A Spline‐Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval‐Censored Data

Abstract. We propose a spline‐based semiparametric maximum likelihood approach to analysing the Cox model with interval‐censored data. With this approach, the baseline cumulative hazard function is approximated by a monotone B‐spline function. We extend the generalized Rosen algorithm to compute the maximum likelihood estimate. We show that the estimator of the regression parameter is asymptotically normal and semiparametrically efficient, although the estimator of the baseline cumulative hazard function converges at a rate slower than root‐n. We also develop an easy‐to‐implement method for consistently estimating the standard error of the estimated regression parameter, which facilitates the proposed inference procedure for the Cox model with interval‐censored data. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated using data from a breast cosmesis study.     

A Class of Semiparametric Transformation Models for Doubly Censored Failure Time Data

Doubly censored failure time data occur in many areas including demographical studies, epidemiology studies, medical studies and tumorigenicity experiments, and correspondingly some inference procedures have been developed in the literature (Biometrika, 91, 2004, 277; Comput. Statist. Data Anal., 57, 2013, 41; J. Comput. Graph. Statist., 13, 2004, 123). In this paper, we discuss regression analysis of such data under a class of flexible semiparametric transformation models, which includes some commonly used models for doubly censored data as special cases. For inference, the non‐parametric maximum likelihood estimation will be developed and in particular, we will present a novel expectation–maximization algorithm with the use of subject‐specific independent Poisson variables. In addition, the asymptotic properties of the proposed estimators are established and an extensive simulation study suggests that the proposed methodology works well for practical situations. The method is applied to an AIDS study.     

Subgroup Analysis with Time‐to‐Event Data Under a Logistic‐Cox Mixture Model

Subgroup detection has received increasing attention recently in different fields such as clinical trials, public management and market segmentation analysis. In these fields, people often face time‐to‐event data, which are commonly subject to right censoring. This paper proposes a semiparametric Logistic‐Cox mixture model for subgroup analysis when the interested outcome is event time with right censoring. The proposed method mainly consists of a likelihood ratio‐based testing procedure for testing the existence of subgroups. The expectation–maximization iteration is applied to improve the testing power, and a model‐based bootstrap approach is developed to implement the testing procedure. When there exist subgroups, one can also use the proposed model to estimate the subgroup effect and construct predictive scores for the subgroup membership. The large sample properties of the proposed method are studied. The finite sample performance of the proposed method is assessed by simulation studies. A real data example is also provided for illustration.     

Estimation of the Association for Bivariate Interval-censored Failure Time Data

Abstract.  Multivariate failure time data frequently occur in medical studies and the dependence or association among survival variables is often of interest ( Biometrics , 51 , 1995, 1384; Stat. Med. , 18 , 1999, 3101; Biometrika , 87 , 2000, 879; J. Roy. Statist. Soc. Ser. B , 65 , 2003, 257). We study the problem of estimating the association between two related survival variables when they follow a copula model and only bivariate interval-censored failure time data are available. For the problem, a two-stage estimation procedure is proposed and the asymptotic properties of the proposed estimator are established. Simulation studies are conducted to assess the finite sample properties of the presented estimate and the results suggest that the method works well for practical situations. An example from an acquired immunodeficiency syndrome clinical trial is discussed.     

A general quantile residual life model for length‐biased right‐censored data

The quantile residual lifetime function provides comprehensive quantitative measures for residual life, especially when the distribution of the latter is skewed or heavy‐tailed and/or when the data contain outliers. In this paper, we propose a general class of semiparametric quantile residual life models for length‐biased right‐censored data. We use the inverse probability weighted method to correct the bias due to length‐biased sampling and informative censoring. Two estimating equations corresponding to the quantile regressions are constructed in two separate steps to obtain an efficient estimator. Consistency and asymptotic normality of the estimator are established. The main difficulty in implementing our proposed method is that the estimating equations associated with the quantiles are nondifferentiable, and we apply the majorize–minimize algorithm and estimate the asymptotic covariance using an efficient resampling method. We use simulation studies to evaluate the proposed method and illustrate its application by a real‐data example.     

