Censored depth quantiles

Quantile regression is a wide spread regression technique which allows to model the entire conditional distribution of the response variable. A natural extension to the case of censored observations has been introduced using a reweighting scheme based on the Kaplan-Meier estimator. The same ideas can be applied to depth quantiles. This leads to regression quantiles for censored data which are robust to both outliers in the predictor and the response variable. For their computation, a fast algorithm over a grid of quantile values is proposed. The robustness of the method is shown in a simulation study and on two real data examples.     

Quantile regression methods with varying-coefficient models for censored data

Considerable intellectual progress has been made to the development of various semiparametric varying-coefficient models over the past ten to fifteen years. An important advantage of these models is that they avoid much of the curse of dimensionality problem as the nonparametric functions are restricted only to some variables. More recently, varying-coefficient methods have been applied to quantile regression modeling, but all previous studies assume that the data are fully observed. The main purpose of this paper is to develop a varying-coefficient approach to the estimation of regression quantiles under random data censoring. We use a weighted inverse probability approach to account for censoring, and propose a majorize–minimize type algorithm to optimize the non-smooth objective function. The asymptotic properties of the proposed estimator of the nonparametric functions are studied, and a resampling method is developed for obtaining the estimator of the sampling variance. An important aspect of our method is that it allows the censoring time to depend on the covariates. Additionally, we show that this varying-coefficient procedure can be further improved when implemented within a composite quantile regression framework. Composite quantile regression has recently gained considerable attention due to its ability to combine information across different quantile functions. We assess the finite sample properties of the proposed procedures in simulated studies. A real data application is also considered.     

Design-based estimation for geometric quantiles with application to outlier detection

Geometric quantiles are investigated using data collected from a complex survey. Geometric quantiles are an extension of univariate quantiles in a multivariate set-up that uses the geometry of multivariate data clouds. A very important application of geometric quantiles is the detection of outliers in multivariate data by means of quantile contours. A design-based estimator of geometric quantiles is constructed and used to compute quantile contours in order to detect outliers in both multivariate data and survey sampling set-ups. An algorithm for computing geometric quantile estimates is also developed. Under broad assumptions, the asymptotic variance of the quantile estimator is derived and a consistent variance estimator is proposed. Theoretical results are illustrated with simulated and real data.     

A quantile estimator under two-phase sampling for stratification

Recently, the estimation of a population quantile has received quite attention. Existing quantile estimators generally assume that values of an auxiliary variable are known for the entire population, and most of them are defined under simple random sampling without replacement. Assuming two-phase sampling for stratification with arbitrary sampling designs in each of the two phases, a new quantile estimator and its variance estimator are defined. The proposed estimators can be used when the population auxiliary information is not available, which is a common situation in practice. Desirable properties such as the unbiasedness are derived. Suggested estimators are compared numerically with an alternative stratification estimator and its variance estimator, and desirable results are observed. Confidence intervals based upon the proposed estimators are also defined, and they are compared via simulation studies with the confidence intervals based upon the stratification estimator. The proposed confidence intervals give desirable coverage probabilities with the smallest interval lengths.     

Support vector censored quantile regression under random censoring

Censored quantile regression models have received a great deal of attention in both the theoretical and applied statistical literature. In this paper, we propose support vector censored quantile regression (SVCQR) under random censoring using iterative reweighted least squares (IRWLS) procedure based on the Newton method instead of usual quadratic programming algorithms. This procedure makes it possible to derive the generalized approximate cross validation (GACV) method for choosing the hyperparameters which affect the performance of SVCQR. Numerical results are then presented which illustrate the performance of SVCQR using the IRWLS procedure.     

Quantile regression with doubly censored data

Quantile regression offers a semiparametric approach to modeling data with possible heterogeneity. It is particularly attractive for censored responses, where the conditional mean functions are unidentifiable without parametric assumptions on the distributions. A new algorithm is proposed to estimate the regression quantile process when the response variable is subject to double censoring. The algorithm distributes the probability mass of each censored point to its left or right appropriately, and iterates towards self-consistent solutions. Numerical results on simulated data and an unemployment duration study are given to demonstrate the merits of the proposed method.     

Estimating quantiles under sampling on two occasions with arbitrary sample designs

A practical problem related to the estimation of quantiles in double sampling with arbitrary sampling designs in each of the two phases is investigated. In practice, this scheme is commonly used for official surveys, in which quantile estimation is often required when the investigation deals with variables such as income or expenditure. A class of estimators for quantiles is proposed and some important properties, such as asymptotic unbiasedness and asymptotic variance, are established. The optimal estimator, in the sense of minimizing the asymptotic variance, is also presented. The proposed class contains several known types of estimators, such as ratio and regression estimators, which are of practical application and are therefore derived. Assuming several populations, the proposed estimators are compared with the direct estimator via an empirical study. Results show that a gain in efficiency can be obtained.     

