Nonparametric tests for multiple regression under progressive censoring

For continuous observations from time-sequential studies, suitable Cramér-von Mises and Kolmogorov-Smirnov types of (nonparametric) statistics (based on linear rank statistics) for testing hypotheses on some multiple-regression models are proposed and studied. The asymptotic theory of these tests is provided for both the null and (local) alternative hypotheses situations and is based on the weak convergence of suitable rank order processes (on the D[0, 1] space) to certain functions of Brownian motions. Bahadur efficiency results are also presented. Empirical values of the percentile points of the null distributions of the proposed test statistics, obtained through simulation studies, are also provided.     

Statistical inference under adaptive progressive censoring scheme

In this paper, a general exponential form of the underlying distribution and a general conjugate prior are used to discuss the maximum likelihood and Bayesian estimation based on an adaptive progressive censored sample. A general procedure for deriving the point and interval Bayesian prediction of the future progressive censored from the same sample as well as that from an unobserved future sample is also developed. The Weibull, Pareto, and Burr Type-XII distributions are then used as illustrative examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.     

On the quantile process under progressive censoring

This work deals with an asymptotic almost-sure representation of the quantile process under type-II progressive censoring. A convergence rate of the law-of-the-iterated-logarithm type is obtained for this representation. To cite this article: S. Alvarez-Andrade, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Statistical Inference for the Rayleigh distribution under progressively Type-II censoring with binomial removal

This paper takes into account the estimation for the unknown parameter of the Rayleigh distribution under Type II progressive censoring with binomial removals, where the number of units removed at each failure time follows a binomial distribution. Maximum likelihood and Bayes procedure are used to derive both point and interval estimates of the parameters involved in the model. The expected termination point to complete the censoring test is computed and analyzed under binomial censoring scheme. Numerical examples are given to illustrate the approach by means of Monte Carlo simulation. A real life data set is used for illustrative purposes in conclusion.     

Inference for a progressive stress model from Weibull distribution under progressive type-II censoring

Based on progressively type-II censored samples, this paper considers progressive stress accelerated life tests when the lifetime of an item under use condition follows the Weibull distribution with a scale parameter satisfying the inverse power law. It is assumed that the progressive stress is directly proportional to time and the cumulative exposure model for the effect of changing stress holds. Point estimation of the model parameters is obtained graphically by using Weibull probability paper plot that serves as a tool for model identification and also by using the maximum likelihood method. Interval estimation is performed by finding approximate confidence intervals (CIs) for the parameters as well as the studentized-t and percentile bootstrap CIs. Monte Carlo simulation study is carried out to investigate the precision of the estimates and compare the performance of CIs obtained. Finally, two examples are presented to illustrate our results.     

Confidence sets for the two-parameter Rayleigh distribution under progressive censoring

Exact confidence intervals and regions are proposed for the location and scale parameters of the Rayleigh distribution. These sets are valid for complete data, and also for standard and progressive failure censoring. Constrained optimization problems are solved to find the minimum-size confidence sets for the Rayleigh parameters with the required confidence level. The smallest-area confidence region is derived by simultaneously solving four nonlinear equations. Three numerical examples regarding remission times of leukemia patients, strength data and the titanium content in an aircraft-grade alloy, as well as a simulation study, are included for illustrative purposes. Further applications in hypothesis testing and the construction of pointwise and simultaneous confidence bands are also pointed out.     

Additive hazards regression under generalized case-cohort sampling

Case-cohort design usually requires the disease rate to be low in large cohort study,although it has been extensively used in practice.However,the disease with high rate is frequently observed in many clinical studies.Under such circumstances,it is desirable to consider a generalized case-cohort design,where only a fraction of cases are sampled.In this article,we propose the inference procedure for the additive hazards regression under the generalized case-cohort sampling.Asymptotic properties of the proposed estimators for the regression coefcients are established.To demonstrate the efectiveness of the generalized case-cohort sampling,we compare it with simple random sampling in terms of asymptotic relative efciency.Furthermore,we derive the optimal allocation of the subsamples for the proposed design.The fnite sample performance of the proposed method is evaluated through simulation studies.     

