# RE: st: evaluating probabilities using kdensity

 From "Nick Cox" To Subject RE: st: evaluating probabilities using kdensity Date Mon, 3 Mar 2008 14:55:42 -0000

```In addition, and to pick up comments Maarten made when this was first
I don't see that you are obliged to work via the density function at all
when the
overall aim is to get an integral like this.

A (possibly smoothed) quantile function and a distribution function are
just
inverses of each other.

Nick
n.j.cox@durham.ac.uk

Maarten buis

*---------- begin example -------------
sysuse auto, clear
kdensity mpg, gen(x d) n(100)
integ d x if x < 20
*------------- end example ------------
(For more on how to use examples I sent to the Statalist, see
http://home.fsw.vu.nl/m.buis/stata/exampleFAQ.html )

Conner Mullally <mullally@primal.ucdavis.edu> wrote:

> Thanks for the help. I think I could have been more specific
> about exactly what I am trying to do.
> I have 20 years of time series, which I have detrended.
> What I want to do is to:
> a) Take the detrended time series and estimate a kernel
> density
> b) Use the estimated kernel density to estimate the expected
> value of shortfalls from the mean of the distribution.  That
> is, I want to calculate the integral of:
> (Mu-Yt)dF(Yt)dYt, for all Yt<Mu
> Where Mu is the mean, Yt is the random variable, and F(Yt)
> is the cumulative density function of Yt.
> The command kdensity returns the estimated density at
> whatever points were used to estimate the density, and I can
> add more points using the at() option.  But these will leave
> gaps at the omitted points in the support of the estimated
> distribution when I sum up the densities.  What I'm
> wondering is if there is a way to integrate under the curve
> of the estimated kernel density. Perhaps some stata
> programming is in my future, but if there is way to do this
> while using kdensity or something similar, I would love to
> know.

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```