ALBERT

Description:    The dependence of large values in a stochastic process is an important topic in risk, insurance and finance. The idea of risk contagion is based on the idea of large value dependence. The Gaussian copula notoriously fails to capture this phenomenon. Two notions in a process or vector context which summarize extremal dependence in a function comparable to a correlation function are the extremal dependence measure (EDM) and the extremogram . We review these ideas and compare the two tools and end with a central limit theorem for a natural estimator of the EDM which allows drawing confidence bands comparable to those provided by Bartlett’s formula in a classical context of sample correlation functions. Content Type Journal Article Pages 1-26 DOI 10.1007/s10687-011-0135-9 Authors Martin Larsson, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA Sidney I. Resnick, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Description:    In many randomized clinical trials, the primary response variable, for example, the survival time, is not observed directly after the patients enroll in the study but rather observed after some period of time (lag time). It is often the case that such a response variable is missing for some patients due to censoring that occurs when the study ends before the patient’s response is observed or when the patients drop out of the study. It is often assumed that censoring occurs at random which is referred to as noninformative censoring; however, in many cases such an assumption may not be reasonable. If the missing data are not analyzed properly, the estimator or test for the treatment effect may be biased. In this paper, we use semiparametric theory to derive a class of consistent and asymptotically normal estimators for the treatment effect parameter which are applicable when the response variable is right censored. The baseline auxiliary covariates and post-treatment auxiliary covariates, which may be time-dependent, are also considered in our semiparametric model. These auxiliary covariates are used to derive estimators that both account for informative censoring and are more efficient then the estimators which do not consider the auxiliary covariates. Content Type Journal Article Pages 1-28 DOI 10.1007/s10985-011-9199-8 Authors Xiaomin Lu, Department of Biostatistics, College of Medicine and College of Public Health and health Professions, University of Florida, Gainesville, FL 32611, USA Anastasios A. Tsiatis, Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203, USA Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870

Description:    In this paper, a new kind of location invariant Weiss-Hill estimator of the extreme value index γ  ∈ ℝ is proposed. The new estimator is a combination of two estimators proposed by Weiss (Nav Res Logist Q 1:111–114, 1971 ) and Fraga Alves (Extremes 4:199–217, 2001a ). The following properties of the new estimator are derived: weak consistency, strong consistency, and asymptotic expansions. A bias corrected version of the proposed estimator is given after determining an optimal sample fraction. Some finite simulation studies are performed. Content Type Journal Article Pages 1-34 DOI 10.1007/s10687-011-0134-x Authors Chengxiu Ling, Department of Actuarial Science, Faculty of Business and Economics, University of Lausanne, Bâtiment Extranef, UNIL-Dorigny, 1015 Lausanne, Switzerland Zuoxiang Peng, School of Mathematics and Statistics, Southwest University, Chongqing, 400715 China Saralees Nadarajah, School of Mathematics, University of Manchester, Manchester, UK Journal Extremes Online ISSN 1572-915X Print ISSN 1386-1999

Description:    This paper considers the analysis of multivariate survival data where the marginal distributions are specified by semiparametric transformation models, a general class including the Cox model and the proportional odds model as special cases. First, consideration is given to the situation where the joint distribution of all failure times within the same cluster is specified by the Clayton–Oakes model (Clayton, Biometrika 65:141–151, l978 ; Oakes, J R Stat Soc B 44:412–422, 1982 ). A two-stage estimation procedure is adopted by first estimating the marginal parameters under the independence working assumption, and then the association parameter is estimated from the maximization of the full likelihood function with the estimators of the marginal parameters plugged in. The asymptotic properties of all estimators in the semiparametric model are derived. For the second situation, the third and higher order dependency structures are left unspecified, and interest focuses on the pairwise correlation between any two failure times. Thus, the pairwise association estimate can be obtained in the second stage by maximizing the pairwise likelihood function. Large sample properties for the pairwise association are also derived. Simulation studies show that the proposed approach is appropriate for practical use. To illustrate, a subset of the data from the Diabetic Retinopathy Study is used. Content Type Journal Article Pages 1-22 DOI 10.1007/s10985-011-9205-1 Authors Chyong-Mei Chen, Department of Statistics and Informatics Science, Providence University, Taichung, Taiwan, ROC Chang-Yung Yu, Department of Financial and Computational Mathematics, Providence University, Taichung, Taiwan, ROC Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870

