Singly and Doubly Censored Current Status Data: Estimation, Asymptotics and Regression

In biostatistical applications interest often focuses on the estimation of the distribution of time between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well-understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, with its distribution over an observed interval [A, B] known; the data is referred to as doubly censored current status data. This model has applications in AIDS partner studies. If the initial time is known to be uniformly distribute d, the model reduces to a submodel of the current status model with the same asymptotic information bounds as in the current status model, but the distribution of interest is essentially the derivative of the distribution of interest in the current status model. As a consequence the non-parametric maximum likelihood estimator is inconsistent. Moreover, this submodel contains only smooth heavy tailed distributions for which no moments exist. In this paper, we discuss the connection between the singly censored current status model and the doubly censored current status model (for the uniform initial time) in detail and explain the difficulties in estimation which arise in the doubly censored case. We propose a regularized MLE corresponding with the current status model. We prove rate results, efficiency of smooth functionals of the regularized MLE, and present a generally applicable efficient method for estimation of regression parameters, which does not rely on the existence of moments. We also discuss extending these ideas to a non-uniform distribution for the initial time.     

A new algorithm for maximum likelihood estimation in normal scale-mixture generalized autoregressive conditional heteroskedastic models

In this paper, we propose a new generalized autoregressive conditional heteroskedastic (GARCH) model using infinite normal scale-mixtures which can suitably avoid order selection problems in the application of finite normal scale-mixtures. We discuss its theoretical properties and develop a two-stage algorithm for the maximum likelihood estimator to estimate the mixing distribution non-parametric maximum likelihood estimator (NPMLE) as well as GARCH parameters (two-stage MLE). For the estimation of a mixing distribution, we employ a fast computational algorithm proposed by Wang [On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. J R Stat Soc Ser B. 2007;69:185–198] under the gradient characterization of the non-parametric mixture likelihood. The GARCH parameters are then estimated either using the expectation-mazimization algorithm or general optimization scheme. In addition, we propose a new forecasting algorithm of value-at-risk (VaR) using the two-stage MLE and the NPMLE. Through a simulation study and real data analysis, we compare the performance of the two-stage MLE with the existing ones including quasi-maximum likelihood estimator based on the standard normal density and the finite normal mixture quasi maximum estimated-likelihood estimator (cf. Lee S, Lee T. Inference for Box–Cox transformed threshold GARCH models with nuisance parameters. Scand J Stat. 2012;39:568–589) in terms of the relative efficiency and accuracy of VaR forecasting.     

Consistency of the NPML Estimator in the Right-Censored Transformation Model

Abstract.  This paper studies the representation and large-sample consistency for non-parametric maximum likelihood estimators (NPMLEs) of an unknown baseline continuous cumulative-hazard-type function and parameter of group survival difference, based on right-censored two-sample survival data with marginal survival function assumed to follow a transformation model, a slight generalization of the class of frailty survival regression models. The paper's main theoretical results are existence and unique a.s. limit, characterized variationally, for large data samples of the NPMLE of baseline nuisance function in an appropriately defined neighbourhood of the true function when the group difference parameter is fixed, leading to consistency of the NPMLE when the difference parameter is fixed at a consistent estimator of its true value. The joint NPMLE is also shown to be consistent. An algorithm for computing it numerically, based directly on likelihood equations in place of the expectation-maximization (EM) algorithm, is illustrated with real data.     

A Nonparametric Estimator of Survival Functions for Arbitrarily Truncated and Censored Data

It is well-known that the nonparametric maximum likelihood estimator (NPMLE) may severely under-estimate the survival function with left truncated data. Based on the Nelson estimator (for right censored data) and self-consistency we suggest a nonparametric estimator of the survival function, the iterative Nelson estimator (INE), for arbitrarily truncated and censored data, where only few nonparametric estimators are available. By simulation we show that the INE does well in overcoming the under-estimation of the survival function from the NPMLE for left-truncated and interval-censored data. An interesting application of the INE is as a diagnostic tool for other estimators, such as the monotone MLE or parametric MLEs. The methodology is illustrated by application to two real world problems: the Channing House and the Massachusetts Health Care Panel Study data sets.     

