Complex moment condition based on the characteristic function.

Description

Computes the moment condition based on the characteristic function as a complexl vector.

Usage

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sampleComplexCFMoment(x, t, theta, pm = 0)

Arguments

x

vector of data where the ecf is computed.

t

Vector of (real) numbers where the CF is evaluated; numeric

theta

Vector of parameters of the stable law; vector of length 4.

pm

Parametrisation, an integer (0 or 1); default: pm=0( the Nolan ‘S0’ parametrisation).

Details

The moment conditions The moment conditions are given by:

g_t(X,θ)=g(t,X;θ)= e^{itX} - φ_{θ}(t)

If one has a sample x_1,…,x_n of i.i.d realisations of the same random variable X, then:

\hat{g}_n(t,θ) = \frac{1}{n}∑_{i=1}^n g(t,x_i;θ) = φ_n(t) -φ_θ(t)

where φ_n(t) is the eCF associated to the sample x_1,…,x_n and defined by φ_n(t)= \frac{1}{n} ∑_{j=1}^n e^{itX_j} The function compute the vector of difference between the eCF and the CF at a set of given point t.

Value

Returns a complex vector of length(t).

See Also

ComplexCF,sampleRealCFMoment

Examples

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## define the parameters
 nt <- 10   
 t <- seq(0.1,3,length.out=nt)
 theta <- c(1.5,0.5,1,0)
pm <- 0

set.seed(222);x=rstable(200,theta[1],theta[2],theta[3],theta[4],pm)

# Compute the characteristic function
CFMC <- sampleComplexCFMoment(x=x,t=t,theta=theta,pm=pm)

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