Maximum Likelihood Estimation in Compound Geometric Failure Model with Changing Parameters From Type-I Two-Stage Progressively Censored and Group Censored Samples

Boardman and Kendell (1970 Boardman , T. J. , Kendell , P. J. ( 1970 ). Estimation in compound failure models . Technometrics 12 : 891 – 908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered the problem of estimation with respect to Type-I censoring when an item is subjected to only one of the two causes of failure assuming exponential model. Patel and Gajjar (1992 Patel , M. N. , Gajjar , A. V. ( 1992 ). Maximum likelihood estimation in compound exponential failure model with changing failure rates from Type-I progressively censored and group censored samples . Commun. Statist. Theor. Meth. 21 ( 10 ): 2899 – 2908 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) considered extension of the Boardman and Kendell's results in case of two-stage progressive censoring. Here we have considered geometric competing risk failure model with two independent causes of failures. Maximum likelihood estimation of the parameters is carried out using Type-I two-stage progressively censored and group censored samples. Asymptotic standard errors of the estimators are obtained for both the cases. Two illustrative examples are cited for ungroup and group competing risk models.     

Nonparametric Estimation of the Multivariate Distribution Function in a Censored Regression Model with Applications

In a regression model with univariate censored responses, a new estimator of the joint distribution function of the covariates and response is proposed, under the assumption that the response and the censoring variable are independent conditionally to the covariates. This estimator is based on the conditional Kaplan–Meier estimator of Beran (1981 Beran , R. ( 1981 ). Nonparametric regression with randomly censored survival data. Technical Report, University of California, Berkeley, California . [Google Scholar]), and happens to be an extension of the multivariate empirical distribution function used in the uncensored case. We derive asymptotic i.i.d. representations for the integrals with respect to the measure defined by this estimated distribution function. These representations hold even in the case where the covariates are multidimensional under some additional assumption on the censoring. Applications to censored regression and to density estimation are considered.     

Inferences for Mixtures of Distributions for Centrally Censored Data with Partial Identification

In this article, several methods to make inferences about the parameters of a finite mixture of distributions in the context of centrally censored data with partial identification are revised. These methods are an adaptation of the work in Contreras-Cristán, Gutiérrez-Peña, and O'Reilly (2003 Contreras-Cristán , A. , Gutiérrez-Peña , E. , O'Reilly , F. ( 2003 ). Inferences using latent variables for mixtures of distributions for censored data with partial identification . Comm. Stat. Theor. Meth. 32 ( 4 ): 749 – 774 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the case of right censoring. The first method focuses on an asymptotic approximation to a suitably simplified likelihood using some latent quantities; the second method is based on the expectation-maximization (EM) algorithm. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to non-Bayesian methods that deal directly with the full likelihood of the mixture appealing to its asymptotic approximation. The third method, from a Bayesian perspective, uses data augmentation to work with an uncensored sample. This last method is related to a recently proposed Bayesian method in Baker, Mengersen, and Davis (2005 Baker , P. , Mengersen , K. , Davis , G. ( 2005 ). A Bayesian solution to reconstructing centrally censored distributions . J. Agr. Biol. Environ. Stat. 1 : 61 – 84 . [Google Scholar]). Our proposal of the three adapted methods is shown to provide similar inferential answers, thus offering alternative analyses.     

Rank Tests for Two-Sample Problems Based on Multiple Type-II Censored Data

In this article, we study the effect of censoring on the asymptotic efficiency of the two-sample rank tests based on multiple Type-II censored data. Since the scores generating functions associated with these test statistics have a finite number of jump discontinuities, we use a slightly modified version of a theorem of Dupac and Hajek (1969 Dupac , V. , Hajek , J. ( 1969 ). Asymptotic normality of simple linear rank statistics under alternatives II . Ann. Math. Statist. 40 : 1992 – 2017 .[Crossref] , [Google Scholar]) to obtain their asymptotic distributions under fixed alternatives. This modified version, which leads to a simpler centering constant, is proved by Dupac (1970 Dupac , V. ( 1970 ). A contribution to the asymptotic normality of simple linear rank statistics . In: Puri , M. L. , ed. Nonparametric Techniques in Statistical Inference . Cambridge : Cambridge University Press . [Google Scholar]) in the light of results of Hoeffding (1968 Hoeffding , W. ( 1968 ). On the Centering of Simple Linear Rank Statistics. Instit. Statist. Mimeo Series No. 585, University of North Carolina . [Google Scholar]), an earlier version of Hoeffding (1973 Hoeffding , W. ( 1973 ). On the centering of simple linear rank statistics . Ann. Statist. 1 : 54 – 66 .[Crossref], [Web of Science ®] , [Google Scholar]). Hence, we obtain the Pitman ARE's of these rank tests relative to the corresponding tests based on the complete samples. The ARE's are computed for some well known rank tests for two-sample location and scale problems, when the combined ordered samples from different underlying distributions are censored using triple and lower order Type-II censoring schemes. The effect of all these censoring schemes on the ARE's of the different tests is examined numerically. It is found that there is a gain in efficiency due to censoring in many of the cases considered here. This suggests that in such cases it is possible to improve the efficiency of rank tests by discarding suitable portions of the data.     

