This is the "truncated exponential" estimate, which is based a simple heuristic extrapolation of d0 (the number of dies not present in the sample) from the first two terms (d1 and d
2) in the actual die frequency distribution, assuming an asymptotically exponential frequency distribution. For actual samples Dte tends to approximate De for quasi-exponential die frequency distributions
(with p values around 1) but gives results closer to D2 for samples with higher p values.
The results to date are summarised below. (Error ranges in the individual calculated estimates of D are not given
but they will be of the order of +/- 40-50% for a 95% coverage). The nominal, or expected, die numbers have been augmented to include known official looking duplicate numbers.
Also given below are the
values of the die frequency distribution shape parameter p which gives the best fit of D to the expected value for each type (using Eqn 2 above with the factor 2d replaced by pd, as in Eqn 5 in "Estimating Die Numbers").
Cr. 357/1 – C. Norbanus:
Nominal obverse dies (both reverses) 259 max (250 max excluding #220-228).
n = 282, d = 137, d1 = 65, d2 = 35; R = n/d = 2.06
De: 266; D2: 221; Dte: 258.
p = 1.2 (250 dies), 1.0 (259 dies)
Cr. 361/1 – P. Crepusius:
Nominal reverse dies 522.
n = 220, d = 166, d1 = 122, d2 = 36; R = n/d = 1.33
De: 676; D2: 509; Dte: 579.
p = 1.8
Cr. 378/1 – C. Marius Capito:
Nominal obverse and reverse dies 151.
n = 163, d = 82, d1 = 38, d2 = 23; R = n/d = 1.96
De: 169; D2: 133; Dte: 146.
p = 1.2
Cr. 382/1 – C. Naevius Balbus:
Nominal reverse dies 240 (+ maybe a couple more dups, say 242?).
n = 207, d = 131, d1 = 77, d2 = 40; R = n/d = 1.58
De: 357; D2: 270; Dte: 279.
p = 3.7
Cr. 383/1 – Ti. Claudius Nero (ordinary series):
Nominal reverse dies 152? ("170 max").
n = 112, d = 64, d1 = 36, d2 = 14; R = n/d = 1.75
De: 149; D2: 121; Dte: 157.
p = 0.9 (152 Dies), 0.7 (170 Dies)
Cr. 383/1 – Ti. Claudius Nero (A series):
Nominal reverse dies 133.
n = 118, d = 72, d1 = 37, d2 = 27; R = n/d = 1.64
De: 185; D2: 132; Dte: 123.
p = 1.9
It will be seen that De always gives the highest estimate, and in most cases overestimates the value of D as compared with the nominal value.
D2 underestimates D in all cases except
one (Naevius), although usually by not too much, and if anything gives a better fit than De overall.
The Dte estimates were above the expected value in three cases, below in one and about equal in two. They mostly
split the other two estimates and overall produced the "best" estimates of the three formula. It will also be seen that as expected the Dte estimate tends toward De for low p values and D2 for higher p values, so that
it effectively selects the more appropriate p value.
Alternatively, choosing the middle of the three values rather than any particular estimator has something to recommend it, and in fact turns out to give
estimates as good as Dte by itself, as where the middle value differs from Dte it produces better results, although only slightly.
One other noticeable result is the range of p values giving the best fit
to the expected values; mostly these lie in the general range of p = 1 to 2, although in the odd case of Naevius the optimal shape parameter is quite large, so that the frequency distribution tends toward a simple
Binomial distribution, which, in principle at least, indicates more or less equal die outputs (although in this case analysis also shows that the sample is clearly a long way from random).
So there you have it - choose your formula, or better, use all three.
* The fuller Negative Binomial formula for the exponential case (Eqn 7 in "Estimating Die Numbers") is:
This includes d1 and hence will give an estimate for D similar to but generally somewhat different from De.
** Alternatively, you can use this formula from Esty: