A risk model needs g(p) = p², the chance that a pair of independent trades both win. You observe 100 independent trades with win rate p. At p = 1/2, what is the Cramér-Rao lower bound on the variance of an unbiased estimator of g(p)? Give a fraction.

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A bound for a function of the rate

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A risk model needs $g (p) = p^{2}$ , the chance that a pair of independent trades both win. You observe $100$ independent trades with win rate $p$ . At $p = \frac{1}{2}$ , what is the Cramér-Rao lower bound on the variance of an unbiased estimator of $g (p)$ ? Give a fraction.

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Hint

Start from the bound for $p$ , then multiply by the squared slope of $g$ .

Solution

The information in $100$ trades is $\frac{100}{\frac{1}{2} \times \frac{1}{2}} = 400$ .
The slope $g^{'} (p) = 2 p$ equals $2 \times \frac{1}{2} = 1$ , so its square is $1$ .
The bound is $1 \times \frac{1}{400} = \frac{1}{400}$ .

Answer: $\frac{1}{400}$

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