Interval censored data of type 1 is also called current status data or binary data. In this case the variable of interest, X, is never observed exactly. We only observe Y and if (X>Y) or not. (The status of X at current time Y).
The Non-parametric Maximum Likelihood Estimator (NPMLE) of distribution of X is a staircase function with the location of the drops randomly placed. The size of the drops also changes.
This JAVA applet plots NPMLE of distributions computed from a sample of current status data. We took both X and Y to be standard exponential random variables. Starting with sample size n=2 and grew to as large as 10,000.
The true distribution, exponential, is plotted in black. Well, actually the survival function, 1-F(t), is plotted. So the black curve is just S(t)= exp(-t).
The NPMLE is shown in red.
The NPMLE converge very slowly, in fact, it is known to converge at a rate of 1/cubic root(n). As opposed to the Kaplan-Meier estimator which converge at a rate of 1/square root(n).
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