Inference Using Latent Variables for Mixtures of Distributions for Censored Data with Partial Identification

Abstract

In this article two methods are proposed to make inferences about the parameters of a finite mixture of distributions in the context of partially identifiable censored data. The first method focuses on a mixture of location and scale models and relies on an asymptotic approximation to a suitably constructed augmented likelihood; the second method provides a full Bayesian analysis of the mixture based on a Gibbs sampler. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to other methods which deal directly with the likelihood of the mixture. This may be crucial if the number of components in the mixture is not small. Our proposals are illustrated on a classical example on failure times for communication devices first studied by Mendenhall and Hader (Mendenhall, W., Hader, R. J. (1958 Mendenhall, W. and Hader, R. J. 1958. Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika, 45: 504–520. [Crossref], [Web of Science ®] , [Google Scholar]). Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45:504–520.). In addition, we study the coverage of the confidence intervals obtained from each of the methods by means of a small simulation exercise.     

Locally Most Powerful Rank Tests for Comparison of Two Failure Rates Based on Multiple Type-II Censored Data

This article deals with the locally most powerful rank tests for testing the hypothesis that two failure rates are equal against the alternative that one failure rate is greater than the other, when the combined ordered sample is multiple Type-II censored. A modified version of the Dupa? and Hájek (1969 Dupa? , V. , Hájek , J. (1969). Asymptotic normality of simple linear rank statistics under alternatives II. Ann. Math. Statist. 40:1992–2017.[Crossref] , [Google Scholar]) theorem is used to establish their asymptotic normality under fixed alternative since the scores generating functions associated with these rank test statistics have a finite number of jump discontinuities. The modified version that leads to a simpler centering constant, is proved by Dupa? (1970 Dupa? , V. ( 1970 ). A contribution to the asymptotic normality of simple linear rank statistics. In: Puri, M. L., ed. Nonparametric Techniques in Statistical Inference. Cambridge: Cambridge University Press . [Google Scholar]) using the results of Hájek (1968 Hájek , J. ( 1968 ). Asymptotic normality of simple linear rank statistics under alternatives . Ann. Math. Statist. 39 : 325 – 346 .[Crossref] , [Google Scholar]). The Pitman AREs of these rank tests based on censored data relative to the corresponding tests based on complete data are obtained under some Lehmann-type alternative distributions such that their failure rates dominate the failure rates of the respective null distributions. The AREs are computed numerically for single (left or right) and double censored data, and the extent of loss due to these censoring schemes is discussed. The rank tests considered here include among them the Mann-Whiney-Wilcoxon (MWW) test, the Savage test, and the linear combination of these two tests. In the case of all the tests, except the MWW test, it is found that the loss of efficiency due to left censoring is considerably less than that due to right censoring. In the case of finite samples, Monte Carlo simulation results showing the empirical levels and empirical powers against some Lehmann alternatives are presented.     

Closeness of k-records to Progressive Type-II Censored Order Statistics for Location-scale Families

In this article, the Pitman closeness of upper and lower k-records to progressive Type-II censored order statistics for location-scale families is investigated. In each case, the special properties of the probability of Pitman closeness are obtained and the corresponding monotonicity properties are discussed. Moreover, the closest k-record to a specific progressive Type-II censored data is obtained. Finally, for the standard exponential and standard uniform distributions, explicit expressions for the probability of Pitman closeness are derived. For various censoring schemes, the results of the numerical computations are displayed in tables. Most of the results in Ahmadi and Balakrishnan (2013) Ahmadi, J., Balakrishnan, N. (2013). On the nearness of record values to order statistics from Pitman measure of closeness. Metrika 76:521–541.[Crossref], [Web of Science ®] , [Google Scholar] can be achieved as special cases.     

Comparison of Confidence Procedures for Type I Censored Exponential Lifetimes

In the model of type I censored exponential lifetimes, coverage probabilities are compared for a number of confidence interval constructions proposed in literature. The coverage probabilities are calculated exactly for sample sizes up to 50 and for different degrees of censoring and different degrees of intended confidence. If not only a fair two-sided coverage is desired, but also fair one-sided coverage's, only few methods are quite satisfactory. A likelihood-based interval and a third root transformation to normality work almost perfectly, but the ²-based method that is perfect under no censoring and under type II censoring can also be advocated.     

