Reduction Algorithm for the NPMLE for the Distribution Function of Bivariate Interval-Censored Data

This article considers computational aspects of the nonparametric maximum likelihood estimator (NPMLE) for the distribution function of bivariate interval-censored data. The computation of the NPMLE consists of a parameter reduction step and an optimization step. This article focuses on the reduction step and introduces two new reduction algorithms: the Tree algorithm and the HeightMap algorithm. The Tree algorithm is mentioned only briefly. The HeightMap algorithm is discussed in detail and also given in pseudo code. It is a fast and simple algorithm of time complexityO(n²). This is an order faster than the best known algorithm thus far by Bogaerts and Lesaffre. We compare the new algorithms to earlier algorithms in a simulation study, and demonstrate that the new algorithms are significantly faster. Finally, we discuss how the HeightMap algorithm can be generalized to d-dimensional data with d > 2. Such a multivariate version of the HeightMap algorithm has time complexity O(n^d).     

An Efficient Implementation of the EMICM Algorithm for the Interval Censored NPMLE

The EMICM algorithm is an established method for computing the interval-censored NPMLE, a generalization of the Kaplan Meier curves for interval censored data. The novel contribution in this work is an efficient implementation, allowing each iteration to be computed in linear time. Using simulated data, it is shown that this new implementation is significantly faster than alternative EMICM implementations or other competing algorithms, allowing for analyses of datasets orders of magnitude larger than previously available.     

A bivariate interval censorship model for partnership formation

We consider a statistical problem of estimating a bivariate age distribution of newly formed partnership. The study is motivated by a type of data that consist of uncensored, right-censored, left-censored, interval-censored and missing observations in the coordinates of a bivariate random vector. A model is proposed for formulating such type of data. A feasible algorithm to estimate the generalized MLE (GMLE) of the bivariate distribution function is also proposed. We establish asymptotic properties for the GMLE under a discrete assumption on the underlying distributions and apply the method to the data set.     

Computationally Efficient Estimation for the Generalized Odds Rate Mixture Cure Model With Interval-Censored Data

For semiparametric survival models with interval-censored data and a cure fraction, it is often difficult to derive nonparametric maximum likelihood estimation due to the challenge in maximizing the complex likelihood function. In this article, we propose a computationally efficient EM algorithm, facilitated by a gamma-Poisson data augmentation, for maximum likelihood estimation in a class of generalized odds rate mixture cure (GORMC) models with interval-censored data. The gamma-Poisson data augmentation greatly simplifies the EM estimation and enhances the convergence speed of the EM algorithm. The empirical properties of the proposed method are examined through extensive simulation studies and compared with numerical maximum likelihood estimates. An R package “GORCure” is developed to implement the proposed method and its use is illustrated by an application to the Aerobic Center Longitudinal Study dataset. Supplementary material for this article is available online.     

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

This paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.     

Computational Algorithms for Censored-Data Problems Using Intersection Graphs

This article presents methods for finding the nonparametric maximum likelihood estimate (NPMLE) of the distribution function of time-to-event data. The basic approach is to use graph theory (in particular intersection graphs) to simplify the problem. Censored data can be represented in terms of their intersection graph. Existing combinatorial algorithms can be used to find the important structures, namely the maximal cliques. When viewed in this framework there is no fundamental difference between right censoring, interval censoring, double censoring, or current status data and hence the algorithms apply to all types of data. These algorithms can be extended to deal with bivariate data and indeed there are no fundamental problems extending the methods to higher dimensional data. Finally this article shows how to obtain the NPMLE using convex optimization methods and methods for mixing distributions. The implementation of these methods is greatly simplified through the graph-theoretic representation of the data.     

比例优势模型在多结局区间截断数据上的应用

探索比例优势模型在临床医学中常见的多结局区间截断数据中的应用.用条件的逻辑回归方法避免讨厌参数的估计,用牛顿-拉普森算法估计回归系数,用＂夹心方差＂估计量作为参数方差的估计.通过随机模型检验模型应用的有效性.     

