psstat: Distribution of S-statistics - c.d.f.

Description

Computes the cumulative distribution function of the S-statistic w.r.t. to a control function in an i.i.d. model with common increasing and continuous c.d.f. $F$ defined on $[0,\infty)$.

Usage

psstat(x, n, cdf, kappa, ...)

Arguments

numeric vector.

sample size.

cdf

a cumulative distribution function $F$, e.g. ppareto2.

kappa

an increasing function, $\kappa$ (see Details), a so-called control function.

...

optional arguments to cdf and pdf.

Value

The value of the c.d.f. at x.

Details

Let $F$ (parameter cdf) be a continuous c.d.f that is strictly increasing on $[a,b]$, where $a=\inf{x: F(x)>0}$ and $b=\sup{x: F(x)<1}$.< p="">

Moreover, let $\kappa:[0,1]\to[c,d]\subseteq[a,b]$ be any control function (parameter kappa), i.e. a function that is continuous and strictly increasing and which fulfills $\kappa(0)=c$ and $\kappa(1)=d$.

The function computes the value of the c.d.f. of an S-statistic w.r.t. to the control function for sample of size n. This result was given in (Gagolewski, Grzegorzewski, 2010).

Note that under certain conditions the distribution of an S-statistic is asymptotically normal with expectation $\rho_\kappa$ (see rho.get) and variance $\rho_\kappa (1-\rho_\kappa)/n/(1+F(\kappa(\rho_\kappa)))^2$.

References

Gagolewski M., Grzegorzewski P., S-Statistics and Their Basic Properties, In: Borgelt C. et al (Eds.), Combining Soft Computing and Statistical Methods in Data Analysis, Springer-Verlag, 2010, 281-288.