A modified chi-square test for Bertholon model with censored data

In this work, we propose the construction of a chi-squared goodness-of-fit test in censored data case, for Bertholon model which can analyse various competing risks of failure or death. This test is based on a modification of the Nikulin-Rao-Robson (NRR) statistic proposed by Bagdonavicius and Nikulin (2011a Bagdonavicius, V., Nikulin, M. (2011a). Chi-squared tests for general composite hypotheses from censored samples. Comptes Rendus Mathématiques: Series I 349(3–4):219–223. [Google Scholar], 2011b Bagdonavicius, V., Nikulin, M. (2011b). Chi-squared goodness-of-fit test for right censored data. International Journal of Applied Mathematics and Statistics 24:30–50. [Google Scholar]) for censored data. We applied this test to numerical examples from simulated samples and real data.     

Empirical Likelihood Ratio for Linear Transformation Models with Doubly Censored Data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. When L is always observed, we consider the empirical likelihood inference for linear transformation models, based on the martingale-type estimating equation proposed by Chen et al. (2002 Chen , K. , Jin , Z. , Ying , Z. ( 2002 ). Semiparametric analysis of transformation models with censored data . Biometrika 89 : 659 – 668 .[Crossref], [Web of Science ®] , [Google Scholar]). It is demonstrated that both the approach of Lu and Liang (2006 Lu , W. , Liang , Y. ( 2006 ). Empirical likelihood inference for linear transformation models . Journal of Multivariate Analysis 97 : 1586 – 1599 .[Crossref], [Web of Science ®] , [Google Scholar]) and that of Yu et al. (2011 Yu , W. , Sun , Y. , Zheng , M. ( 2011 ). Empirical likelihood method for linear transformation models . Annals of the Institute of Statistical Mathematics 63 : 331 – 346 .[Crossref], [Web of Science ®] , [Google Scholar]) can be extended to doubly censored data. Simulation studies are conducted to investigate the performance of the empirical likelihood ratio methods.     

Relations for Moments of Progressively Type-II Censored Order Statistics from Log-Logistic Distribution with Applications to Inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a log-logistic distribution. The use of these relations in a systematic recursive manner would enable the computation of all the means, variances and covariances of progressively Type-II right censored order statistics from the log-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁,…, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Malik (1987 Balakrishnan , N. , Malik , H. J. ( 1987 ). Moments of order statistics from truncated log-logistic distribution . J. Statist. Plann. Infer. 17 : 251 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) and Balakrishnan et al. (1987 Balakrishnan , N. , Malik , H. J. , Puthenpura , S. ( 1987 ). Best linear unbiased estimation of location and scale parameters of the log-logistic distribution . Commun. Statist. Theor. Meth. 16 : 3477 – 3495 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The moments so determined are then utilized to derive best linear unbiased estimators for the scale- and location-scale log-logistic distributions. A comparison of these estimates with the maximum likelihood estimates is made through Monte Carlo simulation. The best linear unbiased predictors of progressively censored failure times is then discussed briefly. Finally, a numerical example is presented to illustrate all the methods of inference developed here.     

Empirical Likelihood Method for Censored Median Regression Models

Based on the semiparametric median regression analysis for the right-censored data developed by Ying et al. (1995 Ying , Z. , Jung , S. H. , Wei , L. J. ( 1995 ). Survival analysis with median regression models . J. Amer. Statist. Assoc. 90 : 178 – 184 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), an empirical likelihood based inferential procedure for the regression coefficients is proposed. The limiting distribution of the proposed log-empirical likelihood ratio test statistic follows a chi-squared distribution, which corresponds to the standard asymptotic results of the empirical likelihood method. The inference about the subsets of the entire regression coefficients vector is discussed. The proposed method is illustrated by some simulation studies.     

Maximum likelihood estimation in Dagum distribution with censored samples

In this work, we show that the Dagum distribution [3 Dagum, C. 1977. A new model of personal income distribution: Specification and estimation. Econ. Appl., XXX: 413–436.  [Google Scholar]] may be a competitive model for describing data which include censored observations in lifetime and reliability problems. Maximum likelihood estimates of the three parameters of the Dagum distribution are determined from samples with type I right and type II doubly censored data. We perform an empirical analysis using published censored data sets: in certain cases, the Dagum distribution fits the data better than other parametric distributions that are more commonly used in survival and reliability analysis. Graphical comparisons confirm that the Dagum model behaves better than a number of competitive distributions in describing the empirical hazard rate of the analyzed data. A probability plot to provide graphical check of the appropriateness of the Dagum model for right censored data is constructed, and the details are given in the appendix. Finally, a simulation study that shows the good performance of the maximum likelihood estimators of the Dagum shape parameters for finite type II doubly censored samples is carried out.     

