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.
Mathematics Subject Classification:
Appendix
A Proof of Main Results
The consistency of can be similarly derived as in Chen et al. (Citation2002) 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. (Citation2002) 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. (Citation2002) 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 (Citation1996, 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 (Citation1998), so its tightness can be obtained by an analogous argument as Spiekerman and Lin (Citation1998). 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. (Citation2000). Therefore,
converges weakly to a mean zero Gaussian process. The proof is complete.