缺失偏态数据下线性回归模型的统计推断

研究缺失偏态数据下线性回归模型的参数估计问题,针对缺失偏态数据,为克服样本分布扭曲缺点和提高模型的回归系数、尺度参数和偏度参数的估计效果,提出了一种适合偏态数据下线性回归模型中缺失数据的修正回归插补方法.通过随机模拟和实例研究,并与均值插补、回归插补、随机回归插补方法比较,结果表明所提出的修正回归插补方法是有效可行的.     

The Additive Nonparametric and Semiparametric Aalen Model as the Rate Function for a Counting Process

We use the additive risk model of Aalen (Aalen, 1980) as a model for the rate of a counting process. Rather than specifying the intensity, that is the instantaneous probability of an event conditional on the entire history of the relevant covariates and counting processes, we present a model for the rate function, i.e., the instantaneous probability of an event conditional on only a selected set of covariates. When the rate function for the counting process is of Aalen form we show that the usual Aalen estimator can be used and gives almost unbiased estimates. The usual martingale based variance estimator is incorrect and an alternative estimator should be used. We also consider the semi-parametric version of the Aalen model as a rate model (McKeague and Sasieni, 1994) and show that the standard errors that are computed based on an assumption of intensities are incorrect and give a different estimator. Finally, we introduce and implement a test-statistic for the hypothesis of a time-constant effect in both the non-parametric and semi-parametric model. A small simulation study was performed to evaluate the performance of the new estimator of the standard error.     

Jackknifing the kaplan-meier survival estimator for censored data: Simulation results and asymptotic analysis

It is shown by simulation t h a t the (arc-sine transformation of the) Kaplan-Meier survival estimator for censored data can be usefully jackknifed to give conservative confidence limits for survival probabilities when samples are small (25 and 50). Mathematical demonstration of the asymptotic, large-sample, validity of the jackknife is included.     

A fast algorithm for univariate log‐concave density estimation

A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate log‐concave density is proposed and studied. It is an extension of the constrained Newton method for nonparametric mixture estimation. In each iteration, the newly extended algorithm includes, if necessary, new knots that are located via a special directional derivative function. The algorithm renews the changes of slope at all knots via a quadratically convergent method and removes the knots at which the changes of slope become zero. Theoretically, the characterisation of the nonparametric maximum likelihood estimate is studied and the algorithm is guaranteed to converge to the unique maximum likelihood estimate. Numerical studies show that it outperforms other algorithms that are available in the literature. Applications to some real‐world financial data are also given.     

Statistical Inference for Single‐index Panel Data Models

We study estimation and hypothesis testing in single‐index panel data models with individual effects. Through regressing the individual effects on the covariates linearly, we convert the estimation problem in single‐index panel data models to that in partially linear single‐index models. The conversion is valid regardless of the individual effects being random or fixed. We propose an estimating equation approach, which has a desirable double robustness property. We show that our method is applicable in single‐index panel data models with heterogeneous link functions. We further design a chi‐squared test to evaluate whether the individual effects are random or fixed. We conduct simulations to demonstrate the finite sample performance of the method and conduct a data analysis to illustrate its usefulness.     

Estimation in semiparametric models using an auxiliary model

Two classes of semiparametric and nonparametric mixture models are defined to represent general kinds of prior information. For these models the nonparametric maximum likelihood estimator (NPMLE) of an unknown probability distribution is derived and is shown to be consistent and relative efficient. Linear functionals are used for the estimation of parameters. Their consistency is proved, the gain of efficiency is derived and asymptotical distributions are given.     

