asymptotic_variance {ambit}

Computing the true asymptotic variance in the CLT of the trawl estimation

Description

This function computes the theoretical asymptotic variance appearing in the CLT of the trawl process for a given trawl function and fourth cumulant.

Usage

asymptotic_variance(t, c4, varlevyseed = 1, trawlfct, trawlfct_par)

Arguments

t

Time point at which the asymptotic variance is computed

c4

The fourth cumulant of the Levy seed of the trawl process

varlevyseed

The variance of the Levy seed of the trawl process, the default is 1

trawlfct

The trawl function for which the asymptotic variance will be computed (Exp, supIG or LM)

trawlfct_par

The parameter vector of the trawl function (Exp: lambda, supIG: delta, gamma, LM: alpha, H)

Details

As derived in Sauri and Veraart (2022), the asymptotic variance in the central limit theorem for the trawl function estimation is given by

\sigma_{a}^{2}(t)=c_{4}(L')a(t)+2\{ \int_{0}^{\infty}a(s)^{2}ds+ \int_{0}^{t}a(t-s)a(t+s)ds-\int_{t}^{\infty}a(s-t)a(t+s)ds\},

for t>0. The integrals in the above formula are approximated numerically.

Value

The function returns \sigma_{a}^{2}(t).

Examples

#Compute the asymptotic variance at time t for an exponential trawl with
#parameter 2; here we assume that the fourth cumulant equals 1.
av<-asymptotic_variance(t=1, c4=1, varlevyseed=1, trawlfct="Exp", trawlfct_par=2)
#Print the av
av$v
#Print the four components of the asymptotic variance separately
av$v1
av$v2
av$v3
av$v4

#Note that v=v1+v2+v3+v4
av$v
av$v1+av$v2+av$v3+av$v4