# 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
`f`

will 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)
```

*Documentation reproduced from package spatstat, version 1.23-3, License: GPL (>= 2)*