ratio_variance

forest.benchmarking.observable_estimation.ratio_variance(a: Union[float, numpy.ndarray], var_a: Union[float, numpy.ndarray], b: Union[float, numpy.ndarray], var_b: Union[float, numpy.ndarray]) → Union[float, numpy.ndarray]

Given random variables ‘A’ and ‘B’, compute the variance on the ratio Y = A/B.

Denote the mean of the random variables as a = E[A] and b = E[B] while the variances are var_a = Var[A] and var_b = Var[B] and the covariance as Cov[A,B]. The following expression approximates the variance of Y

\[Var[Y] \approx (a/b)^2 * ( var_a /a^2 + var_b / b^2 - 2 * Cov[A,B]/(a*b) )\]

We assume the covariance of A and B is negligible, resting on the assumption that A and B are independently measured. The expression above rests on the assumption that B is non-zero, an assumption which we expect to hold true in most cases, but makes no such assumptions about A. If we allow E[A] = 0, then calculating the expression above via numpy would complain about dividing by zero. Instead, we can re-write the above expression as

\[Var[Y] \approx var_a /b^2 + (a^2 * var_b) / b^4\]

where we have dropped the covariance term as noted above.

See the following for more details:

• https://doi.org/10.1002/(SICI)1097-0320(20000401)39:4<300::AID-CYTO8>3.0.CO;2-O
• http://www.stat.cmu.edu/~hseltman/files/ratio.pdf
• https://en.wikipedia.org/wiki/Taylor_expansions_for_the_moments_of_functions_of_random_variables

Parameters:

• a – Mean of ‘A’, to be used as the numerator in a ratio.
• var_a – Variance in ‘A’
• b – Mean of ‘B’, to be used as the numerator in a ratio.
• var_b – Variance in ‘B’