Define distribution function to be zero if ecdf(X) is not defined

I have a distribution of random numbers X in the interval [50,100]. When using the function ecdf(X) to receive the empirical cumulative distribution function, the values for x<50 are not defined.

I the following, I want to integrate (1-ecdf(X)) from 0 to infinity and therefore need the ecdf(X) to be zero.

Is there a possibility to define the ecdf(X) to be zero for x<50?

Answer 1

[Torsten]

[f,x] = ecdf(X);
Ecdf=@(p) interp1(x,f,p)*(p>=50)

Best wishes

Torsten.

[Nina]

Thank your for your answer, it makes sense, but I have trouble implementing this in my specific case, since I already used the interpl-function...would be great if you could help with the following code:

g=0.1;
h=0.2;
a = 50;
b = 100;
R = (b-a).*rand(100,1) + a;

[distribution_neutral,x]=ecdf(R);
distribution_distorted= 1-(1-distribution_neutral).^(1/(1+g)).^(1+h);
x(1)=-realmax;
fun3= @(xq)  interp1(x,distribution_distorted,xq,'previous','extrap'); 
final_value=integral(fun3,0,inf);

[Torsten]

Why not simply

g = 0.1;
h = 0.2;
a = 50;
b = 100;
R = (b-a).*rand(100,1) + a;

[distribution_neutral,x]=ecdf(R);
distribution_distorted= 1-(1-distribution_neutral).^(1/(1+g)).^(1+h);
final_value = trapz(x,distribution_distorted)

?

And note that

a^b^c = a^(b^c).

I don't know if this is really what you want in the expression

(1-distribution_neutral).^(1/(1+g)).^(1+h)

Best wishes

Torsten.

[Nina]

Thank you for your quick answer! Unfortunately, this simple solution is not possible because my distribution function is decreasing from 1 to 0 and not increasing from 0 to 1. Thus for an integration in the interval of [0:inf] the area for x<50 needs to be defined as 1. Sorry for the confusion, would be thankful for any further advice!

[Torsten]

A CDF must always increase from 0 to 1.

And your "distribution_distorted" does:

For x<50, it must be defined as 0, for 50<x<100, it increases from 0 to 1, and for x>100, it must equal 1.

Just use

plot(x,distribution_distorted)

Best wishes

Torsten.

[Nina]

You are totally right, but the problem is that the CDF in my plot seems to be not defined as 0 for x<50. Before integrating I calculate 1-CDF and therefore get the decreasing function...

[Torsten]

If you integrate from the first x-value to the last x-value (as the command

"trapz(x,Distribution_distorted)"

does), you automatically ensure that the CDF is counted to be zero outside the interval where its mass is concentrated (namely outside [x(1);x(end)]). And I guess this is what you want.

Best wishes

Torsten.