Proofs and Additional Numerical Experiments for " Nonuniform Negative Sampling and Log Odds Correction with Rare Events Data "

Slutsky's theorem together with (S.3) and (S.5) implies the result in Theorem 1. Now we check the Lindeberg-Feller condition. 's are non-negative and E S.4 Derivation of corrected model (4) Note that π (x, 1) = 1 and π (x, 0) = π (x) . Slutsky's theorem together with (S.15) and (S.17) implies the result in Theorem 1. 's, whose distribution depends on N . From (S.27) and (S.28), Chebyshev's inequality implies that For sampled data, (5) tell us that the joint density w.r.t. the product counting measure of the responses The outline of the proof is similar to that of the proof of Theorem 2. Write Markov's inequality shows that they are both o The outline of the proof is similar to that of the proof of Theorem 4. The estimator Slutsky's theorem together with (S.38) and (S.40) implies the result in Theorem 1.