stieltjes
Compute Integral of Function Against Cumulative Distribution
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
fwill be integrated. An object of class"fv". - ...
- Additional arguments passed to
f.
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.
Value
- A list containing the value of the Stieltjes integral
computed using each of the versions of the function
M.
See Also
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)