Compute Integral of Function Against Cumulative Distribution

Description

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

Usage

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stieltjes(f, 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".

...

Additional arguments passed to f.

Details

This command computes the Stieltjes integral

I = integral 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 a function value table (object of class "fv", see fv.object). Such objects are returned by the commands Kest, Gest, etc.

Value

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

Author(s)

\adrian

and \rolf

See Also

fv.object, Gest

Examples

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  data(redwood)
  # estimate cdf of nearest neighbour distance
  G <- Gest(redwood)
  # compute estimate of mean nearest neighbour distance
  stieltjes(function(x){x}, G)
  # estimated probability of a distance in the interval [0.1,0.2]
  stieltjes(function(x,a,b){ (x >= a) & (x <= b)}, G, a=0.1, b=0.2)

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