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This paper considers regression models with censored data where the dependent variable T and the censoring variable C are both assumed to follow a multi-index structure with the covariates. An iterative and structure-adaptive procedure is proposed to estimate the sufficient dimension reduction (SDR) spaces for T and C , as well as their joint SDR space, simultaneously. A cross-validation procedure is used to determine the structural dimensions of the individual SDR spaces. Simulation study shows that in terms of estimation efficiency, the proposed method is comparable to parametric models such as the Cox proportional hazards model when the latter is supposed to benefit from correct model specification, and outperforms the latter otherwise. When applied to the popular primary biliary cirrhosis data, the new approach is able to identify an important predictor for the patients’ survival time, which has long been noted by clinicians as a critical indicator but has so far not been picked up by existing statistical analysis.

For censored data, it is very common for the tail of the survival function to be non-identifiable because of the abundance of censored observations in the tail. This is especially prominent in censored regression quantile analysis, and introduces a serious problem with inference, especially near the point of non-identifiability. The lack of readily estimable formulas for asymptotic variances requires the use of resampling methods. Unfortunately, the bootstrap (in any of its versions) generates samples for which the point of non-identifiability has sufficient variability over the pseudo-samples so that (in theory and in practice) an appreciable number of the bootstrap replicates can no longer estimate a quantile that the original data could estimate. This leads to very poor coverage probabilities. Thus, resampling methods that provide less variability over the pseudo-samples may be very helpful. The jackknife is one such method, though it is necessary to use a ''delete-d'' jackknife with d of order larger than n. Another alternative is to use randomly reweighted ''bootstrap'' samples with weights of the form 1+v, with v of order 1/n. These approaches can be justified asymptotically. Furthermore, a simulation experiment shows that randomly sampling a relatively modest number of delete-(2n) jackknifed samples provides quite excellent coverage probabilities, substantially outperforming Bootstrap methods near the point of non-identifiability. This provides a counterexample to the commonly held notion that bootstrap methods are better than jackknife methods, and suggests the possible superiority of jackknife methods for a variety of situations with missing data or other cases of partial identifiability.

The log-ratio methodology represents a powerful set of methods and techniques for statistical analysis of compositional data. These techniques may be used for the estimation of rounded zeros or values below the detection limit in cases when the underlying data are compositional in nature. An algorithm based on iterative log-ratio regressions is developed by combining a particular family of isometric log-ratio transformations with censored regression. In the context of classical regression methods, the equivalence of the method based on additive and isometric log-ratio transformations is proved. This equivalence does not hold for robust regression. Based on Monte Carlo methods, simulations are performed to assess the performance of classical and robust methods. To illustrate the method, a case study involving geochemical data is conducted.

Recently, least absolute deviation (LAD) estimator for median regression models with doubly censored data was proposed and the asymptotic normality of the estimator was established, and the methods based on bootstrap and random weighting were proposed respectively to approximate the distribution of the LAD estimators. But the calculation of the estimators requires solving a non-convex and non-smooth minimization problem, resulting in high computational costs in implementing the bootstrap or random weighting method directly. In this paper, computationally simple resampling methods are proposed to approximate the distribution of the doubly censored LAD estimators. The objective functions in the resampling stage of the new methods are piece-wise linear and convex, and their minimizer can be obtained by the linear programming in the same way as that for the case of uncensored median regression.

Tobit models are extended to allow threshold values which depend on individuals' characteristics. In such models, the parameters are subject to as many inequality constraints as the number of observations, and the maximum likelihood estimation which requires the numerical maximisation of the likelihood is often difficult to be implemented. Using a Bayesian approach, a Gibbs sampler algorithm is proposed and, further, the convergence to the posterior distribution is accelerated by introducing an additional scale transformation step. The procedure is illustrated using the simulated data, wage data and prime rate changes' data.

Additive models in censored regression are considered. A randomly weighted version of the backfitting algorithm that allows for the nonparametric estimation of the effects of the covariates on the response is provided. Given the high computational cost involved, binning techniques are used to speed up the computation in the estimation and testing process. Simulation results and the application to real data reveal that the predictor obtained with the additive model performs well, and that it is a convenient alternative to the linear predictor when some nonlinear effects are expected.

In this paper, a test statistic is constructed to test polynomial relationships in randomly right censored regression models based on the local polynomial smoothing technique. Two bootstrap procedures, namely the residual-based bootstrap and the naive bootstrap procedures, are suggested to derive the p-value of the test. Some simulations are conducted to empirically assess the performance of the two bootstrap procedures. The results demonstrate that the residual-based bootstrap performs much better than the naive bootstrap and the test method with the residual-based bootstrap to derive the p-value works satisfactorily. Although the limiting distribution of the test statistic and the consistency of the bootstrap approximations remain to be investigated, simulation results indicate that the proposed test method may be of some practical use. As a real example, the proposed test is applied to the Stanford heart transplant data.

For the censored regression model, Yang (J. Amer. Statist. Assoc. 92 (1997) 977-984) introduced a new class of estimating functions. These estimating functions produce regression estimators that are asymptotically normal with a density-free asymptotic variance that is simple to estimate reliably. In this paper we further study the estimation function of Yang by considering new classes of weights. Through extensive numerical studies, we find weights that enhance the results of the estimating function and improve upon choices previously recommended by Yang.