Nonparametric estimation of the conditional mean residual life function with censored data

The conditional mean residual life (MRL) function is the expected remaining lifetime of a system given survival past a particular time point and the values of a set of predictor variables. This function is a valuable tool in reliability and actuarial studies when the right tail of the distribution is of interest, and can be more informative than the survivor function. In this paper, we identify theoretical limitations of some semi-parametric conditional MRL models, and propose two nonparametric methods of estimating the conditional MRL function. Asymptotic properties such as consistency and normality of our proposed estimators are established. We investigate via simulation study the empirical properties of the proposed estimators, including bootstrap pointwise confidence intervals. Using Monte Carlo simulations we compare the proposed nonparametric estimators to two popular semi-parametric methods of analysis, for varying types of data. The proposed estimators are demonstrated on the Veteran’s Administration lung cancer trial.     

Nonparametric analysis of recurrent gap time data

Abstract

Recurrent event data are frequently encountered in longitudinal studies. In many applications, the times between successive recurrent events (gap times) are often of interest and lead to problems that have received much attention recently. In this article, using the approach of inverse probability-of-censoring weights (IPCW), we propose nonparametric estimators for the estimation of the bivariate distribution and survival functions for gap times of recurrent event data. We also consider the estimation of Kendall’s tau for two gap times by expressing it as an integral functional of the bivariate survival function. The asymptotic properties of the proposed estimators are established. Simulation studies are conducted to investigate their finite sample performance.     

Nonparametric estimation of the mean function of a stochastic process with missing observations

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.     

A Pointwise Estimator for the k-Fold Convolution of a Distribution Function

ABSTRACT

This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.     

Statistical Analysis of General Degradation Path Model and Failure Time Data with Multiple Failure Modes

ABSTRACT

This article considers degradation and failure time models with multiple failure modes which used to study the problem of longevity and aging in survival analysis and reliability. Degradation process is modeled using general nonparametric, nonlinear path models. Semi-parametric models for the intensities of the traumatic failures are used supposing that these intensities depend on degradation level. Semi-parametric estimators of various reliability characteristics are proposed and asymptotic properties of the estimators are obtained. The theoretical results are illustrated using simulated data.     

Oracally efficient spline-backfitted kernel smoothing of additive partial linear measurement error model

Abstract

We consider statistical inference for additive partial linear models when the linear covariate is measured with error. A bias-corrected spline-backfitted kernel smoothing method is proposed. Under mild assumptions, the proposed component function and parameter estimator are oracally efficient and fast to compute. The nonparametric function estimator’s pointwise distribution is asymptotically equivalent to an function estimator in partial linear model. Finite-sample performance of the proposed estimators is assessed by simulation experiments. The proposed methods are applied to Boston house data set.     

Nonparametric estimators for quantile density function under length-biased sampling

Abstract

In this article, the strong uniform consistency of two nonparametric estimators for the quantile density function is established under length-biased sampling. The rate of the strong approximation of the resulting processes of these estimators will be presented as well. A Monte Carlo simulation study is carried out to compare the proposed estimators with each other in terms of mean squared errors.     

Jackknife Estimation for Smooth Functions of the Parametric Component in Partially Linear Regression Models

Abstract

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Nonparametric estimation with left-truncated and right-censored data when the sample size before truncation is known

In this article, we consider nonparametric estimation of the survival function when the data are subject to left-truncation and right-censoring and the sample size before truncation is known. We propose two estimators. The first estimator is derived based on a self-consistent estimating equation. The second estimator is obtained by using the constrained expectation-maximization algorithm. Simulation results indicate that both estimators are more efficient than the product-limit estimator. When there is no censoring, the performance of the proposed estimators is compared with that of the estimator proposed by Li and Qin [Semiparametric likelihood-based inference for biased and truncated data when total sample size is known, J. R. Stat. Soc. B 60 (1998), pp. 243–254] via simulation study.     

Nonparametric estimation of the bivariate survival function for one modified form of current-status data

In this article, the estimation of the bivariate survival function for one modified form of current-status data is considered. Two types of estimators, which are generalizations of the estimators by Campbell and Földes [G. Campbell and A. Földes, Large sample properties of nonparametric statistical inference, in Nonparametric Statistical Inference, B.V. Gnredenko, M.L. Puri, and I. Vineze, eds., North-Holland, Amsterdam, 1982, pp. 103–122] and Dabrowska [D.M. Dabrowska, Kaplan-Meier estimate on the plane, Ann. Stat. 18 (1988), pp. 1475–1489; D.M. Dabrowska, Kaplan-Meier estimate on the plane: weak convergence, LIL, and the bootstrap, J. Multivariate Anal. 29 (1989), pp. 308–325], are proposed. The consistency of the proposed estimators is established. A simulation study is conducted to investigate the performance of the proposed estimators.     

