asymptotic_variance_est {ambit}

Estimating the asymptotic variance in the trawl function CLT

Description

This function estimates the asymptotic variance which appears in the CLT for the trawl function estimation.

Usage

asymptotic_variance_est(t, c4, varlevyseed = 1, Delta, avector, N = NULL)

Arguments

t

The time point at which to compute the asymptotic variance

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

Delta

The width Delta of the observation grid

avector

The vector (\hat a(0), \hat a(\Delta_n), ..., \hat a((n-1)\Delta_n))

N

The optional parameter to specify the upper bound N_n in the computations of the estimators

Details

As derived in Sauri and Veraart (2022), the estimated asymptotic variance is given by

\hat \sigma^2_a(t)=\hat v_1(t)+\hat v_2(t)+\hat v_3(t)+\hat v_4(t),

where

\hat{v}_{1}(t):=\widehat{c_{4}(L')}\hat{a}(t)=RQ_n\hat{a}(t)/ \hat{a}(0),

for

RQ_n:=\frac{1}{\sqrt{2 n\Delta_{n}}} \sum_{k=0}^{n-2}(X_{(k+1)\Delta_n}-X_{k\Delta_n})^4,

and

\hat{v}_{2}(t):=2\sum_{l=0}^{N_{n}}\hat{a}^{2}(l\Delta_{n}) \Delta_{n},

\hat{v}_{3}(t):=2\sum_{l=0}^{\min\{i,n-1-i\}}\hat{a}((i-l)\Delta_{n}) \hat{a}((i+l)\Delta_{n})\Delta_{n},

\hat{v}_{4}(t):=-2\sum_{l=i}^{N_{n}-i}\hat{a}((l-i)\Delta_{n}) \hat{a}((i+l)\Delta_{n})\Delta_{n}.

Value

The estimated asymptotic variance \hat v=\hat \sigma_a^2(t) and its components \hat v_1, \hat v_2, \hat v_3, \hat v_4.

Examples

##Simulate a trawl process
##Determine the sampling grid
my_n <- 1000
my_delta <- 0.1
my_t <- my_n*my_delta

###Choose the model parameter
#Exponential trawl function:
my_lambda <- 2
#Poisson marginal distribution trawl
my_v <- 1

#Set the seed
set.seed(123)
#Simulate the trawl process
Poi_data <- sim_weighted_trawl(my_n, my_delta,
                               "Exp", my_lambda, "Poi", my_v)$path

#Estimate the trawl function
my_lag <- 100+1
trawl <- nonpar_trawlest(Poi_data, my_delta, lag=my_lag)$a_hat

#Estimate the fourth cumulant of the trawl process
c4_est <- c4est(Poi_data, my_delta)

asymptotic_variance_est(t=1, c4=c4_est, varlevyseed=1,
                        Delta=my_delta, avector=trawl)$v