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MathGroup Archive 2005

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Re: Empirical CDF

  • To: mathgroup at
  • Subject: [mg58926] Re: Empirical CDF
  • From: Mark Fisher <mark at>
  • Date: Sun, 24 Jul 2005 01:21:48 -0400 (EDT)
  • References: <dbt3oh$skn$>
  • Sender: owner-wri-mathgroup at

In the meantime, Piecewise has arrived, which Mathematica *does* know how to 
integrate. Here is an example of how to use it to compute an emprical 
cummulative distribution function given a list of observations.

MakeEDFPiecewise[list_] :=
   With[{d = 1/Length[list]},
     Function @@ {x,
         Transpose[{Range[1, d, -d], Thread[x >= Reverse[Sort[list]]]}]

rand = Table[Random[], {100}];
fun = MakeEDFPiecewise[rand];
Plot[fun[x], {x, -.25, 1.25}, PlotPoints -> 300];

fun2 = Block[{x}, Function @@ {x, Integrate[fun[x], x]}];
Plot[fun2[x], {x, -.25, 1.25}, PlotPoints -> 300, PlotRange -> All]

For faster execution one can replace Function with Compile in the 
definition of MakeEDFPiecewise. (I use an Option to control the choice.) 
However, there are three costs: It takes longer to compute the function 
itself, the resulting object is larger, and it Mathematica complains when one 
tries to integrate it analytically (and returns an uncompiled result).


P.S. Question to Mathematica developers: Why does fun2'[x] sometimes (but not 
always) produce the error message Reduce::ratnz?

David Kahle wrote:

> MathGroup -
> swidrygiello posted on Fri, 13 Sep 2002 a question on how to create the
> empirical cumulative distribution function (ECDF) with Mathematica.  Since
> then, few others (Mark Fisher, , for example) have posted
> responses utilizing the Interpolation[] command.  However, similar results
> can be achieved using the simple code :
> For[i=1,i<(Length[rand]+1),preF[x_,i_]:=UnitStep[x-rand[[i]]],i++]
> F[x_]:=Sum[preF[x,i],{i,Length[rand]}]/Length[rand]
> Where 'rand' is the vector containing the random observations.  The
> resulting function F is right continuous, limits to 0 and 1 as we would
> like it to, and has the correct step sizes.  Even better, Mathematica is
> more comfortable manipulating UnitStep functions than it is piecewise
> functions defined with the Which[] command.  For example, if we would like
> to integrate the ECDF to test for certain orderings (Stochastic dominance,
> etc.), Mathematica understands integrating the F as defined above, but if
> we use the Which[] command it resorts to numerical techniques which
> typically fail for various reasons.  Hope it helps.
> David Kahle
> david.kahle at

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