A class of nonparametric bivariate survival function estimators for randomly censored and truncated data

This paper proposes a class of nonparametric estimators for the bivariate survival function estimation under both random truncation and random censoring. In practice, the pair of random variables under consideration may have certain parametric relationship. The proposed class of nonparametric estimators uses such parametric information via a data transformation approach and thus provides more accurate estimates than existing methods without using such information. The large sample properties of the new class of estimators and a general guidance of how to find a good data transformation are given. The proposed method is also justified via a simulation study and an application on an economic data set.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

Interval estimation of the joint survival function for successive duration times under left truncation and right censoring

In incident cohort studies, survival data often include subjects who have had an initiate event at recruitment and may potentially experience two successive events (first and second) during the follow-up period. Since the second duration process becomes observable only if the first event has occurred, left truncation and dependent censoring arise if the two duration times are correlated. To confront the two potential sampling biases, we propose two inverse-probability-weighted (IPW) estimators for the estimation of the joint survival function of two successive duration times. One of them is similar to the estimator proposed by Chang and Tzeng [Nonparametric estimation of sojourn time distributions for truncated serial event data – a weight adjusted approach, Lifetime Data Anal. 12 (2006), pp. 53–67]. The other is the extension of the nonparametric estimator proposed by Wang and Wells [Nonparametric estimation of successive duration times under dependent censoring, Biometrika 85 (1998), pp. 561–572]. The weak convergence of both estimators are established. Furthermore, the delete-one jackknife and simple bootstrap methods are used to estimate standard deviations and construct interval estimators. A simulation study is conducted to compare the two IPW approaches.     

Nonparametric maximum likelihood estimators for ar and ma time series

The problem of estimating the parameters of moving average or autoregressive time series is studied when the error distribution is completely unknown. Four nonparametric maximum likelihood estimators (NPMLE) are presented for this purpose. These estimators are compared with the classical moment and least squares estimators in a simulation study. The behavior of these NPMLEs is much better than the classical ones, suggesting that they should be used extensively when no parametric information is known in advance about the error distribution. An application of these estimators to coal mining accidents data is also included.     

Partial deconvolution estimation in nonparametric regression

In this article, we propose a class of partial deconvolution kernel estimators for the nonparametric regression function when some covariates are measured with error and some are not. The estimation procedure combines the classical kernel methodology and the deconvolution kernel technique. According to whether the measurement error is ordinarily smooth or supersmooth, we establish the optimal local and global convergence rates for these proposed estimators, and the optimal bandwidths are also identified. Furthermore, lower bounds for the convergence rates of all possible estimators for the nonparametric regression functions are developed. It is shown that, in both the super and ordinarily smooth cases, the convergence rates of the proposed partial deconvolution kernel estimators attain the lower bound. The Canadian Journal of Statistics 48: 535–560; 2020 © 2020 Statistical Society of Canada     

Analysis of left censored data from the generalized exponential distribution

The generalized exponential distribution proposed by Gupta and Kundu [Gupta, R.D and Kundu, D., 1999, Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173–188.] is an important lifetime distribution in survival analysis. In this paper, we consider the maximum likelihood estimation procedure of the parameters of the generalized exponential distribution when the data are left censored. We obtain the maximum likelihood estimators of the unknown para-meters and the Fisher information matrix. Simulation studies are carried out to observe the performance of the estimators in small sample.     

A comparison of estimators for the mean in the inverse gaussian distribution with a known coefficient of variation

In this paper, we discuss an estimation problem of the mean in the inverse Gaussian distribution with a known coefficient of variation. Two types of linear estimators for the mean, the linear minimum variance unbiased estimator and the linear minimum mean squared error estimator, are constructed by using the squared error loss function and their properties are examined. It is observed that, for small samples the performance of the proposed estimators is better than that of the maximum likelihood estimator, when the coefficient of variation is large.     

Maximum Likelihood Estimations and EM Algorithms with Length-biased Data

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.     

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.