Risk analysis under progressive type II censoring with binomial claim numbers

This paper presents a model of actuarial loss events that follow a progressive censoring scheme. Loss events are modelled according to this scheme regarding the claim number and size. Claim events at random time points are assumed to happen progressively in a given period due to each of an m number of claims that occur due to hazardous events, while a fixed number of n claims are anticipated to take place in total. Distribution of the resulting total loss amount is derived, and according to its properties, some risk management issues about reserves and solvency are discussed.     

An efficient computational method for moments of order statistics under progressive censoring

Thomas and Wilson (Technometrics 14 (1972) 679) developed a computational method for calculating the single and product moments of order statistics from progressively censored samples by making use of the corresponding moments of the usual order statistics. The absence of an explicit representation for the marginal and joint density function of order statistics under progressive censoring makes their method extremely tedious. By deriving the required marginal and joint density functions in explicit form, we obtain an alternative, highly efficient, method for computing the desired moments.     

Semiparametric quantile regression with random censoring

This paper considers estimation and inference in semiparametric quantile regression models when the response variable is subject to random censoring. The paper considers both the cases of independent and dependent censoring and proposes three iterative estimators based on inverse probability weighting, where the weights are estimated from the censoring distribution using the Kaplan–Meier, a fully parametric and the conditional Kaplan–Meier estimators. The paper proposes a computationally simple resampling technique that can be used to approximate the finite sample distribution of the parametric estimator. The paper also considers inference for both the parametric and nonparametric components of the quantile regression model. Monte Carlo simulations show that the proposed estimators and test statistics have good finite sample properties. Finally, the paper contains a real data application, which illustrates the usefulness of the proposed methods.

     

Inference of constant-stress accelerated life test for a truncated distribution under progressive censoring

Accelerated life test (ALT) provides a feasible and efficient way to obtain information quickly on lifetime of products by testing them at higher-than-use operating conditions. In this paper, the lifetime of products is assumed to follow a lower truncated family of distributions, when both resilience and threshold parameters are nonconstant and affected by operating stress, inference is discussed for simple constant-stress ALT under progressive Type-II censoring. Point estimates for unknown parameters are presented based on maximum likelihood and pivotal quantities based estimation methods. Meanwhile, generalized, asymptotic and bootstrap confidence intervals for the parameters of interest are constructed as well. Simulation studies and illustrative examples are carried out to investigate the performance of the proposed methods.     

Bayesian analysis for step‐stress accelerated life testing under progressive interval censoring

A step‐stress accelerated life testing model is considered for progressive type‐I censored experiments when the tested items are not monitored continuously but inspected at prespecified time points, producing thus grouped data. The underlying lifetime distributions belong to a general scale family of distributions. The points of stress‐level change are simultaneously inspection points as well while there is the option of assigning additional inspection points in between the stress‐level change points. In a Bayesian framework, the posterior distributions of the parameters of the model are derived for characteristic choices of prior distributions, as conjugate‐like and normal priors; vague or noninformative. The developed approach is illustrated on a simulated example and on a real data set, both known from the literature. The results are compared to previous analyses; frequentist or Bayes.     

Stochastic properties of INID progressive Type-II censored order statistics

Let X_1:n≤X_2:n≤?≤X_n:n denote the order statistics of random variables X₁,X₂,…,X_n which are independent but not necessarily identically distributed (INID), and let K₁,K₂ be two integer-valued random variables, independent of {X₁,…,X_n}, such that 1≤K₁≤K₂≤n. It is shown that if K₁ has a log-concave probability function and SI(K₂|K₁) then RTI(X_K2:n|X_K1:n), and if K₂ has a log-concave probability function and SI(K₁|K₂) then LTD(X_K1:n|X_K2:n), where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI and LTD whenever 1≤i<j≤m, where are progressive Type-II censored order statistics from INID random variables {X₁,…,X_n}. Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.     

