Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 2 / c

Solution

For arbitrary nonnegative raw weights

W_{i}

, let

\overset{ˉ}{W} = K^{- 1} \sum_{i} W_{i}

K

, is

CV^{2} (W) = \frac{K ^{- 1} \sum _{i} ( W _{i} - W ˉ ) ^{2}}{W ˉ ^{2}} = \frac{K \sum _{i} W _{i}^{2}}{( \sum _{i} W _{i} ) ^{2}} - 1.

ESS = \frac{( \sum _{i} W _{i} ) ^{2}}{\sum _{i} W _{i}^{2}} .

For normalized weights

w_{i} = W_{i} / \sum_{j} W_{j}

, this reduces to

ESS = 1/ i \sum w_{i}^{2}

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