Estimate population size based on repeated observation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T00:10:51Z http://mathoverflow.net/feeds/question/14964 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14964/estimate-population-size-based-on-repeated-observation Estimate population size based on repeated observation Pete Bevin 2010-02-11T02:15:50Z 2010-02-11T10:18:14Z <p>I take the bus to work every day. Every bus has a serial number, but unlike in <a href="http://en.wikipedia.org/wiki/German%5Ftank%5Fproblem" rel="nofollow">the German Tank Problem</a>, I don't know if they are numbered uniformly $1...n$.</p> <p>Suppose the first $k$ buses are all different, but on day $k+1$ I take one I've been on before. What is the best estimate for the total number of buses?</p> http://mathoverflow.net/questions/14964/estimate-population-size-based-on-repeated-observation/14988#14988 Answer by Thorny for Estimate population size based on repeated observation Thorny 2010-02-11T10:18:14Z 2010-02-11T10:18:14Z <p>Maximum likelihood estimate is the smallest $n$ for which $$\left( 1+\frac{1}{n} \right)^k \leq \frac{n}{n-k+1},$$ that gives a value of $n$ asymptotically equal to $\frac{k^2}{2}$, consistently with the Birthday Paradox. Not sure whether an unbiased estimate would be better for any practical purpose; maybe you do have an a priori distribution for which a Bayesian estimate makes sense?</p>