The Nelson-Aalen Estimator

The Nelson-Aalen estimator is a non-parametric estimator of the cumulative hazard function based on a sample that are subject to right censoring.

To put it simply, the Nelson-Aalen estimator is a staircase function with (1) the location of the steps placed at each observed death time. (2) the vertical size of the steps is 1/(number at risk). Where (number at risk) is the count of subjects just before the death that are still observed to be alive.

This JAVA applet plots the Nelson-Aalen estimator computed from a sample of iid exponential random variables. The censoring times are also iid exponential random variables-- with a selectable scale parameter. Starting with sample size n=2 and grew to as large as 10,000.

As n grows the Nelson-Aalen estimator (in red) will getting closer to the true cumulative hazard (plotted in black, the one we used to generate the random observations). In this case it is a straight line. The difference of the red and black line is a (local) Martingale. The last segment, plotted in blue, no longer forms a martingale when subtract the black

Following conventional plotting, I have put a little tick on the censored observations. For large samples this just makes the red curve 4 pixels wide.

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Notes:

The y axis is drawn at 0, I am too lazy to put units on the axis. Every time n increases the computation pauses for half a second. (for those want speed instead of detail, I may put up a button for speed adjustment in the future). Also if n exceeds 10000 the computation will be messed up since I have limited the storage to 10000. But after n exceed, say, 3000, the red curve do not change much and almost coincide with the black curve.
Runs under Netscape 3.0 and up. Not (yet) tested under other browser.