We investigate the estimation of the extreme value index when the data are subject to random censorship. We prove, in a unified way, detailed asymptotic normality results for various estimators of the extreme value index and use these estimators as the main building block for estimators of extreme quantiles. We illustrate the quality of these methods by a small simulation study and apply the estimators to medical data.
We make use of the empirical process theory to approximate the adapted Hill estimator, for censored data, in terms of Gaussian processes. Then, we derive its asymptotic normality, only under the usual second-order condition of regular variation. Our methodology allows to relax the assumptions, made in Einmahl, Fils-Villetard and Guillou(2008), on the heavy-tailed distribution functions and the sample fraction of upper order statistics.
In this paper, we propose several statistics for testing uniformity under progressive Type-I interval censoring. We obtain the critical points of these statistics and study the power of the proposed tests against a representative set of alternatives via simulation. Finally, we generalize our methods for continuous and completely specified distributions.
Many insurance premium principles are defined and various estimation procedures introduced in the literature. In this paper, we focus on the estimation of the excess-of-loss reinsurance premium when the risks are randomly right-censored. The asymptotic normality of the proposed estimator is established under suitable conditions and its performance evaluated through sets of simulated data.
We consider nonparametric estimation of cure-rate based on mixture model under Case-1 interval censoring. We show that the nonparametric maximum-likelihood estimator (NPMLE) of cure-rate is non-unique as well as inconsistent, and propose two estimators based on the NPMLE of the distribution function under this censoring model. We present a cross-validation method for choosing a `cut-off' point needed for the estimators. The limiting distributions of the latter are obtained using extreme-value theory. Graphical illustration of the procedures based on simulated data are provided.
The central limit theorem introduced by Stute [The central limit theorem under random censorship. Ann. Statist. 1995; 23: 422-439] does not hold for some class of heavy-tailed distributions. In this paper, we make use of the extreme value theory to propose an alternative estimating approach of the mean ensuring the asymptotic normality property. A simulation study is carried out to evaluate the performance of this estimation procedure
We revisit the estimation of the extreme value index for randomly censored data from a heavy tailed distribution. We introduce a new class of estimators which encompasses earlier proposals given in Worms and Worms (2014) and Beirlant et al. (2018), which were shown to have good bias properties compared with the pseudo maximum likelihood estimator proposed in Beirlant et al. (2007) and Einmahl et al. (2008). However the asymptotic normality of the type of estimators first proposed in Worms and Worms (2014) was still lacking, in the random threshold case. We derive an asymptotic representation a…
In extreme value analysis, the extreme value index plays a vital role as it determines the tail heaviness of the underlying distribution and is the primary parameter required for the estimation of other extreme events. In this paper, we review the estimation of the extreme value index when observations are subject to right random censoring and the presence of covariate information. In addition, we propose some estimators of the extreme value index, including a maximum likelihood estimator from a perturbed Pareto distribution. The existing estimators and the proposed ones are compared through a…
This article describes the inferential procedures and Bayesian optimal life-testing issues under Type-II unified hybrid censoring scheme. First, the explicit expressions of expected number of failures, expected duration of testing and Fisher information matrix for the unknown parameters of the underlying lifetime model are derived. Then, using these quantities, the Bayesian optimal life-testing plans are computed in subsequent section. A cost constraint D-optimal optimization problem has been formulated and the corresponding solution algorithm is provided to obtain optimal plans. Computational…
Chen and Lio (Computational Statistics and Data Analysis 54: 1581-1591, 2010) proposed five methods for estimating the parameters of generalized exponential distribution under progressive type-I interval censoring scheme. Unfortunately, among them, the proposed EM algorithm is incorrect. Here, we propose the correct EM algorithm and compare its performance with the maximum likelihood estimators and that proposed by Chen and Lio (2010) in a simulation study.
