# Phi-Divergences test for diffusion processes Phi-Divergences test for diffusion processes. ```r 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, `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. ## 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, tools:::Rd_expr_doi("10.1016/j.jspi.2009.12.029") . ## Returns - **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 ## Examples ```r ## 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) ```

sdeDiv function