Control Variate Optimal Coefficient

Problem

The optimal coefficient for the control variate estimator X - b * (Y - E[Y]) is b = Cov(X, Y) / Var(Y). Given samples of the estimator values and the control values, return b using population moments, or 0.0 when there are fewer than two pairs or the control variance is zero.

Hint

Regressing the estimator on the control gives the variance-minimizing coefficient.

Test cases

case 1

in: [[1,2,3],[1,2,3]]

expected: 1

case 2

in: [[1,2,3],[3,2,1]]

expected: -1

case 3

in: [[1,2,3,4],[1,1,2,2]]

expected: 2

case 4

in: [[1,2],[3,3]]

expected: 0

case 5

in: [[1,2],[5]]

expected: 0

Starter codePython

def control_variate_optimal_coefficient(values, controls):
    # Your code here
    pass

5 test cases · run your code against them in the editor