Extreme value analysis in the presence of censoring is receiving much attention as it has applications in many disciplines, including survival and reliability studies. Estimation of extreme value index (EVI) is of primary importance as it is a critical parameter needed in estimating extreme events such as quantiles and exceedance probabilities. In this paper, we review several estimators of the extreme value index when data is subject to random censoring. In addition, four estimators are proposed, one based on the exponential regression approximation of log spacings, one based on a Zipf estima…
In this paper, nonparametric estimation of the conditional Weibull-tail coefficient when the variable of interest is right random censored is addressed. A Weissman-type estimator of conditional extreme quantile is also proposed. In addition, a simulation study is conducted to assess the finite-sample behavior of the proposed estimators and a comparison with alternative strategies is provided. Finally, the practical applicability of the methodology is presented using a real datasets of men suffering from a larynx cancer.
Based on a progressively type-II censored sample from the exponential distribution with unknown location and scale parameter, confidence bands are proposed for the underlying distribution function by using confidence regions for the parameters and Kolmogorov-Smirnov type statistics. Simple explicit representations for the boundaries and for the coverage probabilities of the confidence bands are analytically derived, and the performance of the bands is compared in terms of band width and area by means of a data example. As a by-product, a novel confidence region for the location-scale parameter…
In this paper, we focus on the parametric inference based on the Tampered Random Variable (TRV) model for simple step-stress life testing (SSLT) using Type-II censored data. The baseline lifetime of the experimental units, under normal stress conditions, follows the Gumbel Type-II distribution with $α$ and $λ$ being the shape and scale parameters, respectively. Maximum likelihood estimator (MLE) and Bayes estimator of the model parameters are derived based on Type-II censored samples. We obtain asymptotic intervals of the unknown parameters using the observed Fisher information matrix. Bayes e…
The subject of tail estimation for randomly censored data from a heavy tailed distribution receives growing attention, motivated by applications for instance in actuarial statistics. The bias of the available estimators of the extreme value index can be substantial and depends strongly on the amount of censoring. We review the available estimators, propose a new bias reduced estimator, and show how shrinkage estimation can help to keep the MSE under control. A bootstrap algorithm is proposed to construct confidence intervals. We compare these new proposals with the existing estimators through …
In this article, weak convergence of the general non-Markov state transition probability estimator by Titman (2015) is established which, up to now, has not been verified yet for other general non-Markov estimators. A similar theorem is shown for the bootstrap, yielding resampling-based inference methods for statistical functionals. Formulas of the involved covariance functions are presented in detail. Particular applications include the conditional expected length of stay in a specific state, given occupation of another state in the past, as well as the construction of time-simultaneous confi…
In this work, nonparametric log-rank-type statistical tests are introduced in order to verify homogeneity of purely discrete variables subject to arbitrary right-censoring for infinitely many categories. In particular, the Cramér-von Mises test statistics for discrete models under censoring is established. In order to introduce the test, we develop the weighted log-rank statistics in a general multivariate discrete setup which complements previous fundamental results of Gill (1980) and Andersen et al. (1982). Due to the presence of persistent jumps over the unbounded set of categories, the asy…
Linear regression is arguably the most prominent among statistical inference methods, popular both for its simplicity as well as its broad applicability. On par with data-intensive applications, the sheer size of linear regression problems creates an ever growing demand for quick and cost efficient solvers. Fortunately, a significant percentage of the data accrued can be omitted while maintaining a certain quality of statistical inference with an affordable computational budget. The present paper introduces means of identifying and omitting "less informative" observations in an online and data…
The progressive Type-II hybrid censoring scheme introduced by Kundu and Joarder (\textit{Computational Statistics and Data Analysis}, 2509-2528, 2006), has received some attention in the last few years. One major drawback of this censoring scheme is that very few observations (even no observation at all) may be observed at the end of the experiment. To overcome this problem, Cho, Sun and Lee (\textit{Statistical Methodology}, 23, 18-34, 2015) recently introduced generalized progressive censoring which ensures to get a pre specified number of failures. In this paper we analyze generalized progr…
