Asymptotically optimal estimating equation with strongly consistent solutions for longitudinal data

Estimation for recurrent events through conditional estimating equations

“…The following result illustrates condition S(ii) on the leading term in the decomposition of D n (η) (see [5]).…”

“…Proof. The proof is identical to the proof of Theorem 4.2 in [5], once α 1/2+δ n is replaced by n. Conditions S(i) and S(iii) ensure nonsingularity and S(ii) the equicontinuity of the derivatives at η 0 . Remark 3.1.…”

“…Let D n (η) = −∂q n (η)/(∂η T ) be the (p + 1) × (p + 1) matrix of derivatives. The following theorem gives sufficient conditions for the almost sure existence and strong consistency of a sequence of estimators of η 0 (see Theorem 4.2 of [5]). Theorem 3.1.…”

“…In order to apply Theorem 3.1, conditions on the moduli of continuity of functions related to the derivative in Theorem 3.1 are needed. In [5], only the analytical properties of µ are needed, as V ij (θ) is a function of µ ij (θ) and c ij (θ) is a linear function there. As for b ij (η), it is not present since the overdispersion parameter is not considered there.…”

“…The penalty for using a working covariance in lieu of the true one is a decrease in efficiency. Theoretical justification and extensions of the results in [11] were given in [24], [4], [5]. Further applications and examples can be found in [27] and [28].…”