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Register a new userA quantile regression estimator for censored data
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Added by
Chenlei Leng
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
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We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.
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Various events in the nature, economics and in other areas force us to combine the study of extremes with regression and other methods. A useful tool for reducing the role of nuisance regression, while we are interested in the shape or tails of the basic distribution, is provided by the averaged regression quantile and namely by the average extreme regression quantile. Both are weighted means of regression quantile components, with weights depending on the regressors. Our primary interest is the averaged extreme regression quantile (AERQ), its structure, qualities and its applications, e.g. in investigation of a conditional loss given a value exogenous economic and market variables. AERQ has several interesting equivalent forms: While it is originally defined as an optimal solution of a specific linear programming problem, hence is a weighted mean of responses corresponding to the optimal base of the pertaining linear program, we give another equivalent form as a maximum residual of responses from a specific R-estimator of the slope components of regression parameter. The latter form shows that while AERQ equals to the maximum of some residuals of the responses, it has minimal possible perturbation by the regressors. Notice that these finite-sample results are true even for non-identically distributed model errors, e.g. under heteroscedasticity. Moreover, the representations are formally true even when the errors are dependent - this all provokes a question of the right interpretation and of other possible applications.
Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases. Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models. Regression adjustment is based on a new estimating equation that adapts to censoring and leads to quantile score whenever the data do not exhibit censoring. The proposed procedure named {it censored quantile regression forest}, allows us to estimate quantiles of time-to-event without any parametric modeling assumption. We establish its consistency under mild model specifications. Numerical studies showcase a clear advantage of the proposed procedure.
Naive Bayes classifiers have proven to be useful in many prediction problems with complete training data. Here we consider the situation where a naive Bayes classifier is trained with data where the response is right censored. Such prediction problems are for instance encountered in profiling systems used at National Employment Agencies. In this paper we propose the maximum collective conditional likelihood estimator for the prediction and show that it is strongly consistent under the usual identifiability condition.
With the availability of high dimensional genetic biomarkers, it is of interest to identify heterogeneous effects of these predictors on patients survival, along with proper statistical inference. Censored quantile regression has emerged as a powerful tool for detecting heterogeneous effects of covariates on survival outcomes. To our knowledge, there is little work available to draw inference on the effects of high dimensional predictors for censored quantile regression. This paper proposes a novel procedure to draw inference on all predictors within the framework of global censored quantile regression, which investigates covariate-response associations over an interval of quantile levels, instead of a few discrete values. The proposed estimator combines a sequence of low dimensional model estimates that are based on multi-sample splittings and variable selection. We show that, under some regularity conditions, the estimator is consistent and asymptotically follows a Gaussian process indexed by the quantile level. Simulation studies indicate that our procedure can properly quantify the uncertainty of the estimates in high dimensional settings. We apply our method to analyze the heterogeneous effects of SNPs residing in lung cancer pathways on patients survival, using the Boston Lung Cancer Survival Cohort, a cancer epidemiology study on the molecular mechanism of lung cancer.
The processes of the averaged regression quantiles and of their modifications provide useful tools in the regression models when the covariates are not fully under our control. As an application we mention the probabilistic risk assessment in the situation when the return depends on some exogenous variables. The processes enable to evaluate the expected $alpha$-shortfall ($0leqalphaleq 1$) and other measures of the risk, recently generally accepted in the financial literature, but also help to measure the risk in environment analysis and elsewhere.
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