interval censored data using bootstrap method

[zzzc]

I have time Xi that is interval censored and the data consist of time intervals Ii = [Li, Ri] such that Xi is within Ii , Ri>Li, i=1,...,96
My interest is theta=E(Xi)
But I'm not sure what E(Xi) is.
How should I estimate E(Xi )and conduct inference for it?

Thanks

[BGM]

If you already estimate the empirical cumulative distribution function ,
or equivalently the empirical survival function ,

then

The asymptotic approximate standard error of the estimate is available
(at least for the common Kaplan-Meier case)

But if you have only interval censored data, without any observed failures,
traditionally you do not have a unique maximum likelihood nonparametric
estimates. You may search for the Turnbull's interval/algorithm as well.
(I am not an expert in survival analysis)

[zzzc]

If I simply assume X~Unif[I], recall I =[L,R]
and I want to use a kernel density (1/h)*K( ( E(X|I)-x ) / h ) and assume kernel to be Gaussian.
For E(Xi|Ii), do I simply use the mean of Unif[Li,Ri] so my data becomes (Li+Ri)/2 for each i?

Thank you!