The computational prediction algorithm of neural network, or deep learning, has drawn much attention recently in statistics as well as in image recognition and natural language processing. Particularly in statistical application for censored survival data, the loss function used for optimization has been mainly based on the partial likelihood from Cox's model and its variations to utilize existing neural network library such as Keras, which was built upon the open source library of TensorFlow. This paper presents a novel application of the neural network to the quantile regression for survival…
Progressive censoring scheme has received considerable attention in recent years. In this paper we introduce a new type-II progressive censoring scheme for two samples. It is observed that the proposed censoring scheme is analytically more tractable than the existing joint progressive type-II censoring scheme proposed by Rasouli and Balakrishnan \cite{RB:2010}. It has some other advantages also. We study the statistical inference of the unknown parameters based on the assumptions that the lifetime distribution of the experimental units for the two samples follow exponential distribution with d…
An important research topic in survival analysis is related to the modeling and estimation of the cure rate, i.e. the proportion of subjects that will never experience the event of interest. However, most estimation methods proposed so far in the literature do not handle the case of insufficient follow-up, that is when the right end point of the support of the censoring time is strictly less than that of the survival time of the susceptible subjects, and consequently these estimators overestimate the cure rate in that case. We fill this gap by proposing a new estimator of the cure rate that ma…
The i.i.d. censoring model for survival analysis assumes two independent sequences of i.i.d. positive random variables, $(T_i^*)_{1\le i\le n}$ and $(U_i)_{1\le i\le n}$. The data consists of observations on the random sequence $\big(T_i=\min(T_i^*,U_i)$ together with accompanying censor indicators. Values of $T_i$ with $T_i^*\le U_i$ are said to be uncensored, those with $T_i^*> U_i$ are censored. We assume that the distributions of the $T_i^*$ and $U_i$ are in the domain of attraction of the Gumbel distribution and obtain the asymptotic distributions, as sample size $n\to\infty$, of the maxi…
Fiducial Inference, introduced by Fisher in the 1930s, has a long history, which at times aroused passionate disagreements. However, its application has been largely confined to relatively simple parametric problems. In this paper, we present what might be the first time fiducial inference, as generalized by Hannig et al. (2016), is systematically applied to estimation of a nonparametric survival function under right censoring. We find that the resulting fiducial distribution gives rise to surprisingly good statistical procedures applicable to both one sample and two sample problems. In partic…
This paper develops robust confidence intervals in high-dimensional and left-censored regression. Type-I censored regression models are extremely common in practice, where a competing event makes the variable of interest unobservable. However, techniques developed for entirely observed data do not directly apply to the censored observations. In this paper, we develop smoothed estimating equations that augment the de-biasing method, such that the resulting estimator is adaptive to censoring and is more robust to the misspecification of the error distribution. We propose a unified class of robus…
Extreme U-statistics arise when the kernel of a U-statistic has a high degree but depends only on its arguments through a small number of top order statistics. As the kernel degree of the U-statistic grows to infinity with the sample size, estimators built out of such statistics form an intermediate family in between those constructed in the block maxima and peaks-over-threshold frameworks in extreme value analysis. The asymptotic normality of extreme U-statistics based on location-scale invariant kernels is established. Although the asymptotic variance coincides with the one of the Hájek proj…
In the recent information-theoretic literature, the concept of extropy has been studied for order statistics. In the present communication we consider a cumulative analogue of extropy in the same vein of cumulative residual (past) entropy and study it in context with extreme order statistics. A dynamic version of cumulative residual (past) extropy for smallest (largest) order statistic is also studied here. It is shown that the proposed measures (and their dynamic versions) of extreme order statistics determine the distribution uniquely. Some characterizations of the generalized Pareto and pow…
For a bivariate random vector (X,Y), symmetry conditions are presented that yield stochastic orderings among |X|, |Y|, |max(X,Y)|, and | min(X, Y)|. Partial extensions of these results for multivariate random vectors (X1,...,Xn) are also given.