The Two-interval Line-segment Problem

In this paper we define and study the non-parametric maximum likelihood estimator (NPMLE) in the one-dimensional line-segment problem, where we observe line-segments on the real line through an interval with a gap which is smaller than the two remaining intervals. We define the self-consistency equations for the NPMLE and provide a quick algorithm for solving them. We prove supremum norm weak convergence to a Gaussian process and efficiency of the NPMLE. The problem has a geological application in the study of the lifespan of species     

The NPMLE for Doubly Censored Current Status Data

In biostatistical applications interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, but with its distribution over an observed interval ' A, B ' known to be uniformly distributed; the data is referred to as doubly censored current status data. These authors used this model to handle application in AIDS partner studies focusing on the NPMLE of the distribution G of T . The model is a submodel of the current status model, but the distribution G is essentially the derivative of the distribution of interest F in the current status model. In this paper we establish that the NPMLE of G is uniformly consistent and that the resulting estimators for the n ^1/2-estimable parameters are efficient. We propose an iterative weighted pool-adjacent-violator-algorithm to compute the estimator. It is also shown that, without smoothness assumptions, the NPMLE of F converges at rate n ^−2/5 in L ²-norm while the NPMLE of F in the non-parametric current status data model converges at rate n ^−1/3 in L ²-norm, which shows that there is a substantial gain in using the submodel information.     

ASYMPTOTIC DEFICIENCY OF THE JACKKNIFE ESTIMATOR

In this paper it is shown that the bias-adjusted maximum likelihood estimator (MLE) is asymptotically equivalent to the jackknife estimator in the variance up to the order n^-1 and the asymptotic deficiency of the jackknife estimator relative to the bias-adjusted MLE is equal to zero.     

A new three-parameter extension of the log-logistic distribution with applications to survival data

In this article, a new three-parameter extension of the two-parameter log-logistic distribution is introduced. Several distributional properties such as moment-generating function, quantile function, mean residual lifetime, the Renyi and Shanon entropies, and order statistics are considered. The estimation of the model parameters for complete and right-censored cases is investigated competently by maximum likelihood estimation (MLE). A simulation study is conducted to show that these MLEs are consistent in moderate samples. Two real datasets are considered; one is a right-censored data to show that the proposed model has a superior performance over several existing popular models.     

Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.     

Firth adjustment for Weibull current-status survival analysis

Abstract

Analysis of right-censored data is problematic due to infinite maximum likelihood estimates (MLE) and potentially biased estimates, especially for small numbers of events. Analyzing current-status data is especially troublesome because of the extreme loss of precision due to large failure intervals. We extend Firth’s method for regular parametric problems to current-status modeling with the Weibull distribution. Firth advocated a bias reduction method for MLE by systematically correcting the score equation. An advantage is that it is still applicable when the MLE does not exist. We present simulation studies and two illustrative analyses involving RFM mice lung tumor data.     

On the Restricted Liu Estimator in the Logistic Regression Model

The logistic regression model is used when the response variables are dichotomous. In the presence of multicollinearity, the variance of the maximum likelihood estimator (MLE) becomes inflated. The Liu estimator for the linear regression model is proposed by Liu to remedy this problem. Urgan and Tez and Mansson et al. examined the Liu estimator (LE) for the logistic regression model. We introduced the restricted Liu estimator (RLE) for the logistic regression model. Moreover, a Monte Carlo simulation study is conducted for comparing the performances of the MLE, restricted maximum likelihood estimator (RMLE), LE, and RLE for the logistic regression model.     

Performance of nonparametric maximum likelihood estimator in a

When the probability of selecting an individual in a population is propor­tional to its lifelength, it is called length biased sampling. A nonparametric maximum likelihood estimator (NPMLE) of survival in a length biased sam­ple is given in Vardi (1982). In this study, we examine the performance of Vardi's NPMLE in estimating the true survival curve when observations are from a length biased sample. We also compute estimators based on a linear combination (LCE) of empirical distribution function (EDF) estimators and weighted estimators. In our simulations, we consider observations from a mix­ture of two different distributions, one from F and the other from G which is a length biased distribution of F. Through a series of simulations with vari­ous proportions of length biasing in a sample, we show that the NPMLE and the LCE closely approximate the true survival curve. Throughout the sur­vival curve, the EDF estimators overestimate the survival. We also consider a case where the observations are from three different weighted distributions, Again, both the NPMLE and the LCE closely approximate the true distribu­tion, indicating that the length biasedness is properly adjusted for. Finally, an efficiency study shows that Vardi's estimators are more efficient than the EDF estimators in the lower percentiles of the survival curves.     

Estimation of common location parameter of two exponential populations based on records

Consider the problem of estimating the common location parameter of two exponential populations using record data when the scale parameters are unknown. We derive the maximum likelihood estimator (MLE), the modified maximum likelihood estimator (MMLE) and the uniformly minimum variance unbiased estimator (UMVUE) of the common location parameter. Further, we derive a general result for inadmissibility of an equivariant estimator under the scaled-squared error loss function. Using this result, we conclude that the MLE and the UMVUE are inadmissible and better estimators are provided. A simulation study is conducted for comparing the performances of various competing estimators.     