A modification of profile empirical likelihood for the exponential-tilt model

We consider semiparametric models whose infinite-dimensional parameter corresponds to a probability distribution. The NPMLE based on the profile empirical likelihood for this kind of semiparametric model has attracted considerable interest. We propose the use of a modified profile empirical likelihood to improve the accuracy of this estimation. We consider applications to the exponential-tilt model and show that the accuracy of the proposed estimator is better than that of the conventional NPMLE by numerical study.     

Nonparametric estimation of interval-censored failure time data in the presence of informative censoring

Nonparametric estimation of a survival function is one of the most commonly asked questions in the analysis of failure time data and for this, a number of procedures have been developed under various types of censoring structures (Kalbfleisch and Prentice, 2002). In particular, several algorithms are available for interval-censored failure time data with independent censoring mechanism (Sun, 2006; Turnbull, 1976). In this paper, we consider the interval-censored data where the censoring mechanism may be related to the failure time of interest, for which there does not seem to exist a nonparametric estimation procedure. It is well-known that with informative censoring, the estimation is possible only under some assumptions. To attack the problem, we take a copula model approach to model the relationship between the failure time of interest and censoring variables and present a simple nonparametric estimation procedure. The method allows one to conduct a sensitivity analysis among others.     

Asymptotic distribution of the nonparametric distribution estimator based on a martingale approach in doubly censored data

For analysis of time-to-event data with incomplete information beyond right-censoring, many generalizations of the inference of the distribution and regression model have been proposed. However, the development of martingale approaches in this area has not progressed greatly, while for right-censored data such an approach has spread widely to study the asymptotic properties of estimators and to derive regression diagnosis methods. In this paper, focusing on doubly censored data, we discuss a martingale approach for inference of the nonparametric maximum likelihood estimator (NPMLE). We formulate a martingale structure of the NPMLE using a score function of the semiparametric profile likelihood. Finally, an expression of the asymptotic distribution of the NPMLE is derived more conveniently without depending on an infinite matrix expression as in previous research. A further useful point is that a variance-covariance formula of the NPMLE computable in a larger sample is obtained as an empirical version of the limit form presented here.     

Generalized MLE of a Joint Distribution Function with Multivariate Interval-Censored Data

We consider the problem of estimation of a joint distribution function of a multivariate random vector with interval-censored data. The generalized maximum likelihood estimator of the distribution function is studied and its consistency and asymptotic normality are established under the case 2 multivariate interval censorship model and discrete assumptions on the censoring random vectors.     

Inferences based on a bivariate distribution with von Mises marginals

There is very little literature concerning modeling the correlation between paired angular observations. We propose a bivariate model with von Mises marginal distributions. An algorithm for generating bivariate angles from this von Mises distribution is given. Maximum likelihood estimation is then addressed. We also develop a likelihood ratio test for independence in paired circular data. Application of the procedures to paired wind directions is illustrated. Employing simulation, using the proposed model, we compare the power of the likelihood ratio test with six existing tests of independence.     

On Convergence of Convex Minorant Algorithms for Distribution Estimation with Interval-Censored Data

Abstract

Local convergence results of the convex minorant (CM) algorithm to obtain the nonparametric maximum-likelihood estimator of a distribution under interval-censored observations are given. We also provide a variation of the CM algorithm, which yields global convergence. The algorithm is illustrated with data on AIDS survival time in 92 members of the U.S. Air Force.     

Modeling heterogeneity for bivariate survival data by the compound Poisson distribution with random scale

We propose a bivariate Weibull regression model with heterogeneity (frailty or random effect) which is generated by compound Poisson distribution with random scale. We assume that the bivariate survival data follow bivariate Weibull of Hanagal (2004). There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations which motivate us to study this particular model. We propose a two stage maximum likelihood estimation procedure for the parameters in the proposed model and develop large sample tests for testing significance of regression parameters.     

Hazard Estimation With Bivariate Survival Data and Copula Density Estimation

Bivariate survival function can be expressed as the composition of marginal survival functions and a bivariate copula and, consequently, one may estimate bivariate hazard functions via marginal hazard estimation and copula density estimation. Leveraging on earlier developments on penalized likelihood density and hazard estimation, a nonparametric approach to bivariate hazard estimation is being explored in this article. The new ingredient here is the nonparametric estimation of copula density, a subject of interest by itself, and to accommodate survival data one needs to allow for censoring and truncation in the setting. A simple copularization process is implemented to convert density estimates into copula densities, and a cross-validation scheme is devised for density estimation under censoring and truncation. Empirical performances of the techniques are investigated through simulation studies, and potential applications are illustrated using real-data examples and open-source software.     