Estimating quantiles under sampling on two occasions with arbitrary sample designs

A practical problem related to the estimation of quantiles in double sampling with arbitrary sampling designs in each of the two phases is investigated. In practice, this scheme is commonly used for official surveys, in which quantile estimation is often required when the investigation deals with variables such as income or expenditure. A class of estimators for quantiles is proposed and some important properties, such as asymptotic unbiasedness and asymptotic variance, are established. The optimal estimator, in the sense of minimizing the asymptotic variance, is also presented. The proposed class contains several known types of estimators, such as ratio and regression estimators, which are of practical application and are therefore derived. Assuming several populations, the proposed estimators are compared with the direct estimator via an empirical study. Results show that a gain in efficiency can be obtained.     

Simple resampling methods of approximating the distribution of LAD estimators for doubly censored regression models

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.     

A weighted quantile regression for randomly truncated data

Quantile regression offers great flexibility in assessing covariate effects on the response. In this article, based on the weights proposed by He and Yang (2003), we develop a new quantile regression approach for left truncated data. Our method leads to a simple algorithm that can be conveniently implemented with R software. It is shown that the proposed estimator is strongly consistent and asymptotically normal under appropriate conditions. We evaluate the finite sample performance of the proposed estimators through extensive simulation studies.     

Model based bootstrap methods for interval censored data

The performance of model based bootstrap methods for constructing point-wise confidence intervals around the survival function with interval censored data is investigated. It is shown that bootstrapping from the nonparametric maximum likelihood estimator of the survival function is inconsistent for the current status model. A model based smoothed bootstrap procedure is proposed and proved to be consistent. In fact, a general framework for proving the consistency of any model based bootstrap scheme in the current status model is established. In addition, simulation studies are conducted to illustrate the (in)-consistency of different bootstrap methods in mixed case interval censoring. The conclusions in the interval censoring model would extend more generally to estimators in regression models that exhibit non-standard rates of convergence.     

Median Regression Tree for Analysis of Censored Survival Data

Median linear regression modeling is a natural approach for analyzing censored survival or failure time data. A median linear model lends itself to a simple interpretation that is particularly suitable for making direct predictions of survival or failure times. We propose a new, unique, and efficient algorithm for tree-structured median regression modeling that combines the merits of both a median regression model and a tree-structured model. We propose and discuss loss functions for constructing this tree-structured median model and investigate their effects on the determination of tree size. We also propose a split covariate selection algorithm by using residual analysis (RA) rather than loss function reduction. The RA approach allows for the selection of the correct split covariate fairly well, regardless of the distribution of covariates. The loss function with the transformed data performs well in comparison to that with raw or uncensored data in determining the right tree size. Unlike other survival trees, the proposed median regression tree is useful in directly predicting survival or failure times for partitioned homogeneous patient groups as well as in revealing and interpreting complex covariate structures. Furthermore, a median regression tree can be easily generalized to a quantile regression tree with a user-chosen quantile between 0 and 1. We have demonstrated the proposed method with two real data sets and have compared the results with existing regression trees.     

Conditional mean estimation under asymmetric and heteroscedastic error by linear combination of quantile regressions

In this paper we propose a new estimator for regression problems in the form of the linear combination of quantile regressions. The proposed estimator is helpful for the conditional mean estimation when the error distribution is asymmetric and heteroscedastic.It is shown that the proposed estimator has the consistency under heteroscedastic regression model: Y=μ(X)+σ(X)·e, where X is a vector of covariates, Y is a scalar response, e is a zero mean random variable independent of X and σ(X) is a positive value function. When the error term e is asymmetric, we show that the proposed estimator yields better conditional mean estimation performance than the other estimators. Numerical experiments both in synthetic and real data are shown to illustrate the usefulness of the proposed estimator.     

Quantile regression neural networks: Implementation in R and application to precipitation downscaling

The qrnn package for R implements the quantile regression neural network, which is an artificial neural network extension of linear quantile regression. The model formulation follows from previous work on the estimation of censored regression quantiles. The result is a nonparametric, nonlinear model suitable for making probabilistic predictions of mixed discrete-continuous variables like precipitation amounts, wind speeds, or pollutant concentrations, as well as continuous variables. A differentiable approximation to the quantile regression error function is adopted so that gradient-based optimization algorithms can be used to estimate model parameters. Weight penalty and bootstrap aggregation methods are used to avoid overfitting. For convenience, functions for quantile-based probability density, cumulative distribution, and inverse cumulative distribution functions are also provided. Package functions are demonstrated on a simple precipitation downscaling task.     

