Highest Serial Number 105PN?
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Highest Serial Number 105PN?
I see a new listing from a Japanese seller on eBay with a 105mm PrintingNikkor having Serial Number 501281. This is the highest serial number I've seen yet for the 105PN, indicating there were at least 1281 of them made. Has anyone seen a higher number?
Still, 1281 is not a huge number.
Still, 1281 is not a huge number.
Re: Highest Serial Number 105PN?
Ray 
There must have been a few more  mine is 501521. Acknowledging my less than perfect memory, I think I got it from chendata on eBay a few years ago.
Cheers, David
There must have been a few more  mine is 501521. Acknowledging my less than perfect memory, I think I got it from chendata on eBay a few years ago.
Cheers, David
Re: Highest Serial Number 105PN?
If all we know are these two datapoints, the unbiased estimate of the highest existing serial number is 501761. With more samples, this estimate could be improved.

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Re: Highest Serial Number 105PN?
I'm all for using statistics appropriately, but I don't agree with such analysis in this case.
Re: Highest Serial Number 105PN?
It's often surprising how much we can know from sparse data sets.
This is mathematically the same problem as estimating the age of a plant trait given the age of the oldest recovered fossil showing that trait. This is an important problem, because we need such estimates in order to calibrate phylogenetic trees constructed by molecular analyses. I've been working on this problem for some time. It has an exact solution under some reasonable assumptions. These assumptions are surely not exactly satisfied in this case, but the result is fairly robust to minor deviations.
We assume that your list of serial numbers is a random selection of all PN105 serial numbers, with each possible serial number being equally probable. This process is a Bernoulli process. The gaps between successive observed serial numbers are unbiased estimators of the gap between the highest observed number and the true highest number. Of course, the uncertainty in the exact value of the highest serial number is large. But if you were to repeat this process of discovery and analysis many times, the average of your estimates (using my rule) would exactly equal the true value! (That's the technical definition of an unbiased estimator).
This is mathematically the same problem as estimating the age of a plant trait given the age of the oldest recovered fossil showing that trait. This is an important problem, because we need such estimates in order to calibrate phylogenetic trees constructed by molecular analyses. I've been working on this problem for some time. It has an exact solution under some reasonable assumptions. These assumptions are surely not exactly satisfied in this case, but the result is fairly robust to minor deviations.
We assume that your list of serial numbers is a random selection of all PN105 serial numbers, with each possible serial number being equally probable. This process is a Bernoulli process. The gaps between successive observed serial numbers are unbiased estimators of the gap between the highest observed number and the true highest number. Of course, the uncertainty in the exact value of the highest serial number is large. But if you were to repeat this process of discovery and analysis many times, the average of your estimates (using my rule) would exactly equal the true value! (That's the technical definition of an unbiased estimator).

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Re: Highest Serial Number 105PN?
The problem with the analysis is the assumption of a random sample. Having a sample of the highest two numbers that members are willing to divulge is a very nonrandom sample.Lou Jost wrote: ↑Mon Jun 28, 2021 2:36 pmIt's often surprising how much we can know from sparse data sets.
This is mathematically the same problem as estimating the age of a plant trait given the age of the oldest recovered fossil showing that trait. This is an important problem, because we need such estimates in order to calibrate phylogenetic trees constructed by molecular analyses. I've been working on this problem for some time. It has an exact solution under some reasonable assumptions. These assumptions are surely not exactly satisfied in this case, but the result is fairly robust to minor deviations.
We assume that your list of serial numbers is a random selection of all PN105 serial numbers, with each possible serial number being equally probable. This process is a Bernoulli process. The gaps between successive observed serial numbers are unbiased estimators of the gap between the highest observed number and the true highest number. Of course, the uncertainty in the exact value of the highest serial number is large. But if you were to repeat this process of discovery and analysis many times, the average of your estimates (using my rule) would exactly equal the true value! (That's the technical definition of an unbiased estimator).