Parameter estimation of three-parameter Weibull distribution based on progressively Type-II censored samples

In this paper, the estimation of parameters for a three-parameter Weibull distribution based on progressively Type-II right censored sample is studied. Different estimation procedures for complete sample are generalized to the case with progressively censored data. These methods include the maximum likelihood estimators (MLEs), corrected MLEs, weighted MLEs, maximum product spacing estimators and least squares estimators. We also proposed the use of a censored estimation method with one-step bias-correction to obtain reliable initial estimates for iterative procedures. These methods are compared via a Monte Carlo simulation study in terms of their biases, root mean squared errors and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of inference developed here.     

A Mixture Model for Longitudinal Partially Ranked Data

This article discusses the use of mixture models in the analysis of longitudinal partially ranked data, where respondents, for example, choose only the preferred and second preferred out of a set of items. To model such data we convert it to a set of paired comparisons. Covariates can be incorporated into the model. We use a nonparametric mixture to account for unmeasured variability in individuals over time. The resulting multi-valued mass points can be interpreted as latent classes of the items. The work is illustrated by two questions on (post)materialism in three sweeps of the British Household Panel Survey.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

A semiparametric approach for modelling multivariate nonlinear time series

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach. The Canadian Journal of Statistics 47: 668–687; 2019 © 2019 Statistical Society of Canada     

Generalized Additive Models for Current Status Data

Current status data arise in studies where the target measurement is the time of occurrence of some event, but observations are limited to indicators of whether or not the event has occurred at the time the sample is collected - only the current status of each individual with respect to event occurrence is observed. Examples of such data arise in several fields, including demography, epidemiology, econometrics and bioassay. Although estimation of the marginal distribution of times of event occurrence is well understood, techniques for incorporating covariate information are not well developed. This paper proposes a semiparametric approach to estimation for regression models of current status data, using techniques from generalized additive modeling and isotonic regression. This procedure provides simultaneous estimates of the baseline distribution of event times and covariate effects. No parametric assumptions about the form of the baseline distribution are required. The results are illustrated using data from a demographic survey of breastfeeding practices in developing countries, and from an epidemiological study of heterosexual Human Immunodeficiency Virus (HIV) transmission. This revised version was published online in July 2006 with corrections to the Cover Date.     

An Additive–Multiplicative Cox–Aalen Regression Model

We propose an additive–multiplicative intensity model that extends the Cox regression model as well as the additive Aalen model. Instead of having a simple baseline intensity the extended model uses an additive Aalen model as its covariate dependent baseline. Approximate maximum likelihood estimators of the baseline intensity functions and the relative risk parameters of the Cox model are suggested by solving the score equations. The derived estimator is efficient. We establish the large sample properties of the estimator. The model provides a simple pragmatic way of including time-varying covariate effects.     

A Likelihood Based Estimating Equation for the Clayton–Oakes Model with Marginal Proportional Hazards

Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In this paper, we consider the Clayton–Oakes model with marginal proportional hazards and use the full model structure to improve on efficiency compared with the independence analysis. We derive a likelihood based estimating equation for the regression parameters as well as for the correlation parameter of the model. We give the large sample properties of the estimators arising from this estimating equation. Finally, we investigate the small sample properties of the estimators through Monte Carlo simulations.     

Mixture Model Analysis of Partially Rank‐Ordered Set Samples: Age Groups of Fish from Length‐Frequency Data

We present a novel methodology for estimating the parameters of a finite mixture model (FMM) based on partially rank‐ordered set (PROS) sampling and use it in a fishery application. A PROS sampling design first selects a simple random sample of fish and creates partially rank‐ordered judgement subsets by dividing units into subsets of prespecified sizes. The final measurements are then obtained from these partially ordered judgement subsets. The traditional expectation–maximization algorithm is not directly applicable for these observations. We propose a suitable expectation–maximization algorithm to estimate the parameters of the FMMs based on PROS samples. We also study the problem of classification of the PROS sample into the components of the FMM. We show that the maximum likelihood estimators based on PROS samples perform substantially better than their simple random sample counterparts even with small samples. The results are used to classify a fish population using the length‐frequency data.