We calculate finite sample and asymptotic distributions for the largest censored and uncensored survival times, and some related statistics, from a sample of survival data generated according to an iid censoring model. These statistics are important for assessing whether there is sufficient followup in the sample to be confident of the presence of immune or cured individuals in the population. A key structural result obtained is that, conditional on the value of the largest uncensored survival time, and knowing the number of censored observations exceeding this time, the sample partitions into…
Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile cal…
In the context of right-censored and interval-censored data we develop asymptotic formulas to compute pseudo-observations for the survival function and the Restricted Mean Survival Time (RMST). Those formulas are based on the original estimators and do not involve computation of the jackknife estimators. For right-censored data, Von Mises expansions of the Kaplan-Meier estimator are used to derive the pseudo-observations. For interval-censored data, a general class of parametric models for the survival function is studied. An asymptotic representation of the pseudo-observations is derived invo…
Recent advances in extreme value theory have established $\ell$-Pareto processes as the natural limits for extreme events defined in terms of exceedances of a risk functional. Here we provide methods for the practical modelling of data based on a tractable yet flexible dependence model. We introduce the class of elliptical $\ell$-Pareto processes, which arise as the limit of threshold exceedances of certain elliptical processes characterized by a correlation function and a shape parameter. An efficient inference method based on maximizing a full likelihood with partial censoring is developed. …
The inactivity time, or lost lifespan specifically for mortality data, concerns time from occurrence of an event of interest to the current time point and has recently emerged as a new summary measure for cumulative information inherent in time-to-event data. This summary measure provides several benefits over the traditional methods, including more straightforward interpretation yet less sensitivity to heavy censoring. However, there exists no systematic modeling approach to inferring the quantile inactivity time in the literature. In this paper, we propose a regression method for the quantil…
We present new estimators for the statistical analysis of the dependence of the mean gap time length between consecutive recurrent events, on a set of explanatory random variables and in the presence of right censoring. The dependence is expressed through regression-like and overdispersion parameters, estimated via conditional estimating equations. The mean and variance of the length of each gap time, conditioned on the observed history of prior events and other covariates, are known functions of parameters and covariates. Under certain conditions on censoring, we construct normalized estimati…
In this paper, we focus on stochastic comparisons of extreme order statistics stemming from multiple-outlier scale models with dependence. Archimedean copula is used to model dependence structure among nonnegative random variables. Sufficient conditions are obtained for comparison of the largest order statistics in the sense of the usual stochastic, reversed hazard rate, star and Lorenz orders. The smallest order statistics are also compared with respect to the usual stochastic, hazard rate, star and Lorenz orders. To illustrate the theoretical establishments, some examples are provided.
Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail-dependence. In areas such as the modeling of financial and insurance risks, or as the modeling of spatial variables, extreme value models in high dimensions (up to fifty or more) with their statistical inference procedures are needed. In this paper, we consider max-stable models for which the spectral random vectors have absolutely continuous distributions. For random samples with max-stable distributions we provide quasi-explicit analytical expre…
The analysis left truncated and right censored data is very common in survival and reliability analysis. In lifetime studies patients often subject to left truncation in addition to right censoring. For example, in bone marrow transplant studies based on International Bone Marrow Transplant Registry (IBMTR), the patients who die while waiting for the transplants will not be reported to the IBMTR. In this paper, we develop novel U-statistics under left truncation and right censoring. We prove the $\sqrt{n}$-consistency of the proposed U-statistics. We derive the asymptotic distribution of the U…
In this paper we consider the statistical inference of the unknown parameter of an exponential distribution based on the time truncated data. The time truncated data occurs quite often in the reliability analysis for type-I or hybrid censoring cases. All the results available today are based on the conditional argument that at least one failure occurs during the experiment. In this paper we provide some inferential results based on the unconditional argument. We extend the results for some two-parameter distributions also.
