spatstat (version 1.23-1)

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, ...)

Arguments

f

The integrand. A function in the Rlanguage.

M

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

...

Additional arguments passed to f.

Value

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

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 Rlanguage. 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 link{Kest}, Gest, etc.

See Also

fv.object, Gest

Examples

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)