# Simple estimating function based on the infinitesimal generator a the diffusion process

### 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 |

`drift` |
an expression for the drift coefficient; see details. |

`sigma` |
an expression for the diffusion coefficient; see details. |

`h` |
an expression of |

`h.x` |
an expression of |

`h.xx` |
an expression of |

`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))
``` |