| sdeDiv {sde} | R Documentation |
Phi-Divergences test for diffusion processes
Description
Phi-Divergences test for diffusion processes.
Usage
sdeDiv(X, theta1, theta0, phi= expression( -log(x) ), C.phi, K.phi,
b, s, b.x, s.x, s.xx, B, B.x, H, S, guess, ...)
Arguments
X |
a ts object containing a sample path of an sde. |
theta1 |
a vector parameters for the hypothesis H1. If not given, |
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 |
K.phi |
the value of second derivative of |
b |
drift coefficient of the model as a function of |
s |
diffusion coefficient of the model as a function of |
b.x |
partial derivative of |
s.x |
partial derivative of |
s.xx |
second-order partial derivative of |
B |
initial value of the parameters; see details. |
B.x |
partial derivative of |
H |
function of |
S |
function of |
guess |
initial value for the parameters to be estimated; optional. |
... |
passed to the |
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, Journal of Statistical Planning and Inference, 140(7), 1744-1753, doi:10.1016/j.jspi.2009.12.029.
Examples
## Not run:
set.seed(123)
theta0 <- c(0.89218*0.09045,0.89218,sqrt(0.032742))
theta1 <- c(0.89218*0.09045/2,0.89218,sqrt(0.032742/2))
# we test the true model against two competing models
b <- function(x,theta) theta[1]-theta[2]*x
b.x <- function(x,theta) -theta[2]
s <- function(x,theta) theta[3]*sqrt(x)
s.x <- function(x,theta) theta[3]/(2*sqrt(x))
s.xx <- function(x,theta) -theta[3]/(4*x^1.5)
X <- sde.sim(X0=rsCIR(1, theta1), N=1000, delta=1e-3, model="CIR",
theta=theta1)
sdeDiv(X=X, theta0 = theta0, b=b, s=s, b.x=b.x, s.x=s.x,
s.xx=s.xx, method="L-BFGS-B",
lower=rep(1e-3,3), guess=c(1,1,1))
sdeDiv(X=X, theta0 = theta1, b=b, s=s, b.x=b.x, s.x=s.x,
s.xx=s.xx, method="L-BFGS-B",
lower=rep(1e-3,3), guess=c(1,1,1))
lambda <- -1.75
myphi <- expression( (x^(lambda+1) -x - lambda*(x-1))/(lambda*(lambda+1)) )
sdeDiv(X=X, theta0 = theta0, phi = myphi, b=b, s=s, b.x=b.x,
s.x=s.x, s.xx=s.xx, method="L-BFGS-B",
lower=rep(1e-3,3), guess=c(1,1,1))
sdeDiv(X=X, theta0 = theta1, phi = myphi, b=b, s=s, b.x=b.x,
s.x=s.x, s.xx=s.xx, method="L-BFGS-B",
lower=rep(1e-3,3), guess=c(1,1,1))
## End(Not run)