# 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 R language.

- M
The cumulative function against which

`f`

will be integrated. An object of class`"fv"`

or`"stepfun"`

.- …
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 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"`

, see `fv.object`

).
Objects of class `"stepfun"`

are returned by
`ecdf`

, `ewcdf`

,
`spatialcdf`

and other utilities.
Objects of class `"fv"`

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`

.

##### See Also

##### Examples

```
# NOT RUN {
# estimate cdf of nearest neighbour distance in redwood data
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)
# stepfun example
H <- spatialcdf(bei.extra$elev, normalise=TRUE)
stieltjes(function(x){x}, H)
# }
```

*Documentation reproduced from package spatstat, version 1.56-1, License: GPL (>= 2)*