Akaike's information criterion for diffusion processes

Description

Implementation of the AIC statistics for diffusion processes.

Usage

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sdeAIC(X, theta, 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.

theta

a vector or estimates of the parameters.

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 sdeAIC evaluates the AIC statistics for diffusion processes using Dacunha-Castelle and Florens-Zmirou approximations of the likelihood.

The parameter theta 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, theta is estimated using the minimum contrast estimator derived from the locally Gaussian approximation of the density. If both theta and guess are missing, nothing can be calculated.

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

the value of the AIC statistics

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.

Uchida, M., Yoshida, N. (2005) AIC for ergodic diffusion processes from discrete observations, preprint MHF 2005-12, march 2005, Faculty of Mathematics, Kyushu University, Fukuoka, Japan.

Examples

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## Not run: 
set.seed(123)
# true model generating data
dri <- expression(-(x-10))
dif <- expression(2*sqrt(x)) 
sde.sim(X0=10,drift=dri, sigma=dif,N=1000,delta=0.1) -> X

# we test the true model against two competing models
b <- function(x,theta) -theta[1]*(x-theta[2])
b.x <- function(x,theta)  -theta[1]+0*x

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)
# AIC for the true model
sdeAIC(X, NULL, b, s, b.x, s.x, s.xx, guess=c(1,1,1),
       lower=rep(1e-3,3), method="L-BFGS-B")

s <- function(x,theta) sqrt(theta[3]*+theta[4]*x)
s.x <- function(x,theta) theta[4]/(2*sqrt(theta[3]+theta[4]*x))
s.xx <- function(x,theta) -theta[4]^2/(4*(theta[3]+theta[4]*x)^1.5)
# AIC for competing model 1
sdeAIC(X, NULL, b, s, b.x, s.x, s.xx, guess=c(1,1,1,1),
       lower=rep(1e-3,4), method="L-BFGS-B")

s <- function(x,theta) (theta[3]+theta[4]*x)^theta[5]
s.x <- function(x,theta) 
           theta[4]*theta[5]*(theta[3]+theta[4]*x)^(-1+theta[5])
s.xx <- function(x,theta) (theta[4]^2*theta[5]*(theta[5]-1)
                 *(theta[3]+theta[4]*x)^(-2+theta[5]))
# AIC for competing model 2
sdeAIC(X, NULL, b, s, b.x, s.x, s.xx, guess=c(1,1,1,1,1),
       lower=rep(1e-3,5), method="L-BFGS-B")

## End(Not run)

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