# simple.ef: Simple estimating functions of types I and II In sde: Simulation and Inference for Stochastic Differential Equations

## Description

Apply a simple estimating function to find estimates of the parameters of a process solution of a stochastic differential equation.

## Usage

 `1` ```simple.ef(X, f, guess, lower, upper) ```

## Arguments

 `X` a ts object containing a sample path of an sde. `f` a list of expressions of `x` and/or `y` and the parameters to be estimated; see details. `guess` initial value of the parameters; see details. `lower` lower bounds for the parameters; see details. `upper` upper bounds for the parameters; see details.

## Details

The function `simple.ef` minimizes a simple estimating function of the form `sum_i f_i(x,y;theta) = 0` or `sum_i f_i(x;theta)` as a function of `theta`. The index `i` varies in `1:length(theta)`.

The list `f` is a list of expressions in `x` or `(x,y)`.

## Value

 `x` a vector of estimates

## Author(s)

Stefano Maria Iacus

## References

Kessler, M. (1997) Estimation of an ergodic diffusion from discrete observations, Scand. J. Statist., 24, 211-229.

Kessler, M. (2000) Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process, Scand. J. Statist., 27, 65-82.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```set.seed(123); # Kessler's estimator for O-H process K.est <- function(x) { n.obs <- length(x) n.obs/(2*(sum(x^2))) } # Least squares estimators for the O-H process LS.est <- function(x) { n <- length(x) -1 k.sum <- sum(x[1:n]*x[2:(n+1)]) dt <- deltat(x) ifelse(k.sum>0, -log(k.sum/sum(x[1:n]^2))/dt, NA) } d <- expression(-1 * x) s <- expression(1) x0 <- rnorm(1,sd=sqrt(1/2)) sde.sim(X0=x0,drift=d, sigma=s,N=2500,delta=0.1) -> X # Kessler's estimator as estimating function f <- list(expression(2*theta*x^2-1)) simple.ef(X, f, lower=0, upper=Inf) K.est(X) # Least Squares estimator as estimating function f <- list(expression(x*(y-x*exp(-0.1*theta)))) simple.ef(X, f, lower=0, upper=Inf) LS.est(X) ```

sde documentation built on May 31, 2017, 3:58 a.m.