here some further testing on ambus with an exp mixture.
if I use a 2 phase model and assume truncation at 50 PED (so not a shift by 50) I get the following estimates. (I changed from MLE to EM estimation, but thats only for those that are interested how to estimate).
Code:
m2 =
p m pt pe
0.05108 226.02 0.8015 0.01764
0.94892 37.912 0.2674 0.98235
p is the relative frequency of the respective exp dist within global loot. m is the respective mean, pt is the rel. freq. within the truncated observations as expected from the respective exp distribution and finally
pe is the effective rel. freq without truncation.
With the 2 phase model we get one exp with mean 37.9 and a relative frequency within global loot of 95%. The second has mean 226 and a rel. freq of 5%. An exp dist. with mean 37.9 would have about 27% of all observations after the truncation point of 50 PED. For the second dist this is 80%.
So for every dist we have
pt_i * pe_i = p_i
this can be resolved to pe_i = p_i/pt_i * n
where n is a normalizing constant and is equal to sum(p_i/pt_i).
This gives the above depicted
pe. So the real rel. frequency of the first dist. with mean 37.9 is 98% and that of the second with mean 226 is 2%.
So what does that mean?
If the assumption "that we observe truncated data" is right, then there are two loot distributions. The first has mean 37.9 PED and is triggered in 98% of cases. Only about 27% of the values from dist dist lead tio globals. The second has mean 226 and is triggered in about 2% of all loots. About 80% of the values from this dist. lead to globals. So it looks as we are able to estimate from global loot also one part of the non observable loot.
One further thing. It might be that
MA sets an upper limit. With an exp distribution you would be able to generate an infinity high loot. Therfore I guess they truncate the propability at .001 or .0001. The respective multipliers would be abou 7 and 9. So maximally 259-333 PED would be generated by the first dist and 1582-2034 from the second.
Loot expectation for the m2 model is 41.23 PED
For a 3 phase model I get the following:
Code:
m3 =
p m pt pe
0.65368 57.103 0.4166 0.17237
0.34144 16.164 0.0453 0.82706
0.00488 957.13 0.9491 0.00056
the 3 phase model fits quite better. However, I have some diffculties to understand the exp dist with mean 16. If this dist. existst, then there is so much in the global data, that we are able to extract it. Maybe this is the distribution for the pedders.
Loot expectation for the m3 model is 23.75 PED