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Download a pdf copy of this help file here |
See also: VosePoissonLambda
The confidence we have that the true value of the intensity is less than or equal to l is:
P(X>a;l) + ½ P(X=a;l)
where X is the random variable of the number of events one could observe in a period with an expected rate of l. This is saying that the greater the true value of the intensity l, the more confident we would be in observing a particular number of events a, or more. Translated into Excel formula, the confidence we have that the true Poisson intensity is less than any specific tested value l is given by:
=1-VosePoissonProb(a,l,1)+0.5*VosePoissonProb(a,l,0)
We can use this to construct a cumulative confidence distribution by testing a range of values of l:
Figure 1: Cumulative distributions of estimate of l for varying number of observations a
The points used to construct the above plot can be fed into a VoseCumulA distribution so that you can generate values from it.
ModelRisk offers the function VosePoissonLambda that will perform this whole process automatically.
Looking at Figure 1 you'll see that the cumulative distribution for a = 0 starts at 0.5. That means the distribution is assigning 50% confidence to l = 0, and the remaining 50% confidence to all other values of l. That is equivalent to saying that when there have been no observations, we are equally confident that no such stochastic process exists.
In the ModelRisk VosePoissonLambda(a,t,ProcessExists,U) function you input the counts a and time over which they have been observed t, and in the situation where a = 0 you have the option to specify whether you know that the intensity is non-zero (ProcessExists = TRUE), similarly to the BinomialP function. The U parameter also allows you to specify a cumulative percentile – if omitted the function simulates random values of what the value a might be. So, for example:
VosePoissonLambda(2,3, TRUE, 0.2) = VosePoissonLambda(2,3, FALSE, 0.4) = 0.36634
VosePoissonLambda(0,3, TRUE, 0.2) = 0.203324 (it assumes that a cannot be zero)
VoseBinomialP(0,20,FALSE,0.2) = 0 (it allows that a could be zero)