same, since there is no way to control frequency given a sample path.

References

Mazur S, Otryakhin D, Podolskij M (2020). “Estimation of the linear fractional stable motion.” Bernoulli, 26(1), 226–252. https://doi.org/10.3150/19-BEJ1124.

Examples

m<-45; M<-60; N<-2^10-M
alpha<-0.8; H<-0.8; sigma<-0.3
p<-0.3; k=3

lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm
R_hl(p=p,k=k,path=lfsm)