Description:    Life tables used in life insurance determine the age of death distribution only at integer ages. Therefore, actuaries make fractional age assumptions to interpolate between integer age values when they have to value payments that are not restricted to integer ages. Traditional fractional age assumptions as well as the fractional independence assumption are easy to apply but result in a non-intuitive overall shape of the force of mortality. Other approaches proposed either require expensive optimization procedures or produce many discontinuities. We suggest a new, computationally inexpensive algorithm to select the parameters within the LFM-family introduced by Jones and Mereu (Insur Math Econ 27:261–276, 2000 ). In contrast to previously suggested methods, our algorithm enforces a monotone force of mortality between integer ages if the mortality rates are monotone and keeps the number of discontinuities small. Content Type Journal Article Pages 1-13 DOI 10.1007/s10985-011-9211-3 Authors Christiane Barz, Anderson School of Management, University of California at Los Angeles, 110 Westwood Plaza, Los Angeles, CA 90095, USA Alfred Müller, Department Mathematik, Universität Siegen, 57072 Siegen, Germany Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870

Description:    In many biomedical studies, it is common that due to budget constraints, the primary covariate is only collected in a randomly selected subset from the full study cohort. Often, there is an inexpensive auxiliary covariate for the primary exposure variable that is readily available for all the cohort subjects. Valid statistical methods that make use of the auxiliary information to improve study efficiency need to be developed. To this end, we develop an estimated partial likelihood approach for correlated failure time data with auxiliary information. We assume a marginal hazard model with common baseline hazard function. The asymptotic properties for the proposed estimators are developed. The proof of the asymptotic results for the proposed estimators is nontrivial since the moments used in estimating equation are not martingale-based and the classical martingale theory is not sufficient. Instead, our proofs rely on modern empirical process theory. The proposed estimator is evaluated through simulation studies and is shown to have increased efficiency compared to existing methods. The proposed method is illustrated with a data set from the Framingham study. Content Type Journal Article Pages 1-23 DOI 10.1007/s10985-011-9209-x Authors Yanyan Liu, School of Mathematics and Statistics, Wuhan University, Wuhan, 430072 Hubei, China Zhongshang Yuan, School of Mathematics and Statistics, Wuhan University, Wuhan, 430072 Hubei, China Jianwen Cai, Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7420, USA Haibo Zhou, Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7420, USA Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870

Description:    In this article, we propose a parametric model for the distribution of time to first event when events are overdispersed and can be properly fitted by a Negative Binomial distribution. This is a very common situation in medical statistics, when the occurrence of events is summarized as a count for each patient and the simple Poisson model is not adequate to account for overdispersion of data. In this situation, studying the time of occurrence of the first event can be of interest. From the Negative Binomial distribution of counts, we derive a new parametric model for time to first event and apply it to fit the distribution of time to first relapse in multiple sclerosis (MS). We develop the regression model with methods for covariate estimation. We show that, as the Negative Binomial model properly fits relapse counts data, this new model matches quite perfectly the distribution of time to first relapse, as tested in two large datasets of MS patients. Finally we compare its performance, when fitting time to first relapse in MS, with other models widely used in survival analysis (the semiparametric Cox model and the parametric exponential, Weibull, log-logistic and log-normal models). Content Type Journal Article Pages 1-18 DOI 10.1007/s10985-011-9207-z Authors Paola Siri, Department of Mathematics (DIMAT), Polytechnic of Turin, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Eric Henninger, Merck Serono S.A., 9 Chemin des Mines, 1202 Geneva, Switzerland Maria Pia Sormani, Biostatistics Unit, Department of Health Sciences (DISSAL), University of Genova, Via Pastore 1, 16132 Genova, Italy Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870

