Semiparametric regression for restricted mean residual life under right censoring

A mean residual life function (MRLF) is the remaining life expectancy of a subject who has survived to a certain time point. In the presence of covariates, regression models are needed to study the association between the MRLFs and covariates. If the survival time tends to be too long or the tail is not observed, the restricted mean residual life must be considered. In this paper, we propose the proportional restricted mean residual life model for fitting survival data under right censoring. For inference on the model parameters, martingale estimating equations are developed, and the asymptotic properties of the proposed estimators are established. In addition, a class of goodness-of-fit test is presented to assess the adequacy of the model. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and the approach is applied to a set of real life data collected from a randomized clinical trial.     

Aspects of the mean residual life order for weighted distributions

In this paper, we first provide conditions for preservation of the mean residual life (mrl) order under weighting. Then we apply the obtained results to establish our results about preservation of the decreasing mrl class by weighted distributions. In addition, we present some results for comparing the original random variable to its weighted version in terms of the mrl order. Also, some examples are given to illustrate the results.     

A class of mean residual life regression models with censored survival data

When describing a failure time distribution, the mean residual life is sometimes preferred to the survival or hazard rate. Regression analysis making use of the mean residual life function has recently drawn a great deal of attention. In this paper, a class of mean residual life regression models are proposed for censored data, and estimation procedures and a goodness-of-fit test are developed. Both asymptotic and finite sample properties of the proposed estimators are established, and the proposed methods are applied to a cancer data set from a clinic trial.     

An Additive–Multiplicative Restricted Mean Residual Life Model

The mean residual life measures the expected remaining life of a subject who has survived up to a particular time. When survival time distribution is highly skewed or heavy tailed, the restricted mean residual life must be considered. In this paper, we propose an additive–multiplicative restricted mean residual life model to study the association between the restricted mean residual life function and potential regression covariates in the presence of right censoring. This model extends the proportional mean residual life model using an additive model as its covariate dependent baseline. For the suggested model, some covariate effects are allowed to be time‐varying. To estimate the model parameters, martingale estimating equations are developed, and the large sample properties of the resulting estimators are established. In addition, to assess the adequacy of the model, we investigate a goodness of fit test that is asymptotically justified. The proposed methodology is evaluated via simulation studies and further applied to a kidney cancer data set collected from a clinical trial.     

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.     

Weighted exponential distribution: properties and different methods of estimation

In this article, we investigate various properties and methods of estimation of the Weighted Exponential distribution. Although, our main focus is on estimation (from both frequentist and Bayesian point of view) yet, the stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics are derived first time for the said distribution. Different types of loss functions are considered for Bayesian estimation. Furthermore, the Bayes estimators and their respective posterior risks are computed and compared using Gibbs sampling. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and two real data sets have been analysed for illustrative purposes.     

Semiparametric Proportional Mean Residual Life Model With Censoring Indicators Missing at Random

For right-censored survival data, the information that whether the observed time is survival or censoring time is frequently lost. This is the case for the competing risk data. In this article, we consider statistical inference for the right-censored survival data with censoring indicators missing at random under the proportional mean residual life model. Simple and augmented inverse probability weighted estimating equation approaches are developed, in which the nonmissingness probability and some unknown conditional expectations are estimated by the kernel smoothing technique. The asymptotic properties of all the proposed estimators are established, while extensive simulation studies demonstrate that our proposed methods perform well under the moderate sample size. At last, the proposed method is applied to a data set from a stage II breast cancer trial.     

A New Kernel Distribution Function Estimator Based on a Non-parametric Transformation of the Data

Abstract.  A new kernel distribution function (df) estimator based on a non-parametric transformation of the data is proposed. It is shown that the asymptotic bias and mean squared error of the estimator are considerably smaller than that of the standard kernel df estimator. For the practical implementation of the new estimator a data-based choice of the bandwidth is proposed. Two possible areas of application are the non-parametric smoothed bootstrap and survival analysis. In the latter case new estimators for the survival function and the mean residual life function are derived.     

