Single-index regression models with right-censored responses

In this article, we propose some new generalizations of M-estimation procedures for single-index regression models in presence of randomly right-censored responses. We derive consistency and asymptotic normality of our estimates. The results are proved in order to be adapted to a wide range of techniques used in a censored regression framework (e.g. synthetic data or weighted least squares). As in the uncensored case, the estimator of the single-index parameter is seen to have the same asymptotic behavior as in a fully parametric scheme. We compare these new estimators with those based on the average derivative technique of Lu and Burke [2005. Censored multiple regression by the method of average derivatives. J. Multivariate Anal. 95, 182–205] through a simulation study.     

A jackknife-based versatile test for two-sample problems with right-censored data

For testing the equality of two survival functions, the weighted logrank test and the weighted Kaplan–Meier test are the two most widely used methods. Actually, each of these tests has advantages and defects against various alternatives, while we cannot specify in advance the possible types of the survival differences. Hence, how to choose a single test or combine a number of competitive tests for indicating the diversities of two survival functions without suffering a substantial loss in power is an important issue. Instead of directly using a particular test which generally performs well in some situations and poorly in others, we further consider a class of tests indexed by a weighted parameter for testing the equality of two survival functions in this paper. A delete-1 jackknife method is implemented for selecting weights such that the variance of the test is minimized. Some numerical experiments are performed under various alternatives for illustrating the superiority of the proposed method. Finally, the proposed testing procedure is applied to two real-data examples as well.     

Power and sample size calculation for paired right-censored data based on survival copula models

Sample size determination is essential during the planning phases of clinical trials. To calculate the required sample size for paired right-censored data, the structure of the within-paired correlations needs to be pre-specified. In this article, we consider using popular parametric copula models, including the Clayton, Gumbel, or Frank families, to model the distribution of joint survival times. Under each copula model, we derive a sample size formula based on the testing framework for rank-based tests and non-rank-based tests (i.e., logrank test and Kaplan–Meier statistic, respectively). We also investigate how the power or the sample size was affected by the choice of testing methods and copula model under different alternative hypotheses. In addition to this, we examine the impacts of paired-correlations, accrual times, follow-up times, and the loss to follow-up rates on sample size estimation. Finally, two real-world studies are used to illustrate our method and R code is available to the user.     

Small sample study on estimators of life distributions and of mean survival times using randomly censored data

Whereas large-sample properties of the estimators of survival distributions using censored data have been studied by many authors, exact results for small samples have been difficult to obtain. In this paper we obtain the exact expression for the ath moment (a > 0) of the Bayes estimator of survival distribution using the censored data under proportional hazard model. Using the exact expression we compute the exact mean, variance and MSE of the Bayes estimator. Also two estimators ofthe mean survival time based on the Kaplan-Meier estimator and the Bayes estimator are compared for small samples under proportional hazards.     

Testing Censoring Point Independence

Identification in censored regression analysis and hazard models of duration outcomes relies on the condition that censoring points are conditionally independent of latent outcomes, an assumption which may be questionable in many settings. This article proposes a test for this assumption based on a Cramer–von-Mises-like test statistic comparing two different nonparametric estimators for the latent outcome cdf: the Kaplan–Meier estimator, and the empirical cdf conditional on the censoring point exceeding (for right-censored data) the cdf evaluation point. The test is consistent and has power against a wide variety of alternatives. Applying the test to unemployment duration data from the NLSY, the SIPP, and the PSID suggests the assumption is frequently suspect.     

ON THE ESTIMATION OF SURVIVAL FUNCTION UNDER RANDOM CENSORSHIP

ABSTRACT

The paper deals with an improvement of the well-known Kaplan–Meier estimator of survival function when the censoring mechanism is random and independent of the failure times. Small sample size properties of the new estimator, as well as the original Kaplan–Meier estimator are inspected by means of Monte Carlo simulations. It follows from the simulations that the proposed estimator prevails with respect to some basic statistical characteristics.     

Nonparametric regression estimates with censored data based on block thresholding method

Here we consider wavelet-based identification and estimation of a censored nonparametric regression model via block thresholding methods and investigate their asymptotic convergence rates. We show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes, and in particular enjoy those rates without the extraneous logarithmic penalties that are usually suffered by term-by-term thresholding methods. This work is extension of results in Li et al. (2008). The performance of proposed estimator is investigated by a numerical study.     

Function-on-function regression for two-dimensional functional data

ABSTRACT

We present methods for modeling and estimation of a concurrent functional regression when the predictors and responses are two-dimensional functional datasets. The implementations use spline basis functions and model fitting is based on smoothing penalties and mixed model estimation. The proposed methods are implemented in available statistical software, allow the construction of confidence intervals for the bivariate model parameters, and can be applied to completely or sparsely sampled responses. Methods are tested to data in simulations and they show favorable results in practice. The usefulness of the methods is illustrated in an application to environmental data.     

