ksmooth {sde} | R Documentation |
Nonparametric invariant density, drift, and diffusion coefficient estimation
Description
Implementation of simple Nadaraya-Watson nonparametric estimation of drift and diffusion coefficient, and plain kernel density estimation of the invariant density for a one-dimensional diffusion process.
Usage
ksdrift(x, bw, n = 512)
ksdiff(x, bw, n = 512)
ksdens(x, bw, n = 512)
Arguments
x |
a |
bw |
bandwidth. |
n |
number of points in which to calculate the estimates. |
Details
These functions return the nonparametric estimate of the drift or
diffusion coefficients for data x
using the Nadaraya-Watson estimator
for diffusion processes.
ksdens
returns the density estimates of the invariant density.
If not provided, the bandwidth bw
is calculated using Scott's rule (i.e.,
bw = len^(-1/5)*sd(x)
) where len=length(x)
is the number of observed points of the diffusion path.
Value
val |
an invisible list of |
Author(s)
Stefano Maria Iacus
References
Ait-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 527-560.
Bandi, F., Phillips, P. (2003) Fully nonparametric estimation of scalar diffusion models, Econometrica, 71, 241-283.
Florens-Zmirou, D. (1993) On estimating the diffusion coefficient from discrete observations, Journal of Applied Probability, 30, 790-804.
Examples
set.seed(123)
theta <- c(6,2,1)
X <- sde.sim(X0 = rsCIR(1, theta), model="CIR", theta=theta,
N=1000,delta=0.1)
b <- function(x)
theta[1]-theta[2]*x
sigma <- function(x)
theta[3]*sqrt(x)
minX <- min(X)
maxX <- max(X)
par(mfrow=c(3,1))
curve(b,minX,maxX)
lines(ksdrift(X),lty=3)
curve(sigma,minX, maxX)
lines(ksdiff(X),lty=3)
f <-function(x) dsCIR(x, theta)
curve(f,minX,maxX)
lines(ksdens(X),lty=3)