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The German tank problem is about estimating the total size of a set of objects from random samples of serial numbers.

It can simply be done by:

$$N \approx m+\frac{m}{k}-1$$

Where $m$ is the highest observed serial number, and $k$ is the number of samples.

I want to find the growth rate, (assumed to be linear), and that is easily found by comparing an older estimate to a newer.

However:

The newest estimate should partly be considering older data.

Say I observed a few serial numbers yesterday, and a few today, and I want to find the growth. The naive solution is to do an independent estimate for each day, and compare. But it is clear that yesterday's observations are also valuable for a new estimate, for example if the highest serial number observed was higher.

How do I account for the aged data, while still compensating for the fact that it comes from a smaller total number of serial numbers?

Clarification edit:

This is referring to the original tank estimation problem, and the production of new tanks is assumed to be constant (same number added each day). A negative production is not possible, and potential loss of vehicles are ignored. What I am looking for is a model with a production rate that does not change, although a generalization is welcome.

That boils down to: Given a list of tank serial number observations for different days, how do I find a production rate?

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    $\begingroup$ Growth rate in what? The population parameter? Can you describe the situation more clearly please? If the parameter of the discrete uniform ($N$) is changing, you should distinguish different values. If you have an $N_t$ you would need to discuss your assumptions for how it might change over time - does it only increase or stay steady for example, or could it decrease? Can it jump by arbitrarily large amounts or intermittently -- or will it tend to increase fairly steadily? That is, how does $N_{t}-N_{t-1}$ behave? Are successive increments dependent in some way? $\endgroup$
    – Glen_b
    Commented Feb 15, 2016 at 0:22
  • $\begingroup$ @Glen_b I see I have not been as precise as I could have been. This is referring to the original tank estimation problem, and the production of new tanks is assumed to be constant (same number added each day). A negative production is not possible, and potential loss of vehicles are ignored. What I am looking for is a model with a production rate that does not change, although a generalization is welcome. $\endgroup$
    – SE - stop firing the good guys
    Commented Feb 15, 2016 at 0:31
  • $\begingroup$ Doesn't it make sense to use all the data available each day? That is if you observe 10 data points per day, then day 1 uses $k=10,$ day 2 uses $k=20,$ etc.? Starting from day 2 you can estimate the growth. Then you just need a method to deal with how to average your growth estimates and handle the cases where no new maximum was observed. If you don't use all the historical data each day, I think you will be sacrificing your sample sizes unnecessarily. $\endgroup$
    – soakley
    Commented Feb 16, 2016 at 18:40
  • $\begingroup$ @soakley A small sampling size is indeed one of the key elements of the German tank problem. An use-all-data-points-equally approach is therefore of interest, but will always produce an underestimate. The case handling you describe can benefit from the signed nature of the problem. (the real number is always growing.) $\endgroup$
    – SE - stop firing the good guys
    Commented Feb 16, 2016 at 20:46

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A reasonable approach may be to estimate the production rate by always using the maximum time period available. That is, create an estimate of $N$ every day, but use today's estimate along with the day 1 estimate and the number of days that have passed to get the estimated production rate.

For $i \ge 2,$ your day $i$ growth rate estimate $\hat{G}$ will be $$\hat{G}={{\hat{N_i}-\hat{N_1}} \over {i-1}}$$

The idea is similar to observing a stochastic process and estimating the rate as the number of observed events divided by the total time.

Because your daily estimates $\hat{N_i}$ are unbiased for the total number of tanks, the difference $\hat{N_i}-\hat{N_1}$ is unbiased for the number of tanks produced in that time period, and your overall production rate estimate $\hat{G}$ will also be unbiased.

It is true that you can get negative estimates in the early periods. So you will have to decide if you want to cap those at zero or use some other method to handle those cases.

Note that your estimator is unbiased only if the sampling is without replacement. If your sampling is with replacement, you will need to consider another estimator.

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