The new robust two-sample test for randomly right-censored data

Using a minimum p-value principle, a new two-sample test MIN3 is proposed in the paper. The cumulative distribution function of the MIN3 test statistic is studied and approximated by the Beta distribution of the third kind. Lower percentage points of the distribution of the new test statistic under the null hypothesis are computed. Also the test power for a lot of types of alternative hypotheses (with 0, 1 and 2 point(-s) of the intersection(-s) of survival functions) is studied and we found that the usage of the MIN3 test is a preferred strategy by the Wald and Savage decision-making criteria under risk and uncertainty. The results of application of the MIN3 test are shown for two examples from lifetime data analysis.     

On the goodness-of-fit test based on the ratio of cumulative hazard functions with the Type II censored data

For comparing two cumulative hazard functions, we consider an extension of the Kullback–Leibler information to the cumulative hazard function, which is concerning the ratio of cumulative hazard functions. Then we consider its estimate as a goodness-of-fit test with the Type II censored data. For an exponential null distribution, the proposed test statistic is shown to outperform other test statistics based on the empirical distribution function in the heavy censoring case against the increasing hazard alternatives.     

A large-scale monte carlo study of the buckley-james estimator with censored data

Estimation in the presence of censoring is an important problem. In the linear model, the Buckley-James method proceeds iteratively by estimating the censored values than re-estimating the regression coeffi- cients. A large-scale Monte Carlo simulation technique has been developed to test the performance of the Buckley-James (denoted B-J) estimator. One hundred and seventy two randomly generated data sets, each with three thousand replications, based on four failure distributions, four censoring patterns, three sample sizes and four censoring rates have been investigated, and the results are presented. It is found that, except for Type I1 censoring, the B-J estimator is essentially unbiased, even when the data sets with small sample sizes are subjected to a high censoring rate. The variance formula suggested by Buckley and James (1979) is shown to be sensitive to the failure distribution. If the censoring rate is kept constant along the covariate line, the sample variance of the estimator appears to be insensitive to the censoring pattern with a selected failure distribution. Oscillation of the convergence values associated with the B-J estimator is illustrated and thoroughly discussed.     

A Bezier curve method in interval-censored data

In this paper we propose a Bezier curve method to estimate the survival function and the median survival time in interval-censored data. We compare the proposed estimator with other existing methods such as the parametric method, the single point imputation method, and the nonparametric maximum likelihood estimator through extensive numerical studies, and it is shown that the proposed estimator performs better than others in the sense of mean squared error and mean integrated squared error. An illustrative example based on a real data set is given.     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

Central Limit Theorem for ISE of Kernel Density Estimators in Censored Dependent Model

In some long-term studies, a series of dependent and possibly censored failure times may be observed. Suppose that the failure times have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the failure times and its kernel estimate f _n is the integrated square error(ISE). In this article, we derive a central limit theorem for the integrated square error of the kernel density estimators under a censored dependent model.     

A Nonparametric Test Procedure Based on Quantiles for the Right Censored Data

In this study, we propose nonparametric tests using the several quantile statistics simultaneously for the right censored data. First of all, we consider statistics of the quadratic form with estimated covariance matrices. Then we derive the limiting distribution using the large sample approximation theory. Also we consider different forms of statistics such as the maximal and summing types with their limiting distributions. Then we illustrate our procedure with examples and compare performance among tests with empirical powers through a simulation study. Also we comment briefly on some interesting features including re-sampling methods as concluding remarks. Finally in Appendices, we provide proofs for the theoretic results needed for the derivation of the limiting distributions of the proposed test statistics.     

Strong Gaussian Approximations of Product-Limit and Quantile Processes for Strong Mixing and Censored Data

In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)^?λ] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.     

A flexible semiparametric regression model for bimodal,asymmetric and censored data

In this paper, we propose a new semiparametric heteroscedastic regression model allowing for positive and negative skewness and bimodal shapes using the B-spline basis for nonlinear effects. The proposed distribution is based on the generalized additive models for location, scale and shape framework in order to model any or all parameters of the distribution using parametric linear and/or nonparametric smooth functions of explanatory variables. We motivate the new model by means of Monte Carlo simulations, thus ignoring the skewness and bimodality of the random errors in semiparametric regression models, which may introduce biases on the parameter estimates and/or on the estimation of the associated variability measures. An iterative estimation process and some diagnostic methods are investigated. Applications to two real data sets are presented and the method is compared to the usual regression methods.