Estimation of the parameters of a truncated gamma distribution

This paper deals with the estimation of the parameters of a truncated gamma distribution over (0,τ), where τ is assumed to be a real number. We obtain a necessary and sufficient condition for the existence of the maximum likelihood estimator(MLE). The probability of nonexistence of MLE is observed to be positive. A simulation study indicates that the modified maximum likelihood estimator and the mixed estimator, which exist with probability one,are to be preferred over MLE. The bias, the mean square error, and the probability of nearness form a basis of our simulation study.     

Estimating the survival function based on the semi-Markov model for dependent censoring

In this paper, we study a nonparametric maximum likelihood estimator (NPMLE) of the survival function based on a semi-Markov model under dependent censoring. We show that the NPMLE is asymptotically normal and achieves asymptotic nonparametric efficiency. We also provide a uniformly consistent estimator of the corresponding asymptotic covariance function based on an information operator. The finite-sample performance of the proposed NPMLE is examined with simulation studies, which show that the NPMLE has smaller mean squared error than the existing estimators and its corresponding pointwise confidence intervals have reasonable coverages. A real example is also presented.     

Computing the log concave NPMLE for interval censored data

In analyzing interval censored data, a non-parametric estimator is often desired due to difficulties in assessing model fits. Because of this, the non-parametric maximum likelihood estimator (NPMLE) is often the default estimator. However, the estimates for values of interest of the survival function, such as the quantiles, have very large standard errors due to the jagged form of the estimator. By forcing the estimator to be constrained to the class of log concave functions, the estimator is ensured to have a smooth survival estimate which has much better operating characteristics than the unconstrained NPMLE, without needing to specify a parametric family or smoothing parameter. In this paper, we first prove that the likelihood can be maximized under a finite set of parameters under mild conditions, although the log likelihood function is not strictly concave. We then present an efficient algorithm for computing a local maximum of the likelihood function. Using our fast new algorithm, we present evidence from simulated current status data suggesting that the rate of convergence of the log-concave estimator is faster (between \(n^{2/5}\) and \(n^{1/2}\)) than the unconstrained NPMLE (between \(n^{1/3}\) and \(n^{1/2}\)).     

Confidence Intervals for Current Status Data

Abstract.  The likelihood ratio statistic for testing pointwise hypotheses about the survival time distribution in the current status model can be inverted to yield confidence intervals (CIs). One advantage of this procedure is that CIs can be formed without estimating the unknown parameters that figure in the asymptotic distribution of the maximum likelihood estimator (MLE) of the distribution function. We discuss the likelihood ratio-based CIs for the distribution function and the quantile function and compare these intervals to several different intervals based on the MLE. The quantiles of the limiting distribution of the MLE are estimated using various methods including parametric fitting, kernel smoothing and subsampling techniques. Comparisons are carried out both for simulated data and on a data set involving time to immunization against rubella. The comparisons indicate that the likelihood ratio-based intervals are preferable from several perspectives.     

Estimation of the parameters in a truncated normal distribution

This paper deals with the estimation of the parameters of doubly truncated and singly truncated normal distributions when truncation points are known. We derive, for these families, a necessary and sufficient condition for the maximum likelihood estimator(MLE) to be finite. Furthermore, the probability of the MLE being infinite is positive. A simulation study for single truncation is carried out to compare the modified maximum likelihood estimator, and the mixed estimator.     

Semi-parametric modelling and likelihood estimation with estimating equations

This paper proposes a semi-parametric modelling and estimating method for analysing censored survival data. The proposed method uses the empirical likelihood function to describe the information in data, and formulates estimating equations to incorporate knowledge of the underlying distribution and regression structure. The method is more flexible than the traditional methods such as the parametric maximum likelihood estimation (MLE), Cox's (1972) proportional hazards model, accelerated life test model, quasi-likelihood (Wedderburn, 1974) and generalized estimating equations (Liang & Zeger, 1986). This paper shows the existence and uniqueness of the proposed semi-parametric maximum likelihood estimates (SMLE) with estimating equations. The method is validated with known cases studied in the literature. Several finite sample simulation and large sample efficiency studies indicate that when the sample size is larger than 100 the SMLE is compatible with the parametric MLE; and in all case studies, the SMLE is about 15% better than the parametric MLE with a mis-specified underlying distribution.     

Life table estimator revisited

We explore the standard life table (actuarial) estimator for grouped right-censored survival data and its extensions in order to consider its relationship with the Kaplan–Meier estimator, and to investigate the critical properties of the extended life table estimators (ELTEs). We discuss certain conditions for the ELTE to be consistent and develop a characterization of the standard life table estimator using the consistency property under any choice of at least two observation times of a finite interval. We also perform a comparative analysis of the ELTEs with the corresponding maximum likelihood estimators for grouped right-censored survival data.