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Empirical likelihood inference for general transformation models with right censored data

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Abstract

In this work, we consider empirical likelihood inference for general transformation models with right censored data. The models are a class of flexible semiparametric survival models and include many popular survival models as their special cases. Based on the marginal likelihood function, we define an empirical likelihood ratio statistic. Under some regularity conditions, we show that the empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution. Through some simulation studies and a real data application, we show that our proposed procedure can work fairly well even for relatively small sample size and high censoring.

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Acknowledgements

We sincerely thank the AE and two reviewers for their very insightful and detailed comments which lead to improvements of the manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu, 221116, P.R. China

    Jianbo Li

  2. School of Sciences, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, P.R. China

    Zhensheng Huang

  3. Division of Mathematical Sciences, SPMS, Nanyang Technological University, Singapore, 637371, Singapore

    Heng Lian

Authors
  1. Jianbo Li
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  2. Zhensheng Huang
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  3. Heng Lian
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Corresponding author

Correspondence to Heng Lian.

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Appendix

Appendix

In this paper, we assume that the cumulative density function of failure time T, F 0(t)=1−S 0(t), is continuous and F 0(t)=0 for t<0. The following regularity conditions will be used in the proofs of Theorems 1–2.

(A0) \(\lim_{n\rightarrow\infty}\frac{k_{n}}{n}=c>0\). Censoring time C and failure time T are conditionally independent given Z. Moreover, P(T<C)>0 and P(|T−S|<ε)>0 for any fixed S>0 and any small ε>0.

(A1) Suppose that β∈Θ, a compact subset of the Euclidean space R p and the true value β 0 is an interior point of Θ. The covariate vector Z is assumed to be bounded, that is, there exists a constant M 1>0 such that P(∥Z∥≤M 1)=1. For all u∈(0,1), ∥v∥≤M 1 and w∈Θ,

$$\begin{aligned} &\phi_3(u,v,w)=\frac{\partial\phi(u,v,w)}{\partial w},\\ &\phi _{33}(u,v,w)= \frac{\partial^2\phi(u,v,w)}{\partial w\partial w'} \end{aligned}$$

exist and are continuous with respect to w∈Θ. The condition also holds with ϕ(u,v,w) replaced by Φ(u,v,w). We use ∥⋅∥ for both the Euclidean norm of a vector and the operator norm of a matrix.

(A2) For any v satisfying ∥v∥≤M 1, there are functions F 1(u,v) and F 2(u,v), integrable with respect to u over (0,1) such that

$$\begin{aligned} &\bigl\Vert \phi_3(u,v,w)\bigr\Vert <F_1(u,v),\qquad \bigl\Vert \phi_{33}(u,v,w)\bigr\Vert <F_2(u,v),\\ &\quad\text{for all}\ w\in\varTheta. \end{aligned}$$

This condition also holds when ϕ(u,v,w) is replaced by Φ(u,v,w).

(A3) Denote \(\varPsi(u,v,w)=\frac{\varPhi_{3}(u,v,w)}{\varPhi(u,v,w)}\), \(\varPhi_{3}(u,v,w)= \frac{\partial\varPhi(u,v,w)}{\partial w}\), U=F 0(T). For any \(\boldsymbol {\beta}\in\mathcal{O}_{\boldsymbol {\beta}_{0}}\), an neighborhood of β 0 in Θ, E\(_{\boldsymbol {\beta}_{0}}[\psi (1-U,Z,\boldsymbol {\beta})]\), E\(_{\boldsymbol {\beta}_{0}}[\psi^{2}(1-U,Z,\boldsymbol {\beta})]\) and E\(_{\boldsymbol {\beta}_{0}}[-\frac{\partial\psi(1-U,Z,\boldsymbol {\beta})}{\partial \boldsymbol {\beta}'}]\) exist. These expectations also exist with ψ(u,v,w) replaced by Ψ(u,v,w) and the matrix

$$\begin{aligned} V(\boldsymbol {\beta})&=\operatorname{Var}_{\boldsymbol {\beta}_0}\bigl[\psi \bigl(1-F_0(Y),Z,\boldsymbol {\beta}\bigr)\delta \\ &\quad {}+\varPsi \bigl(1-F_0(Y),Z, \boldsymbol {\beta}\bigr) (1-\delta)\bigr] \end{aligned}$$
(14)

is positive definite at β=β 0. The subscript β 0 means that the expectation or variance is evaluated at the true parameter value.

(A4) \(\text{E}_{\boldsymbol {\beta}_{0}}[\varPsi^{2}(1-F_{0}(C),Z,\boldsymbol {\beta}_{0})]\) exists and there exists a constant M 2>0 such that

$$\begin{aligned} &\max \bigl\{\text{E}_{\boldsymbol {\beta}_0}\bigl[\psi^2(1-U,Z,\boldsymbol {\beta}_0)\bigr],\text{E}_{\boldsymbol {\beta}_0} \bigl[\varPsi^2(1-U,Z, \boldsymbol {\beta}_0) \bigr],\\ &\quad \text{E}_{\boldsymbol {\beta} _0}\bigl[\varPsi^2 \bigl(1-F_0(C),Z,\boldsymbol {\beta}_0\bigr)\bigr] \bigr \}<M_2. \end{aligned}$$

(A5) The function ψ(u,v,w) is continuous for u∈(0,1) and satisfies the Lipschitz condition with respect to u in any closed subset of (0,1). More specifically, for any [L,R]⊂(0,1), there exists a constant M 3(L,R), dependent on [L,R], such that for all u 1,u 2∈[L,R], |v|≤M 1 and \(w\in\mathcal{O}_{\boldsymbol {\beta}_{0}}\),

$$\bigl|\psi(u_1,v,w)-\psi(u_2,v,w)\bigr|\leq M_3|u_1-u_2|. $$

This condition also holds when ψ(u,v,w) is replaced by Ψ(u,v,w).

(A6) The function \(\frac{\partial\psi(u,v,w)}{\partial w'}\) satisfies the Lipschitz condition with respect to u as in condition (A5) and this condition holds when ψ(u,v,w) is replaced by Ψ(u,v,w).

(A7) For any fixed discretization of failure time support (0,∞) (the discretization technique can be found in Sect. 2.3 of Wu (2008)) and later in the proof of this paper, and the number of discretization points m is fixed), there exists N such that when n>N,

$$ \text{E}_{\boldsymbol {\beta}_0} \biggl[-\frac{\partial}{\partial \boldsymbol {\beta}'}S_n( \boldsymbol {\beta}) \biggr]\geq\text{E}_{\boldsymbol {\beta}_0} \biggl[-\frac{\partial}{\partial \boldsymbol {\beta}'}S_{n,m}( \boldsymbol {\beta}) \biggr] $$
(15)

where S n,m (β) is a discretized version of S n (β) given the fixed discretization.

Conditions (A1) and (A3) are smoothness conditions for model (1) while (A2) implies that the order for differentiation and integration can be changed. (A0) and (A4)–(A7) are used to prove the asymptotic normality of the score function S n (β 0). The boundedness of covariates in condition (A2) is often used to simplify the proof. The inequality (15) is general true, otherwise, use of discretized data is better than use of original data.

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Li, J., Huang, Z. & Lian, H. Empirical likelihood inference for general transformation models with right censored data. Stat Comput 24, 985–995 (2014). https://doi.org/10.1007/s11222-013-9415-3

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