Principal Component Analysis of Two-Dimensional Functional Data

This article presents and compares two approaches of principal component (PC) analysis for two-dimensional functional data on a possibly irregular domain. The first approach applies the singular value decomposition of the data matrix obtained from a fine discretization of the two-dimensional functions. When the functions are only observed at discrete points that are possibly sparse and may differ from function to function, this approach incorporates an initial smoothing step prior to the singular value decomposition. The second approach employs a mixed effects model that specifies the PC functions as bivariate splines on triangulations and the PC scores as random effects. We apply the thin-plate penalty for regularizing the function estimation and develop an effective expectation–maximization algorithm for calculating the penalized likelihood estimates of the parameters. The mixed effects model-based approach integrates scatterplot smoothing and functional PC analysis in a unified framework and is shown in a simulation study to be more efficient than the two-step approach that separately performs smoothing and PC analysis. The proposed methods are applied to analyze the temperature variation in Texas using 100 years of temperature data recorded by Texas weather stations. Supplementary materials for this article are available online.     

Skewed bivariate models and nonparametric estimation for the CTE risk measure

In this paper, we illustrate the use of the Conditional Tail Expectation (CTE) risk measure on a set of bivariate real data consisting of two types of auto insurance claim costs. Several continuous bivariate distributions (normal, lognormal, skew-normal with the alternative log-skew-normal) are fitted to the data. Besides, a bivariate nonparametric transformed kernel estimation is presented. CTE formulas are given for all these, and numerical results on the real data are discussed and compared.     

Skewed bivariate models and nonparametric estimation for the CTE risk measure

In this paper, we illustrate the use of the Conditional Tail Expectation (CTE) risk measure on a set of bivariate real data consisting of two types of auto insurance claim costs. Several continuous bivariate distributions (normal, lognormal, skew-normal with the alternative log-skew-normal) are fitted to the data. Besides, a bivariate nonparametric transformed kernel estimation is presented. CTE formulas are given for all these, and numerical results on the real data are discussed and compared.     

Projective interpolation of polynomial vectors and improved key recovery attack on SFLASH

SFLASH is an instance of the famous C* \(^{-}\) multivariate public key cryptographic schemes and it was chosen by the NESSIE cryptographic project of the European Consortium in 2003 as a candidate signature algorithm used for digital signatures on limited-resource devices. Recently, a successful private key recovery attack on SFLASH was proposed by Bouillaguet, Fouque and Macario-Rat by uncovering the kernel properties of quadratic forms of the central map. The most expensive step in the attack is the calculation of kernel vectors of skew-symmetric matrices over a bivariate polynomial ring. Bouillaguet et al. proposed two methods to accomplish this computation. Both methods involve symbolic computation on bivariate polynomials. The first method computes characteristic polynomials of matrices of polynomials and is very expensive. The second method involves a Gröbner basis computation and so its complexity is difficult to estimate. In this paper, we show this critical step of calculating kernel vectors can be done by numerical computation on field elements instead of symbolic computation. Our method uses a nondeterministic interpolation of polynomial vectors called projective interpolation, and its complexity can be explicitly evaluated. Experiments show that it is much faster, making the total attack on SFLASH about 30 times faster (the critical step is about 100 times faster) than the first method of Bouillaguet et al. The new method is also slighter faster than their second method.     

Indirect assessment of the bivariate survival function

Estimating the bivariate survival function has been a major goal of many researchers. For that purpose many methods and techniques have been published. However, most of these techniques and methods rely heavily on bivariate failure data. There are situations in which failure time data are difficult to obtain and thus there is a growing need to assess the bivariate survival function for such cases. In this paper we propose two techniques for generating families of bivariate processes for describing several variables that can be used to indirectly assess the bivariate survival function. An estimation procedure is provided and a simulation study is conducted to evaluate the performance of our proposed estimator.