I have a problem which is Serial Numbered Population (SNP) problem. I want to estimate the size (N) of SNP with a sample of size n. The sample are x1,x2,,...,xn. It says that we will use X=max(x1,x2,,...,xn) as a statistic to estimate N. How do I prove that X is a appropriate to estimate N?
I attached the journal.
There are many methods of calculating an estimator of a parameter. All the methods have some optimal criteria with respect of which you can calculate the estimate. Maximum likelihood estimate maximises the probability with which we already have the data. Minimum Distance Estimates minimises the distance form the real parameter. Finally there is the famous Rao-Blackwell Theorem (http://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell_theorem) with which you can convert any estimator into a good one. (Raoblackwellisation)
Statistic is nothing but a sample measurement and parameter is population measurement e.g. sample mean is a statistic and population mean is corresponding parameter. A statistic can correctly estimate the corresponding parameter if and only if it satisfies certain characteristics., that a good statistic must be
1. unbiased
2. consistent
3. sufficient
4. efficient.
The above suggestions could offer pivot direction. in addition, Since the problem is to use X=max(x1,x2,...xn) as the statistic to estimate N, you clearly have order statistics problem. Further, you should explore the properties of good estimator while focusing on the order statistic aspect.
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