First root of the empirical characteristic function

Computes the first root of the real part of the empirical characteristic function.

ComputeFirstRootRealeCF(x, ..., tol = 0.001, maxIter = 100,
                        lowerBand = 1e-04, upperBand = 30)

Arguments

x

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

...

other arguments to pass to the optimisation function.

tol

tolerance to accept the solution; default = 1e-3.

maxIter

maximum number of iteration in the Welsh algorithm; default = 100.

lowerBand

lower band of the domain where the graphical seach is performed; default = 1e-4.

upperBand

Lower band of the domain where the graphical seach is performed; default = 30.

Details

The Welsh algorithm is first applied. If it fails to provide a satisfactory value (< tol), a graphical/ numerical approach is used. We first plot the real part of the eCF vs t in order to determine the first zero directly and use it as the initial guess of a numerical minimisation routine.

Value

numeric: first zero of the real part of the eCF.

References

Welsh AH (1986). “Implementing empirical characteristic function procedures.” Statistics & probability letters, 4(2), 65--67.

See also

ComplexCF

Examples

set.seed(345)
x <- rstable(500, 1.5, 0.5)
ComputeFirstRootRealeCF(x)
#> [1] 2.302698