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I am interested in the following sampling problem, which I will try to describe by a motivating example.

Suppose we want to estimate how many people in a certain area, has blue eyes, how many have them has brown eyes etc.(think of a color scale) We do have an estimate on total number eye colors that the population contains (let's say 100 colors).

We then sequentially observe samples - in a motivating context let's say a cruise arrives from that region each month. We disrupt those passengers and examine the eye color of each of them. The cruise does not always contain same number of passengers, and we don't know the decision rule on putting those people on that cruise. A reasonable assumption is that each person has an equal chance (p) of being in that cruise and reasonably we tend to see more the people with common eye colors.

The problem is

  1. Can we estimate total number of people living on the area?

  2. Can we estimate the proportion or chance "p" ?

MånsT
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Roark
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  • This seems to be a capture-recapture experiment. [The Wikipedia page on the topic](http://en.wikipedia.org/wiki/Mark_and_recapture) is a pretty good place to start reading more about it. – MånsT Jan 14 '13 at 12:12
  • @MånsT Yes, it starts off sounding like capture-recapture, but at the end it apparently is not. After all, people are not marked when they are encountered, so it is not possible to determine who has been recaptured. Because we don't know the rates at which people go on cruises, it looks like question (1) is not answerable with these data, and therefore neither is (2). – whuber Jan 14 '13 at 15:46
  • @whuber if we assume that the cruise goes back to the land with the same people that we examined, and we sometimes see the same people on the following cruises : would that make it capture-recapture? Thanks – Roark Jan 15 '13 at 04:45
  • Superficially, yes, at least in terms of the apparent mechanics of the observations. But effectively, no: for capture-recapture estimates to apply, each cruise must be a *random sample* of the entire population. Because cruises don't work like that, you cannot even get started. – whuber Jan 15 '13 at 13:08

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