Semiparametric median residual life model and inference

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada     

On estimating the mean of the selected normal population in two-stage adaptive designs

Adaptive design is widely used in clinical trials. In this paper, we consider the problem of estimating the mean of the selected normal population in two-stage adaptive designs. Under the LINEX and L₂ loss functions, admissibility and minimax results are derived for some location invariant estimators of the selected normal mean. The naive sample mean estimator is shown to be inadmissible under the LINEX loss function and to be not minimax under both loss functions.     

Reliability Studies of the Log-Exponential Inverse Gaussian Distribution

In this article, we investigate the monotonicity of the density, failure rate, and mean residual life functions of the log-exponential inverse Gaussian distribution. It turns out that, in this case, the monotonicity of the density, failure rate, and mean residual life functions take different forms depending on the range of the parameters. Maximum likelihood estimators of the critical points of the density, failure rate, and mean residual life functions of the model are evaluated using Monte Carlo simulations. An example of a published data set is used to illustrate the estimation of the critical points.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

Nonparametric inference on mean residual life function with length-biased right-censored data

Abstract

In survival or reliability studies, the mean residual life (MRL) function is an important characteristic in understanding the survival or ageing process. In this article, we consider the problem of nonparametric MRL function estimation with length-biased right-censored data. Two nonparametric estimators of the MRL are proposed and their weak convergence is presented. In order to evaluate the performance of these estimators, small Monte Carlo simulations are carried out. Results show that the proposed estimators work well especially when the sample size is small and their calculations are simple. Finally, a real data example is provided.     

Estimators based on ranks for arma models

In this paper we introduce a new family of robust estimators for ARMA models. These estimators are defined by replacing the residual sample autocovariances in the least squares equations by autocovariances based on ranks. The asymptotic normality of the proposed estimators is provided. The efficiency and robustness properties of these estimators are studied. An adequate choice of the score functions gives estimators which have high efficiency under normality and robustness in the presence of outliers. The score functions can also be chosen so that the resulting estimators are asymptotically as efficient as the maximum likelihood estimators for a given distribution.     

Nonparametric smooth estimation of the expected inactivity time function

The expected inactivity time (EIT) function (also known as the mean past lifetime function) is a well known reliability function which has application in many disciplines such as survival analysis, actuarial studies and forensic science, to name but a few. In this paper, we use a fixed design local polynomial fitting technique to obtain estimators for the EIT function when the lifetime random variable has an unknown distribution. It will be shown that the proposed estimators are asymptotically unbiased, consistent and also, when standardized, has an asymptotic normal distribution. An optimal bandwidth, which minimizes the AMISE (asymptotic mean integrated squared error) of the estimator, is derived. Numerical examples based on simulated samples from various lifetime distributions common in reliability studies will be presented to evaluate the performances of these estimators. Finally, three real life applications will also be presented to further illustrate the wide applicability of these estimators.     

Nonparametric Tests for Comparing Two Mean Residual Life Functions

We present nonparametric tests for the comparison of two mean residual life functions. We propose a new graphical approach for the simultaneous comparison of two mean residual life functions and their corresponding failure rate functions. We also consider the problem of testing against crossing mean residual life functions.     

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.     

Generalized ratio-type and ratio-exponential-type estimators for population mean under modified Horvitz-Thompson estimator in adaptive cluster sampling

In the present article, we propose the generalized ratio-type and generalized ratio-exponential-type estimators for population mean in adaptive cluster sampling (ACS) under modified Horvitz-Thompson estimator. The proposed estimators utilize the auxiliary information in combination of conventional measures (coefficient of skewness, coefficient of variation, correlation coefficient, covariance, coefficient of kurtosis) and robust measures (tri-mean, Hodges-Lehmann, mid-range) to increase the efficiency of the estimators. Properties of the proposed estimators are discussed using the first order of approximation. The simulation study is conducted to evaluate the performances of the estimators. The results reveal that the proposed estimators are more efficient than competing estimators for population mean in ACS under both modified Hansen-Hurwitz and Horvitz-Thompson estimators.     

Improved precision in the analysis of randomized trials with survival outcomes,without assuming proportional hazards

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan–Meier and inverse probability weighted estimators, under identifiability conditions; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan–Meier estimator) when either the censoring or survival distributions, conditional on covariates, are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan–Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)–(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.

     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.