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
\(x_1,\dots,x_n\), defined by \(\phi_n(t)= \frac{1}{n}
\sum_{j=1}^n e^{itX_j}\).
Objective function
Following @@carrasco2007efficientCont, Proposition 3.4;textualStableEstim,
the objective function to minimise is given by:
$$obj(\theta)=\overline{\underline{v}^{\prime}}(\theta)[\alpha_{Reg} \mathcal{I}_n+C^2]^{-1}\underline{v}(\theta)$$
where:
-
\(\underline{v} = [v_1,\ldots,v_n]^{\prime}\);
\(v_i(\theta)
= \int_I \overline{g_i}(t;\hat{\theta}^1_n) \hat{g}(t;\theta) \pi(t) dt\).
- \(I_n\)
is the identity matrix of size \(n\).
- \(C\)
is a \(n \times n\) matrix with \((i,j)\)th
element given by
\(c_{ij} = \frac{1}{n-4}\int_I
\overline{g_i}(t;\hat{\theta}^1_n) g_j(t;\hat{\theta}^1_n)
\pi(t) dt\).
To compute \(C\) and \(v_i()\) we will use the function
IntegrateRandomVectorsProduct.
The IterationControl
If type = "IT" or type = "Cue", the user can control
each iteration using argument IterationControl, which should be
a list which contains the following elements:
NbIter:maximum number of iterations.
The loop stops when NBIter is reached; default = 10.
PrintIterlogical:if set to TRUE the values of the
current parameter estimates are printed to the screen at each
iteration; default = TRUE.
RelativeErrMax:the loop stops if the relative error
between two consecutive estimation steps is smaller then
RelativeErrMax; default = 1e-3.