Real moment condition based on the characteristic function

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

sampleRealCFMoment(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: $$g_t(X,\theta) = g(t,X;\theta) = e^{itX} - \phi_{\theta}(t) . $$ 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.

Value

a vector of length 2 * length(t).

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
CFMR <- sampleRealCFMoment(x = x, t = t, theta = theta, pm = pm)
CFMR
#>  [1]  0.006242431 -0.020703369 -0.072449040 -0.050660290 -0.040969284
#>  [6] -0.028959753 -0.029889636 -0.015320422 -0.032306259 -0.017038430
#> [11] -0.004761586  0.057950623  0.064189392  0.051092006  0.022624526
#> [16]  0.036014037  0.060445754  0.058259580  0.050194423 -0.014834322