a vector parameters for the hypothesis H1. If not given, theta1 is estimated from the data.
theta0
a vector parameters for the hypothesis H0.
phi
an expression containing the phi function of the phi-divergence.
C.phi
the value of first derivtive of phi at point 1. If not given, it is calculated within this function.
K.phi
the value of second derivative of phi at point 1. If not given, it is calculated within this function.
b
drift coefficient of the model as a function of x and theta.
s
diffusion coefficient of the model as a function of x and theta.
b.x
partial derivative of b as a function of x and theta.
s.x
partial derivative of s as a function of x and theta.
s.xx
second-order partial derivative of s as a function of x and theta.
B
initial value of the parameters; see details.
B.x
partial derivative of B as a function of x and theta.
H
function of (x,y), the integral of B/s; optional.
S
function of (x,y), the integral of 1/s; optional.
guess
initial value for the parameters to be estimated; optional.
...
passed to the optim function; optional.
Details
The sdeDiv estimate the phi-divergence for diffusion processes defined as
D(theta1, theta0) = phi( f(theta1)/f(theta0) ) where f is the
likelihood function of the process. This function uses the Dacunha-Castelle
and Florens-Zmirou approximation of the likelihood for f.
The parameter theta1 is supposed to be the value of the true MLE estimator
or the minimum contrast estimator of the parameters in the model. If missing
or NULL and guess is specified, theta1 is estimated using the
minimum contrast estimator derived from the locally Gaussian approximation
of the density. If both theta1 and guess are missing, nothing can
be calculated.
The function always calculates the likelihood ratio test and the p-value of the
test statistics.
In some cases, the p-value of the phi-divergence test statistics is obtained by simulation. In such
a case, the out$est.pval is set to TRUE
Dy default phi is set to -log(x). In this case the phi-divergence and the
likelihood ratio test are equivalent (e.g. phi-Div = LRT/2)
For more informations on phi-divergences for discretely observed diffusion processes
see the references.
If missing, B is calculated as b/s - 0.5*s.x provided that s.x
is not missing.
If missing, B.x is calculated as b.x/s - b*s.x/(s^2)-0.5*s.xx, provided
that b.x, s.x, and s.xx are not missing.
If missing, both H and S are evaluated numerically.
Value
x
a list containing the value of the divergence, its pvalue, the likelihood ratio test
statistics and its p-value
Author(s)
Stefano Maria Iacus
References
Dacunha-Castelle, D., Florens-Zmirou, D. (1986) Estimation of the coefficients
of a diffusion from discrete observations, Stochastics, 19, 263-284.
De Gregorio, A., Iacus, S.M. (2008) Divergences Test Statistics for
Discretely Observed Diffusion Processes. Available at http://arxiv.org/abs/0808.0853