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The gamma-frailty Poisson model for the nonparametric estimation of panel count data

BIOMETRICS 2003年第4期59卷 1099-1106页

作者： Zhang, Y Jamshidian, M Univ Cent Florida Dept Stat & Actuarial Sci Orlando FL 32816 USA Calif State Univ Dept Math Fullerton CA 92834 USA

In this article, we study nonparametric estimation of the mean function of a counting process with panel observations. We introduce the gamma frailty variable to account for the intracorrelation between the panel counts of the counting process and construct a maximum pseudo-likelihood estimate with the frailty variable. Three simulated examples are given to show that this estimation procedure, while preserving the robustness and simplicity of the computation, improves the efficiency of the nonparametric maximum pseudo-likelihood estimate studied in Wellner and Zhang (2000, Annals of Statistics 28, 779-814). A real example from a bladder tumor study is used to illustrate the method.

关键词： iterative convex minorant algorithm EM algorithm isotonic regression Monte-Carlo nonparametric maximum pseudo-likelihood estimator

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Dimension-reduced nonparametric maximum likelihood computation for interval-censored data

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2008年第5期52卷 2388-2402页

作者： Wang, Yong Univ Auckland Dept Stat Auckland 1142 New Zealand

A general technique is proposed for efficient computation of the nonparametric maximum likelihood estimate (NPMLE) of a survival function. The main idea is to include a new support interval that has the largest gradient value between inclusively every two neighbouring support intervals in the support set at each iteration. It is thus able to expand the support set exponentially fast during the initial stage of computation and tends to produce the same support set of the NPMLE afterward. The use of the proposed technique needs to be combined with an algorithm that can effectively find and remove redundant support intervals, for example, the constrained Newton method, the iterative convex minorant algorithm and the subspace-based Newton method. Numerical studies show that the dimension-reducing technique works very well, especially for purely interval-censored data, where a significant computational improvement via dimension reduction is possible. Strengths and weaknesses of various algorithms are also discussed and demonstrated. (c) 2007 Elsevier B.V. All rights reserved.

关键词： censored data constrained Newton method iterative convex minorant algorithm nonparametric maximum likelihood subspace-based Newton method survival function

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Estimation of the incubation time distribution for COVID-19

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STATISTICA NEERLANDICA 2021年第2期75卷 161-179页

作者： Groeneboom, Piet Delft Univ Technol Delft Inst Appl Math Delft Netherlands

We consider smooth nonparametric estimation of the incubation time distribution of COVID-19, in connection with the investigation of researchers from the National Institute for Public Health and the Environment (Dutch: RIVM) of 88 travelers from Wuhan: Backer et al. (2020). The advantages of the smooth nonparametric approach with respect to the parametric approach, using three parametric distributions (Weibull, log-normal and gamma) in Backer et al. (2020) is discussed. It is shown that the typical rate of convergence of the smooth estimate of the density is n(2/7) in a continuous version of the model, where n is the sample size. The (nonsmoothed) nonparametric maximum likelihood estimator itself is computed by the iterative convex minorant algorithm (Groeneboom and Jongbloed (2014)). All computations are available as R scripts in Groeneboom (2020a).

关键词： incubation time iterative convex minorant algorithm nonparametric MLE smooth nonparametric density estimation Weibull distribution

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A k-sample test with interval censored data

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BIOMETRIKA 2006年第2期93卷 315-328页

作者： Yuen, Kam-Chuen Shi, Jian Zhu, Lixing Univ Hong Kong Dept Stat & Actuarial Sci Hong Kong Hong Kong Peoples R China Chinese Acad Sci Acad Math & Syst Sci Beijing Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

The problem of testing for the equality of k distribution functions under Case 2 interval censoring is studied and a supremum-type test statistic is proposed based on the differences between the nonparametric maximum likelihood estimator and the so-called leveraged bootstrap estimator of the k underlying distributions. The proposed test is distributionfree and consistent against all alternatives. As the main results hold for a wide range of resampling sizes, a data-driven method is suggested for determining the size of each leveraged bootstrap sample. Another advantage of the test is that it can detect different distributions with equal means or heavy crossover. Simulation studies indicate that the test performs quite well with a moderate sample size. Finally, a slightly modified version of the test is applied to breast cosmesis data.

关键词： goodness of fit interval censoring iterative convex minorant algorithm leveraged bootstrap nonparametric maximum likelihood

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Computing maximum likelihood estimators of a log-concave density function

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JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION 2007年第7期77卷 561-574页

作者： Rufibach, Kaspar Univ Bern Inst Math Stat & Actuarial Sci CH-3012 Bern Switzerland

We consider the problem of estimating a density function that is assumed to be log- concave. This semi-parametric model includes many well- known parametric classes;such as Normal, Gamma, Laplace, Logistic, Beta or Extreme value distributions, for specific parameter ranges. It is known that the maximum likelihood estimator for the log- density is always a piecewise linear function with at most as many knots as observations, but typically much less. We show that this property can be exploited to design a linearly constrained optimization problem whose iteratively calculated solution yields the estimator. We compare several standard and one recently proposed algorithm regarding their performance on this problem.

