# simple.ef2: Simple estimating function based on the infinitesimal... In sde: Simulation and Inference for Stochastic Differential Equations

## Description

Apply a simple estimating function based on the infinitesimal generator of a diffusion to find estimates of the parameters of a process solution of that particular stochastic differential equation.

## Usage

 ```1 2``` ```simple.ef2(X, drift, sigma, h, h.x, h.xx, guess, lower, upper) ```

## Arguments

 `X` a `ts` object containing a sample path of an sde. `drift` an expression for the drift coefficient; see details. `sigma` an expression for the diffusion coefficient; see details. `h` an expression of `x` and the parameters to be estimated; see details. `h.x` an expression of `x` containing the first derivative of `h`; see details. `h.xx` an expression of `x` containing the second derivative of `h`; 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.ef2` minimizes the simple estimating function of the form `sum_i f_i(x;theta) = 0`, where `f` is the result of applying the infinitesimal generator of the diffusion to the function `h`. This involves the drift and diffusion coefficients plus the first two derivatives of `h`. If not provided by the user, the derivatives are calculated by the function.

## 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``` ```set.seed(123) d <- expression(10 - x) s <- expression(sqrt(x)) x0 <- 10 sde.sim(X0=x0,drift=d, sigma=s,N=1500,delta=0.1) -> X # rather difficult problem unless a good initial guess is given d <- expression(alpha + theta*x) s <- expression(x^gamma) h <- list(expression(x), expression(x^2), expression(x^2)) simple.ef2(X, d, s, h, lower=c(0,-Inf,0), upper=c(Inf,0,1)) ```

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