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The following estimators can be used for the german-tank-like problems.

If we collect a sample of size $k$ with sample mean $\bar{x}$ and highest number $m$, then we can estimate the population size with either $\hat{x}_{1}$ or $\hat{x}_{2}$:

$$\hat{x}_{1} = m + \frac{m}{k} - 1$$ $$\hat{x}_{2} = 2 \bar{x}$$

After computationally testing both on random datasets, with various sample sizes and various number of samples, it appears that:

  • both are unbiased (both hit the true value on average)
  • $\hat{x}_{1}$ is much more accurate than $\hat{x}_{2}$, especially when $k$ is relatively small ($k < 30$)

The reasoning behind $\hat{x}_{2}$ is obvious, while $\hat{x}_{1}$ is a little more obscure.

What I don't get is, how comes $\hat{x}_{1}$ is the more accurate of the two?

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