Bootstrapping a ratio that has no clean standard error

Question

You estimate a strategy's "payoff ratio" as average win size divided by average loss size, computed from paired trade outcomes. You need a confidence interval, but the ratio of two random averages has an awkward, skewed distribution.

Describe how to bootstrap the interval, and the danger to watch for when the denominator can be small.

Accepted Answer

Why the ratio is awkward. A ratio \hat R = \bar A / \bar B of two estimated averages is a nonlinear function of two random quantities. Its sampling distribution is generally skewed, and analytic standard errors require approximations (the delta method gives \operatorname{Var}(\hat R) \approx \frac{1}{\bar B^2}[\operatorname{Var}(\bar A) + R^2\operatorname{Var}(\bar B) - 2R\operatorname{Cov}(\bar A,\bar B)]) that assume near-normality and can be poor when the ratio is skewed. Bootstrapping avoids the algebra and captures the skew. The procedure (pairs bootstrap). The numerator and denominator …

Bootstrapping a ratio that has no clean standard error

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