A simulation-based approach to the study of coefficient of variation of Gompertz distribution under progressive first-failure censoring

In applied statistics, the coefficient of variation is widely used. However, inference concerning the coefficient of variation of non-normal distributions are rarely reported. In this article, a simulation-based Bayesian approach is adopted to estimate the coefficient of variation (CV) under progressive first-failure censored data from Gompertz distribution. The sampling schemes such as, first-failure censoring, progressive type II censoring, type II censoring and complete sample can be obtained as special cases of the progressive first-failure censored scheme. The simulation-based approach will give us a point estimate as well as the empirical sampling distribution of CV. The joint prior density as a product of conditional gamma density and inverted gamma density for the unknown Gompertz parameters are considered. In addition, the results of maximum likelihood and parametric bootstrap techniques are also proposed. An analysis of a real life data set is presented for illustrative purposes. Results from simulation studies assessing the performance of our proposed method are included.     

Nonparametric prediction intervals for progressive Type-II censored order statistics based on k-records

In this paper, we consider the prediction problem in two-sample case and study the non-parametric predicting future progressively Type-II censored order statistics based on observed $k$ -records from the same distribution. Also, prediction intervals for progressively Type-II censored spacings are obtained based on $k$ -record spacings. It is shown that the coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Moreover, optimal prediction intervals are derived for each case. Finally, for illustrating the proposed procedure, we consider a real data set and numerical computations are given. The results of Ahmadi and Balakrishnan (Statistics 44:417–430, 2010) can be achieved as special cases of our results.     

Daniels's association measures under right censoring

This paper considers extensions of the Daniels generalized measures of association (Daniels 1944) to censored data. A number of existing tests for the detection of association in the presence of right-censoring (Gehan 1965; Efron 1967; Brown, Hollander and Korwar 1974; Weier and Basu 1980) are seen to be special cases of the Daniels coefficient under particular scoring schemes. The Daniels coefficient provides a very general framework for the construction of tests and a number of other scoring schemes, simplifying to give well known tests in the non-censored case, are considered. In this paper we focus attention on tests which are, in some sense, non-parametric with respect to both variables, i.e. the explanatory variable as well as the failure time variable, such a property, though, not necessarily implying rank invariance. Parametric tests can be seen to fit in with the same formulation although no exploration of this approach is given here. Moving away from the null hypothesis of no association, it is also possible to obtain population measures of association applicable to right censored data. These will not in general converge to their non-censored equivalents although, for a large family of cases studied, they do come very close. We illustrate these findings by some simulations.     

Empirical likelihood for median regression model with designed censoring variables

We propose a new and simple estimating equation for the parameters in median regression models with designed censoring variables, and then apply the empirical log likelihood ratio statistic to construct confidence region for the parameters. The empirical log likelihood ratio statistic is shown to have a standard chi-square distribution, which makes this method easy to implement. At the same time, another empirical log likelihood ratio statistic is proposed based on an existing estimating equation and the limiting distribution of the empirical likelihood ratio statistic is shown to be a sum of weighted chi-square distributions. We compare the performance of the empirical likelihood confidence region based on the new estimating equation, with that based on the existing estimating equation and a normal approximation method by simulation studies.     

The quantile process under random censoring

In this paper we discuss the asymptotic properties of quantile processes under random censoring. In contrast to most work in this area we prove weak convergence of an appropriately standardized quantile process under the assumption that the quantile regression model is only linear in the region, where the process is investigated. Additionally, we also discuss properties of the quantile process in sparse regression models including quantile processes obtained from the Lasso and adaptive Lasso. The results are derived by a combination of modern empirical process theory, classical martingale methods and a recent result of Kato (2009).     

Versatile estimation in censored single-index hazards regression

One attractive advantage of the presented single-index hazards regression is that it can take into account possibly time-dependent covariates. In such a model formulation, the main theme of this research is to develop a theoretically valid and practically feasible estimation procedure for the index coefficients and the induced survival function. In particular, compared with the existing pseudo-likelihood approaches, our one proposes an automatic bandwidth selection and suppresses an influence of outliers. By making an effective use of the considered versatile survival process, we further reduce a substantial finite-sample bias in the Chambless-Diao type estimator of the most popular time-dependent accuracy summary. The asymptotic properties of estimators and data-driven bandwidths are also established under some suitable conditions. It is found in simulations that the proposed estimators and inference procedures exhibit quite satisfactory performances. Moreover, the general applicability of our methodology is illustrated by two empirical data.