David's mention of getting a 105PN from chendata jogged my memory though...prompting me to pull out all the 105PN's I currently own. Indeed I found one that I purchased from chendata that has S/N 1520. So in the spirit of a (possibly more) random sample, here are the serial numbers of all the 105PNs I've handled over the years. The distribution is very nonuniform, so I expect the analysis to still show a high estimated maximum:
153
168
177
221
241
247
263
284
285
319
328
355
358
365
372
379
402
411
414
419
420
425
429
438
444
463
467
668
1281 (eBay)
1520
1521 (David)
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Re: Highest Serial Number 105PN?
Interesting! Then, simplifying, I notice that the average gap is equal to (maxObservedminObserved)/(numObservations1), so the estimated maximum must depend on nothing more than the two extreme observations and the number of observations. Have I done that correctly?Lou Jost wrote:The gaps between successive observed serial numbers are unbiased estimators of the gap between the highest observed number and the true highest number.
If so, then taking Ray's numbers I would calculate as 1521 + (1521153)/(291) = 1570.
Plotting the data does show extreme nonuniformity of the sampling, so clearly something is wrong with the model in this case.
Rik
Re: Highest Serial Number 105PN?
Ray:
However, seeing this list, it is definitely not a random sample. It has huge gaps, but the sample from 153467 is remarkably complete. This strongly suggests that Nikon did not always use successive serial numbers, and for some reason the later numbers are comparatively poorly sampled. If Nikon was at least consecutively numbering the lenses from 1271 onward, my estimate would still be valid (the two last numbers, differing only by 1, are likely not independent, so one of them should be ignored).
Rik:
Yes, that is right. But as you say, the model fails because the numbers are too clustered to have come from a random sample of consecutively numbered lenses. As I mentioned above, though, if the last few lenses come from a random sample of consecutively numbered lenses, the analysis still applies, though with a large uncertainty.
The assumption of random sampling applies to your whole list, not specifically the two highest ones. Given a list based on random sampling, the gap between the true max serial number and the highest number on your list will, on the average, be equal to the gap between the highest and secondhighest numbers on your list.The problem with the analysis is the assumption of a random sample. Having a sample of the highest two numbers that members are willing to divulge is a very nonrandom sample.
However, seeing this list, it is definitely not a random sample. It has huge gaps, but the sample from 153467 is remarkably complete. This strongly suggests that Nikon did not always use successive serial numbers, and for some reason the later numbers are comparatively poorly sampled. If Nikon was at least consecutively numbering the lenses from 1271 onward, my estimate would still be valid (the two last numbers, differing only by 1, are likely not independent, so one of them should be ignored).
Rik:
Yes, that is right. But as you say, the model fails because the numbers are too clustered to have come from a random sample of consecutively numbered lenses. As I mentioned above, though, if the last few lenses come from a random sample of consecutively numbered lenses, the analysis still applies, though with a large uncertainty.

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Re: Highest Serial Number 105PN?
Agreed, not a random sample, but I don't agree with your conclusion about nonsuccessive serial numbers. This might be the case if this were a random sample but you begin by correctly stating it is not, invalidating this assumption. It's more likely that Nikon serialized these consecutively as they were made, and that each and every serial number between ~1 and 1521 was produced, with possibly some higher numbers as well. The nonrandom nature of the sampling is probably related to the ebb and flow of surplus industrial lenses. I purchased most of the lower numbers from a closed IBM inspection facility, while the later numbers came from chendata, presumably from closed faciities in China. It seems likely the "middle" serial numbers were purchased by some other entity(ies) which have either remained in business or simply threw those lenses away when they closed. Perhaps the lenses are about to come to market. Who knows?Lou Jost wrote: ↑Mon Jun 28, 2021 3:54 pmHowever, seeing this list, it is definitely not a random sample. It has huge gaps, but the sample from 153467 is remarkably complete. This strongly suggests that Nikon did not always use successive serial numbers, and for some reason the later numbers are comparatively poorly sampled. If Nikon was at least consecutively numbering the lenses from 1271 onward, my estimate would still be valid (the two last numbers, differing only by 1, are likely not independent, so one of them should be ignored).
