Estimating total value of population based upon a sample

I'm trying to estimate the total value of a population of a collection of assets given information on the N largest (assuming that the information on the N largest is certain).

For example, if I am told the 10 largest values in a population are 100, 10, 1, ... 10^-8, I can be fairly certain that the total value of the population is 111.11111...

How can I formalize this to a series of observations like 100, 90, 80,..10? (or any other description of observations)?

• You can't say anything (apart from the trivial fact that the total assets in the population can be no less than those you have observed, assuming assets cannot be negative) unless you also have (1) information about the size of the population and (2) information about how the smaller unobserved assets might be distributed. – whuber Jan 29 '18 at 16:04
• What if I was to put some limit on the total population size? EG, if I told you total number of assets was Y, and I knew the N largest by size, we should be able to bracket the result. As an extreme example of what I am thinking, in the latter example above, if Y was 20, I could draw some conclusions about the range of total sizes. – GPB Jan 29 '18 at 21:01
• Yes, you can bracket the result: but this is just simple arithmetic. It amounts to knowing only that all the remaining $Y-N$ assets have values between the minimum possible and the $N^{\text{th}}$ largest. If $Y$ is large compared to $N$, that could be such a wide interval as to be useless. – whuber Jan 29 '18 at 21:11