sampleComplexCFMoment {StableEstim} | R Documentation |
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: |
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 to 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
.
Value
a complex vector of length(t)
.
See Also
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