I suppose my original question of "who has the highest serial number 105PN?" needs to be expanded to "does anyone have 105PNs in the serial numbers above 467, and where did they get them?"
Re: Highest Serial Number 105PN?
Yes, I agree with you Ray, if you got most of these from a single source, they aren't independent samples.
But I think it is neat that there is an unbiased estimator for this serial number problem, when the sample really is random.
But I think it is neat that there is an unbiased estimator for this serial number problem, when the sample really is random.
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Re: Highest Serial Number 105PN?
Probably best known as the German Tank Problem. I imagine you can guess the application.
Re: Highest Serial Number 105PN?
Rik, I derived it. I needed it for calibrating the phylogenetic tree of the Teagueia orchids I discovered (you featured one of them in the Administrator's Appreciation Gallery last year). I talked to biologists about it and was surprised to find that it was not known to them. I was surprised because the result follows quickly from basic probability theory.I agree. I do not recall having run into this estimator before. Do you have a reference for a publication about it?
Scarodactyl, this is great! I did not know this. The Wikipedia article you linked to includes much more detail than I had derived. There is one difference between my approach and the one described in the article, though. The article says that the solution for a single sample is unstable. Nevertheless I found that my solution is an unbiased estimator of the max serial number even if there is only one sample. Maybe "unstable" in this case just means that the variance can't be calculated from a single observation. Maybe that would be true. But the estimator still works (i.e, is unbiased) with just one sample, under the stated assumptions.Probably best known as the German Tank Problem.
Re: Highest Serial Number 105PN?
Here are some other interesting thing that seems unknowable but which can be calculated:
Nikon has made x different lens models throughout its history, and we don't know x. If eBay listings can be considered as a random sample of the population of Nikon lenses in the world, then we can figure out what percentage of the world's Nikon lenses belong to models that have never appeared on eBay. The answer is:
[the number of Nikon lens models that have appeared exactly once on eBay]/[the total number of Nikon lenses that have everappeared on eBay].
We can also put a lower limit on the actual number of Nikon lens models that have never appeared on eBay:
{[the number of Nikon lens models that have appeared exactly once on eBay]^2}/{2*the number of Nikon lens models that have appeared exactly twice on eBay}.
Nikon has made x different lens models throughout its history, and we don't know x. If eBay listings can be considered as a random sample of the population of Nikon lenses in the world, then we can figure out what percentage of the world's Nikon lenses belong to models that have never appeared on eBay. The answer is:
[the number of Nikon lens models that have appeared exactly once on eBay]/[the total number of Nikon lenses that have everappeared on eBay].
We can also put a lower limit on the actual number of Nikon lens models that have never appeared on eBay:
{[the number of Nikon lens models that have appeared exactly once on eBay]^2}/{2*the number of Nikon lens models that have appeared exactly twice on eBay}.

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Re: Highest Serial Number 105PN?
With the scope of eBay, and its seeming ability to bring all types of items to market, it should be an excellent sampling medium, though far as I know it is not really searchable in a way to answer these questions. Does anyone know of a historical database for eBay listings/sales?Lou Jost wrote: ↑Wed Jun 30, 2021 7:41 amHere are some other interesting thing that seems unknowable but which can be calculated:
Nikon has made x different lens models throughout its history, and we don't know x. If eBay listings can be considered as a random sample of the population of Nikon lenses in the world, then we can figure out what percentage of the world's Nikon lenses belong to models that have never appeared on eBay. The answer is:
[the number of Nikon lens models that have appeared exactly once on eBay]/[the total number of Nikon lenses that have everappeared on eBay].
We can also put a lower limit on the actual number of Nikon lens models that have never appeared on eBay:
{[the number of Nikon lens models that have appeared exactly once on eBay]^2}/{2*the number of Nikon lens models that have appeared exactly twice on eBay}.