spatstat.explore (version 3.2-7)

stieltjes: Compute Integral of Function Against Cumulative Distribution

Description

Computes the Stieltjes integral of a function \(f\) with respect to a function \(M\).

Usage

stieltjes(f, M, ...)

Value

A list containing the value of the Stieltjes integral computed using each of the versions of the function M.

Arguments

f

The integrand. A function in the R language.

M

The cumulative function against which f will be integrated. An object of class "fv" or "stepfun".

...

Additional arguments passed to f.

Author

Adrian Baddeley , Rolf Turner and Ege Rubak .

Details

This command computes the Stieltjes integral $$I = \int f(x) dM(x)$$ of a real-valued function \(f(x)\) with respect to a nondecreasing function \(M(x)\).

One common use of the Stieltjes integral is to find the mean value of a random variable from its cumulative distribution function \(F(x)\). The mean value is the Stieltjes integral of \(f(x)=x\) with respect to \(F(x)\).

The argument f should be a function in the R language. It should accept a numeric vector argument x and should return a numeric vector of the same length.

The argument M should be either a step function (object of class "stepfun") or a function value table (object of class "fv" ). Objects of class "stepfun" are returned by ecdf, ewcdf, and other utilities.

Examples

  x <- runif(100)
  w <- runif(100)
  H <- ewcdf(x, w)
  stieltjes(function(x) { x^2 }, H)