sampleComplexCFMoment: Complex moment condition based on the characteristic function

In StableEstim: Estimate the Four Parameters of Stable Laws using Different Methods

Complex moment condition based on the characteristic function

Description

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

Usage

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 (Nolan's ‘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

a complex vector of length(t).

See Also

ComplexCF, sampleRealCFMoment

Examples

## 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)
CFMC

StableEstim documentation built on Aug. 7, 2022, 5:17 p.m.