关键词： non-parametric density estimation shape restriction interior point method iterative convex minorant algorithm Newton method on subspace

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Inference and Modeling with Log-concave Distributions

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STATISTICAL SCIENCE 2009年第3期24卷 319-327页

作者： Walther, Guenther Stanford Univ Dept Stat Stanford CA 94305 USA

Log-concave distributions are an attractive choice for modeling and inference, for several reasons: The class of log-concave distributions contains most of the commonly used parametric distributions and thus is a rich and flexible nonparametric class of distributions. Further, the MLE exists and can be computed with readily available algorithms. Thus, no tuning parameter, such as a bandwidth, is necessary for estimation. Due to these attractive properties, there has been considerable recent research activity concerning the theory and applications of log-concave distributions. This article gives a review of these results.

关键词： Nonparametric density estimation shape constraint log-concave density Polya frequency function strongly unimodal iterative convex minorant algorithm active set algorithm

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Saddlepoint approximations for rank-invariant permutation tests and confidence intervals with interval-censoring

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CANADIAN JOURNAL OF STATISTICS-REVUE CANADIENNE DE STATISTIQUE 2014年第2期42卷 308-324页

作者： Abd-Elfattah, Ehab Butler, Ronald Ain Shams Univ Fac Educ Dept Math Cairo Egypt So Methodist Univ Dallas TX 75275 USA

Interval-censored data occur when subjects are assessed by using regular follow-up. In such instances, we consider rank-invariant permutation tests to test the significance of a treatment versus a control. For a wide class of such tests, which includes the Peto & Peto class, we present saddlepoint approximations for the exact permutation mid-P-values which achieve extremely small relative errors. The speed and stability of these saddlepoint computations make them practicable for inverting the permutation tests and we compute nominal 100(1-)% confidence intervals for the treatment effect. Such confidence intervals are of substantial clinical importance since, more than simply stating the level of statistical significance, they quantify the significant benefit of the treatment by providing a confidence interval for the percentage increase in mean (or median) treatment survival time as compared to control. Our methodology makes heavy use of nonparametric MLEs (NPMLEs) for survival functions and some limitations of existing algorithms, such as the hybrid ICM algorithm, are noted and accommodated. The Canadian Journal of Statistics 42: 308-324;2014 (c) 2014 Statistical Society of Canada ResumeLes donnees obtenues lorsque des sujets sont evalues lors de visites regulieres sont souvent censurees par intervalle. Dans de telles circonstances, les auteurs etudient des tests de permutation invariants aux transformations preservant les rangs pour tester la significativite d'un traitement par rapport au controle. Pour une large classe de tests, incluant la classe de Peto & Peto, ils proposent des approximations en point de selle pour le p-value central qui presentent une erreur relative tres faible. La vitesse et la stabilite du calcul de ces p-value centraux de permutation en font une methode applicable pour inverser les tests de permutation et en tirer des intervalles de confiance a 100(1-)% pour la taille de l'effet du traitement. D'une importance clinique incontestable, de t

关键词： Expectation maximization algorithm interval censoring iterative convex minorant algorithm permutation test saddlepoint approximation two-sample test

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On algorithms for the nonparametric maximum likelihood estimator of the failure function with censored data

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2004年第1期13卷 123-140页

作者： Zhang, Y Jamshidian, M Univ Cent Florida Dept Stat & Actuarial Sci Orlando FL 32816 USA Calif State Univ Fullerton Dept Math Fullerton CA 92834 USA

In this article, we study algorithms for computing the nonparametric maximum likelihood estimator (NPMLE) of the failure function with two types of censored data: doubly censored data and (type 2) interval-censored data. We consider two projection methods, namely the iterative convex minorant algorithm (ICM) and a generalization of the Rosen algorithm (GR) and compare these methods to the well-known EM algorithm. The comparison conducted via simulation studies shows that the hybrid algorithms that alternately use the EM and GR for doubly censored data or, alternately, use the EM and ICM for (type 2) interval-censored data appear to be much more efficient than the EM, especially in large sample situation.

关键词： double censoring EM algorithm gradient projection algorithm interval censoring iterative convex minorant algorithm Rosen method

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Efficient algorithms for computing the non and semi-parametric maximum likelihood estimates with panel count data

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JOURNAL OF NONPARAMETRIC STATISTICS 2011年第2期23卷 567-579页

作者： Cheng, Gang Zhang, Ying Lu, Liqiang Univ Iowa Dept Biostat Iowa City IA 52242 USA Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Shanghai Key Lab Contemporary Appl Math Shanghai 200433 Peoples R China

Nonparametric and semi-parametric analysis of panel count data have recently been active research topics in statistical literature. The maximum likelihood method based on the non-homogeneous Poisson process has been proved an efficient inference procedure for such analysis. However, computing the non- and semi-parametric maximum likelihood estimates (MLEs) can be very intensive numerically and the available methods are not efficient. In this manuscript, we develop an efficient numerical algorithm stemming from the Newton-Raphson method to compute the non- and semi-parametric MLEs for panel count data. Simulation studies are carried out to demonstrate the numerical efficiency of the proposed algorithm compared to the existing methods in the literature.

关键词： quadratic programming panel count data isotonic regression iterative convex minorant algorithm Monte-Carlo

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