Transformation-based estimation

To alleviate the computational burden of making the relevant estimation algorithms stable for nonlinear and semiparametric regression models with, particularly, high-dimensional data, a transformation-based method combining sufficient dimension reduction approach is proposed. To this end, model-independent transformations are introduced to models under study. This generic methodology can be applied to transformation models; generalized linear models; and their corresponding quantile regression variants. The constructed estimates almost have closed forms in certain sense such that the above goals can be achieved. Simulation results show that, in finite sample cases with high-dimensional predictors and long-tailed distributions of error, the new estimates often exhibit a smaller degree of variance, and have much less computational burden than the classical methods such as the classical least squares and quantile regression estimation.     

Time-adaptive quantile regression

An algorithm for time-adaptive quantile regression is presented. The algorithm is based on the simplex algorithm, and the linear optimization formulation of the quantile regression problem is given. The observations have been split to allow a direct use of the simplex algorithm. The simplex method and an updating procedure are combined into a new algorithm for time-adaptive quantile regression, which generates new solutions on the basis of the old solution, leading to savings in computation time. The suggested algorithm is tested against a static quantile regression model on a data set with wind power production, where the models combine splines and quantile regression. The comparison indicates superior performance for the time-adaptive quantile regression in all the performance parameters considered.¹     

Implementing lifetime performance index of products with two-parameter exponential distribution

The manufacturing industry has prioritised enhancing the quality, lifetime and conforming rate of products. Process capability indices (PCIs) are used to measure process potential and performance. The process capability is evaluated with product survival time and a longer lifetime implies a better process capability and a higher reliability. In order to save experimental time and cost, a censored sample arises in practice. In the case of product possessing a two-parameter exponential distribution, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of the lifetime performance index based on the type II right-censored sample. Then the UMVUE of the lifetime performance index is utilised to develop the new hypothesis testing procedure in the condition of known lower specification limit. Finally, two practical examples are illustrated to employ the testing procedure to determine whether the product is reliable.     

Robust accelerated failure time regression

Robust estimators for accelerated failure time models with asymmetric (or symmetric) error distribution and censored observations are proposed. It is assumed that the error model belongs to a log-location-scale family of distributions and that the mean response is the parameter of interest. Since scale is a main component of mean, scale is not treated as a nuisance parameter. A three steps procedure is proposed. In the first step, an initial high breakdown point S estimate is computed. In the second step, observations that are unlikely under the estimated model are rejected or down weighted. Finally, a weighted maximum likelihood estimate is computed. To define the estimates, functions of censored residuals are replaced by their estimated conditional expectation given that the response is larger than the observed censored value. The rejection rule in the second step is based on an adaptive cut-off that, asymptotically, does not reject any observation when the data are generated according to the model. Therefore, the final estimate attains full efficiency at the model, with respect to the maximum likelihood estimate, while maintaining the breakdown point of the initial estimator. Asymptotic results are provided. The new procedure is evaluated with the help of Monte Carlo simulations. Two examples with real data are discussed.     

Log-Burr XII regression models with censored data

In survival analysis applications, the failure rate function may frequently present a unimodal shape. In such case, the log-normal or log-logistic distributions are used. In this paper, we shall be concerned only with parametric forms, so a location-scale regression model based on the Burr XII distribution is proposed for modeling data with a unimodal failure rate function as an alternative to the log-logistic regression model. Assuming censored data, we consider a classic analysis, a Bayesian analysis and a jackknife estimator for the parameters of the proposed model. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the log-logistic and log-Burr XII regression models. Besides, we use sensitivity analysis to detect influential or outlying observations, and residual analysis is used to check the assumptions in the model. Finally, we analyze a real data set under log-Burr XII regression models.     

Asymptotically efficient estimation of the conditional expected shortfall

A procedure for efficient estimation of the trimmed mean of a random variable conditional on a set of covariates is proposed. For concreteness, the focus is on a financial application where the trimmed mean of interest corresponds to the conditional expected shortfall, which is known to be a coherent risk measure. The proposed class of estimators is based on representing the estimator as an integral of the conditional quantile function. Relative to the simple analog estimator that weights all conditional quantiles equally, asymptotic efficiency gains may be attained by giving different weights to the different conditional quantiles while penalizing excessive departures from uniform weighting. The approach presented here allows for either parametric or nonparametric modeling of the conditional quantiles and the weights, but is essentially nonparametric in spirit. The asymptotic properties of the proposed class of estimators are established. Their finite sample properties are illustrated through a set of Monte Carlo experiments and an empirical application¹.