Maximum likelihood (ML) method

Uses the numerical ML approach described by Nolan to estimate the 4 parameters of stable law. The method may be slow for large sample size due to the use of numerical optimisation routine.

Usage

MLParametersEstim(x, theta0 = NULL, pm = 0, PrintTime = FALSE, ...)

Arguments

x

data used to perform the estimation: vector of length n.

theta0

initial guess for the 4 parameters values: If NULL, the Kogon-McCulloch method is called, see IGParametersEstim; a vector of length 4.

pm

parametrisation, an integer (0 or 1); default: pm=0 (Nolan's ‘S0’ parametrisation).

PrintTime

logical flag; if set to TRUE, the estimation duration is printed out to the screen in a readable format (h/min/sec).

...

Other argument to be passed to the optimisation function.

Details

The function performs the minimisation of the numerical (-)log-density of stable laws computed by function dstable from package stabledist.

After testing several optimisation routines, we have found out that the "L-BFGS-B" algorithm performs better with the ML method (faster, more accurate).

Value

a list with the following elements:

Estim

output of the optimisation function,

duration

estimation duration in a numerical format,

method

character describing the method used.

References

Nolan J (2001). “Maximum likelihood estimation and diagnostics for stable distributions.” L'evy processes: theory and applications, pp. 379–400.

See also

Estim

Examples

theta <- c(1.5, 0.4, 1, 0)
pm <- 0
## 50 points does not give accurate estimation
## but it makes estimation fast for installation purposes
## use at least 200 points to get decent results.
set.seed(1333)
x <- rstable(50, theta[1], theta[2], theta[3], theta[4], pm)

## This example takes > 30 sec hence commented out
if (FALSE) { # \dontrun{
  ML <- MLParametersEstim(x = x, pm = pm, PrintTime = TRUE)
} # }
## see the Examples folder for more examples.