alpha_hat {rlfsm}

Statistical estimator for alpha

Description

Defined for the two frequencies as

\widehat \alpha_{high} := \frac{\log | \log \varphi_{high} (t_2; \widehat H_{high} (p,k)_n, k)_n| - \log | \log \varphi_{high} (t_1; \widehat H_{high} (p,k)_n, k)_n|}{\log t_2 - \log t_1}

\widehat \alpha_{low} := \frac{\log | \log \varphi_{low} (t_2;k)_n| - \log | \log \varphi_{low} (t_1; k)_n|}{\log t_2 - \log t_1}

Usage

alpha_hat(t1, t2, k, path, H, freq)

Arguments

t1, t2

real number such that t2 > t1 > 0

k

increment order

path

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

H

Hurst parameter

freq

Frequency of the motion. It can take two values: "H" for high frequency and "L" for the low frequency setting.

Details

The function triggers function phi, thus Hurst parameter is required only in high frequency case. In the low frequency, there is no need to assign H a value because it will not be evaluated.

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<-2^14-M
alpha<-1.8; H<-0.8; sigma<-0.3
freq='H'
r=1; k=2; p=0.4; t1=1; t2=2

# Estimating alpha in the high frequency case
# using preliminary estimation of H
lfsm<-path(N=N,m=m,M=M,alpha=alpha,H=H,
           sigma=sigma,freq='L',disable_X=FALSE,seed=3)$lfsm

H_est<-H_hat(p=p,k=k,path=lfsm)
H_est
alpha_est<-alpha_hat(t1=t1,t2=t2,k=k,path=lfsm,H=H_est,freq=freq)
alpha_est