Compute expectation from empirical CDF

Question

I have a empirical cumulative probability distribution function for a random variable. The random variable is "time to failure". I want to know Mean Time To Failure i.e expectation of that random variable. Is there any standard method to find mean from an empirical distribution.

I am getting the empirical CDF (as discrete values) as output from a "model checking tool" which uses iterative numerical computation techniques to get those probabilities. For example, let F(t)=P(X<=t) is the CDF of the random variable X where X stands for time between failure. To plot the curve of "F(t) vs t" I am varying t with some step size, calculating F(t) for that t using the "model checking tool" and adding the points to get the curve. I can use small step size to get the more accurate curve. So, I have access to only this CDF values at different t. From this values I want to do a good estimate of mean value of X.

Now the parameters to do a good estimate will be:

1) T, the maximum value of t. We need to find this with some precision i.e if F(T1)-F(T2) is less than some epsilon we set T=T1.

2) Once we have found T we need to find suitable step size h at which we will be calculating the CDF values.

3) Suppose for t=h,2h,....,nh (where nh=T) we get the corresponding cumulative probability values from the model checking tool as P1, P2,.....,Pn then

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a) For each interval (a,b] select the leftmost point as the representative of the interval So, E[X]=0.P1+h.(P2-P1)+2h.(P3-P2)+......+(n-1)h(Pn-Pn-1) b) For each interval (a,b] select the rightmost point as the representative of the interval So, E[X]=h.P1+2h.(P2-P1)+........+nh(Pn-Pn-1)

How should I choose those parameters?

Answer 1

For a non-negative random variable $X$ whose expectation exists, $$ E(X) = \int_0^\infty \mathrm{Pr}(X>t) dt.$$ In the case of a non-negative integer random variable, this reduces to $$ E(X) = \sum_{i=1}^\infty ~\mathrm{Pr}(X\ge i),$$ which has a very easy proof.

In principle you can use these directly, but since you are dealing with sampling, you need to seek out the theory that surely exists about how to do these estimations with good statistical properties. Let's hope someone tells us about it.