sampleRealCFMoment: Real moment condition based on the characteristic function

Description

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

Usage

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

Value

a vector of length 2 * length(t).

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: $$g_t(X,\theta) = g(t,X;\theta) = e^{itX} - \phi_{\theta}(t) . $$ If one has a sample $x_1,\dots,x_n$ of i.i.d realisations of the same random variable $X$, then: $$\hat{g}_n(t,\theta) = \frac{1}{n}\sum_{i=1}^n g(t,x_i;\theta) = \phi_n(t) -\phi_\theta(t) , $$ where $\phi_n(t)$ is the eCF associated with the sample $x_1,\dots,x_n$, and defined by $\phi_n(t) = \frac{1}{n} \sum_{j=1}^n e^{itX_j}$.

The function compute the vector of difference between the eCF and the CF at a set of given point t. If length(t) = n, the resulting vector will be of length = 2n, where the first n components will be the real part and the remaining the imaginary part.

Examples

Run this code

## 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
CFMR <- sampleRealCFMoment(x = x, t = t, theta = theta, pm = pm)
CFMR