Sliced Average Variance Estimation for Censored Data

We consider a nonlinear censored regression problem with a vector of predictors. With censoring, high-dimensional regression analysis becomes much more complicated. Since censoring can cause severe bias in estimation, modification to adjust such bias is needed to be made. Based on the weight adjustment, we develop the modification of sliced average variance estimation for estimating the lifetime central subspace without requiring a prespecified parametric model. Our proposed method preserves as much regression information as possible. Simulation results are reported and comparisons are made with the sliced inverse regression of Li et al. (1999 Li , K. C. , Wang , J. L. , Chen , C. H. ( 1999 ). Dimension reduction for censored regression data . Ann. Statist. 27 : 1 – 23 . [Google Scholar]).     

Testing exponentiality based on Kullback—Leibler information for progressively Type II censored data

In many life-testing and reliability experiments, data are often censored in order to reduce the cost and time associated with testing and since the conventional Type-I and Type-II censoring schemes are not flexible enough, progressive censoring is developed by researchers. In this article, we develop a general goodness of fit test by using a new estimate of Kullback–Leibler information based on progressively Type-II censored data. Consistency and other properties of the proposed test are shown. Then, we use the proposed test statistic to test for exponentiality based on progressively Type-II censored data. The power values of the proposed test under different progressively Type-II censoring schemes are computed, through Monte Carlo simulations. It is observed that the proposed test is quite powerful in compared with the test proposed by Balakrishnan et al. (2007 Balakrishnan, N., Habibi Rad, A., and Arghami, N. R. (2007). Testing exponentiality based on Kullback–Leibler information with progressively type-II censored data. IEEE Transactions on Reliability 56:301–307. [Google Scholar]). Two real datasets from progressive censoring literature are finally presented for illustrative purpose.     

Estimation for exponential distribution based on competing risk middle censored data

Abstract

For a general censoring scheme called “middle censoring” scheme which was proposed by Jammalamadaka and Mangalam (2003 Jammalamadaka, S.R., Mangalam, V. (2003). Nonparametric estimation for middle censored data. J. Nonparametric Statist. 15:253–265.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in nonparametric set up. In this article, point and interval estimation problems are considered for the exponential distribution when the failure data is middle censored with two independent competing failure risks. Different methods are introduced to estimate the unknown model parameters such as maximum likelihood estimation, midpoint approximation, equivalent quantities estimation. The Bayesian estimation is also considered with gamma priors. Two numerical examples are analyzed to show the performance of the proposed methods.     

Chi-Squared Type Test for the AFT-Generalized Inverse Weibull Distribution

The generalized inverse Weibull distribution is a newlife time probability distribution which can be used to model a variety of failure characteristics. It has several desirable properties and nice physical interpretations which enable them to be used frequently. In this article, we present a chi-squared goodness-of-fit test for an accelerated failure time (AFT) model with generalized inverse Weibull distribution (GIW) as the baseline distribution, in both of complete and censored data. This test is based on a modification of the NRR (Nikulin-Rao-Robson) statistic Y², proposed by Bagdonavicius and Nikulin (2011 Bagdonavicius, V., Nikulin, M. (2011). Chi-squared tests for general composite hypotheses from censored samples. Comptes Rendus Mathematique, Ser. I, 349(3–4): 219–223.[Crossref], [Web of Science ®] , [Google Scholar]), for censored data. Two applications of real data are given to illustrate the potentiality of the proposed test.     

Semiparametric Efficient Estimation in the Generalized Odds-Rate Class of Regression Models for Right-Censored Time-to-Event Data

The generalized odds-rate class of regression models for time to event data is indexed by a non-negative constant and assumes thatg(S(t|Z)) = (t) + Zwhere g(s) = log(^-1(s-) for > 0, g₀(s) = log(- log s), S(t|Z) is the survival function of the time to event for an individual with qx1 covariate vector Z, is a qx1 vector of unknown regression parameters, and (t) is some arbitrary increasing function of t. When =0, this model is equivalent to the proportional hazards model and when =1, this model reduces to the proportional odds model. In the presence of right censoring, we construct estimators for and exp((t)) and show that they are consistent and asymptotically normal. In addition, we show that the estimator for is semiparametric efficient in the sense that it attains the semiparametric variance bound.