Optimal estimators and tests for reciprocal effects

However, due to nonlinear factors in this model, how to estimate ρ efficiently is a long-standing open problem.</p> <p>We tackle the problem by the cycle count approach. The challenge is, due to the nonlinear factors in the model, for any given type of generalized cycles, the expected count is a complicated function of many parameters in the model, so it is unclear how to use cycle counts to estimate ρ.

However, somewhat surprisingly, we discover that, among many types of generalized cycles with the same length, we can carefully pick a pair of them such that in the ratio between the expected cycle counts of the two types, the non-linear factors <i>cancel out nicely with each other</i>, and as a result, the ratio equals to exp (ρ) exactly. Therefore, though the expected count of cycles of any type is not tractable, the ratio between the expected cycle counts of a (carefully chosen) pair of generalized cycles may have an utterly simple form.</p> <p>We study to what extent such pairs exist, and use our discovery to derive both an estimate for ρ and a testing procedure for testing ρ=ρ0.