# Akaike's information criterion for diffusion processes Implementation of the AIC statistics for diffusion processes. ```r 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. ## 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**. ## Returns - **x**: the value of the AIC statistics ## Author(s) Stefano Maria Iacus ## Examples ```r ## 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) ```

sdeAIC function