test_asymnorm_est: Computing the feasible statistic of the trawl function CLT

In ambit: Simulation and Estimation of Ambit Processes

Computing the feasible statistic of the trawl function CLT

Description

This function computes the feasible statistics associated with the CLT for the trawl function estimation.

Usage

test_asymnorm_est(
  data,
  Delta,
  trawlfct,
  trawlfct_par,
  biascor = FALSE,
  k = NULL
)

Arguments

data	The data set based on observations of X_0, X_{Δ_n}, …, X_{(n-1)Δ_n}
Delta	The width Delta of the observation grid
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)
biascor	A binary variable determining whether a bias correction should be computed, the default is FALSE
k	The optional parameter specifying the time point in 0, 1, …, n-1; the test statistic will be computed for the time point k Δ_n.

Details

As derived in Sauri and Veraart (2022), the feasible statistic, for t>0, is given by

T(t)_n:=\frac{√{nΔ_{n}}}{√{\widehat{σ_{a}^2(t)}}} ≤ft(\hat{a}(t)-a(t)-bias(t)\right).

For t=0, we have

T(t)_n:=\frac{√{nΔ_{n}}}{√{RQ_n}} ≤ft(\hat{a}(0)-a(0)-bias(0)\right),

where

RQ_n:=\frac{1}{√{2 nΔ_{n}}} ∑_{k=0}^{n-2}(X_{(k+1)Δ_n}-X_{kΔ_n})^4.

We set bias(t)=0 in the case when biascor==FALSE and bias(t)=0.5 * Δ * \hat a'(t) otherwise.

Value

The function returns the vector of the feasible statistics (T(0)_n, T((Δ)_n, …, T((n-2)Δ_n)) if no bias correction is required and (T(0)_n, T((Δ)_n, …, T((n-3)Δ_n)) if bias correction is required if k is not provided, otherwise it returns the value T(k Δ_n)_n. If the estimated asymptotic variance is <= 0, the value of the test statistic is set to 999.

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

#Compute the test statistic for time t=0
##Either one can use:
test_asymnorm_est(Poi_data, my_delta,
                  trawlfct="Exp", trawlfct_par=my_lambda)[1]
#or:
test_asymnorm_est(Poi_data, my_delta,
                  trawlfct="Exp", trawlfct_par=my_lambda, k=0)

ambit documentation built on Aug. 19, 2022, 5:19 p.m.