Weighted Estimator for the Linear Transformation Models with Multivariate Failure Time Data

Abstract

In this article, a simple and efficient weighted method is proposed to improve the estimation efficiency for the linear transformation models with multivariate failure time data. Asymptotic properties of the estimators with a closed-form variance-covariance matrix are established. In addition, a goodness-of-fit test is developed to evaluate the adequacy of the model. The performance of proposed method and the comparison on the efficiency between the proposed method and the working independence method (Lu, 2005) are conducted in finite-sample situation by simulation studies. Finally a real data set from the Busselton Population Health Surveys is illustrated to validate the proposed methodology. The related proofs of the theorems are given in the Appendix.

Keywords:

• Weighted average estimator
• Estimating equations
• Linear transformation models
• Martingale
• Model checking
• Multivariate failure time data
• Working independence method

Mathematics Subject Classification:

• 62N01

Appendix

A Proof of Main Results

The consistency of can be similarly derived as in Chen et al. (2002) using the empirical process and weak convergence theory and so the proof is omitted here. For j = 1, 2, …, J, let and are the solution of the following estimating equations: (A1) (A2) and is the true value of parameter, but is its estimator.

Proof of Theorem 2.1.

For j = 1, 2, …, J, t > 0 and x ∈ ( − ∞, ∞), define where a > 0 and b are fixed finite constants to insure that the integrals are finite. We divide the proof into the following three steps.

First, using the analogous argument as given in Step A2 in Chen et al. (2002) and noting that , j = 1, 2, …, J, we can obtain Therefore, for t ∈ [0, τ], we have (A3) Furthermore, since (A4) differentiating (A4) with respect to on both sides and noting again , j = 1, 2, …, J, one can obtain (A5) Then it follows from the law of large numbers and (A5) that Therefore, where “” denotes the convergence in probability.

Secondly, mimicking Step A4 in Chen et al. (2002) and according to the equation (A3), for j = 1, 2, …, J, we have Therefore, Thus, by the multivariate central limit theorem and the Slutsky theorem, it follows that

Last, according to the Taylor expansion, we can obtain Then Theorem 2.1 follows immediately from the Slutsky theorem. The proof is complete.

Proof of Corollary 2.1.

Note that then the result of the corollary is followed immediately from the continuous mapping theorem.

Proof of Theorem 2.2.

Set , recall . Then by the block matrix multiplication, we have Therefore, by the continuous mapping theorem, it follows that But, by the block matrix multiplication, again, we have The proof is complete.

Proof of Theorem 2.3.

Since , then by the Taylor series expansion and (A5), we can get Therefore, it follows that by Theorem 2.2 and (A3) Let then based on the functional delta method (van de Vaart and Wellner (1996, Lemma 3.9.20)), we have which is a sum of random variables with independent identical distribution. Thus, the finite dimensional distribution of converge to the mean zero multivariate normal distribution by multivariate central limit theorem. Furthermore, note that takes a similar form as Spiekerman and Lin (1998), so its tightness can be obtained by an analogous argument as Spiekerman and Lin (1998). The proof is complete.

Proof of Theorem 3.1.

Note that , so Then, according to the Taylor series expansion, we have where

Then, by (A3) and the analogous argument as Theorem 2.1, it follows that where

Moreover, similar to the proof in Theorem 2.1, we have

Therefore, it follows that which is the sum of independent identical distributions random variable, so the finite dimensional distribution of converges to the multivariate normal distribution according to the multivariate central limit theorem. Moreover, the tightness of follows from the similar argument as Lin et al. (2000). Therefore, converges weakly to a mean zero Gaussian process. The proof is complete.