This applet puts the last two applets together: you can change both the speed and the jump size.
The result is a stochastic integration with respect to a time changed Poisson process.
The initial m(s)=1, speed(s)=1, so if you do nothing other than click the "Start" button, then the plot shows a standard Poisson process N(t) (black) and its cumulative intensity t (red).
For whatever speed and jump size you may click, the resulting red line is always the integration of m(s)speed(s)ds from 0 to t. The black line is integration of m(s) d N( speed(s)) from 0 to t.
It is also true that the difference between the black and red plot is always a martingale.
Click on "Start" to begin. Click "Increase jump size" or "Decrease jump size" to see the effects of m(t); click on "Faster" or "Slower" to see the effects of speed(t).
To simulate the Nelson-Aalen estimator based on a sample, you would keep m(s)= 1/R(s) where R(s) is the number of subjects at risk at time s; and speed(s) = sum h_i (s) where the summation is over all subjects that are still at risk at time s. (h_i(s) is the hazard function of the ith subject at time s).
Written by Mai Zhou Copyright © 2005 Mai Zhou. All rights reserved.