Description:    Several authors have indicated that incorrectly classified cause of death for prostate cancer survivors may have played a role in the observed recent peak and decline of prostate cancer mortality. Motivated by the suggestion we studied a competing risks model where other cause of death may be misattributed as a death of interest. We first consider a naïve approach using unconstrained nonparametric maximum likelihood estimation (NPMLE), and then present the constrained NPMLE where the survival function is forced to be monotonic. Surprising observations were made as we studied their small-sample and asymptotic properties in continuous and discrete situations. Contrary to the common belief that the non-monotonicity of a survival function NPMLE is a small-sample problem, the constrained NPMLE is asymptotically biased in the continuous setting. Other isotonic approaches, the supremum (SUP) method and the Pooled-Adjacent-Violators (PAV) algorithm, and the EM algorithm are also considered. We found that the EM algorithm is equivalent to the constrained NPMLE. Both SUP method and PAV algorithm deliver consistent and asymptotically unbiased estimator. All methods behave well asymptotically in the discrete time setting. Data from the Surveillance, Epidemiology and End Results (SEER) database are used to illustrate the proposed estimators. Content Type Journal Article Pages 1-22 DOI 10.1007/s10985-011-9210-4 Authors Jinkyung Ha, Int Med-Geriatric Medicine, University of Michigan, 300 NIB, Ann Arbor, MI 48109, USA Alexander Tsodikov, Department of Biostatistics, University of Michigan, 1415 Washington Heights, Ann Arbor, MI 48109, USA Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870

Description:    In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t from a particular type of risk in the presence of other risks. Assume that the lifetime distributions of two populations are uniformly stochastically ordered. Since this ordering may not hold for the empiricals due to sampling variability, it is natural to estimate these distributions under this constraint. This will in turn affect the estimation of the CIFs. This article considers this estimation problem. We do not assume that the risk sets in the two populations are related, give consistent estimators of all the CIFs and study the weak convergence of the resulting processes. We also report the results of a simulation study that show that our restricted estimators outperform the unrestricted ones in terms of mean square error. A real life example is used to illustrate our theoretical results. Content Type Journal Article Pages 1-17 DOI 10.1007/s10985-011-9204-2 Authors Noriah M. A. Al-Kandari, Kuwait University, Safat, Kuwait Emad-Eldin A. A. Aly, Kuwait University, Safat, Kuwait Hammou El Barmi, Baruch College, The City University of New York, New York, NY, USA Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870

Description:    In this article we present a nonparametric method for constructing confidence bands for the difference of two quantile residual life (qrl) functions. These bands provide evidence for two random variables ordering with respect to the qrl order. The comparison of qrl functions is of importance, specially in the treatment of cancer when there exists a possibility of benefiting from a new secondary therapy. A qrl function is the quantile of the remaining life of a surviving subject, as it varies with time. We show the applicability of this approach in Medicine and Ecology. A simulation study has been carried out to evaluate and illustrate the performance and the consistency of this new methodology. Content Type Journal Article Pages 1-20 DOI 10.1007/s10985-011-9208-y Authors Alba M. Franco-Pereira, Department of Statistics and Operational Research, Universidad de Vigo, 36310 Vigo, Pontevedra, Spain Rosa E. Lillo, Department of Statistics, Universidad Carlos III de Madrid, 28903 Getafe, Madrid, Spain Juan Romo, Department of Statistics, Universidad Carlos III de Madrid, 28903 Getafe, Madrid, Spain Journal Lifetime Data Analysis Online ISSN 1572-9249 Print ISSN 1380-7870