Nonparametric Estimation of the Average Availability

The average availability of a repairable system is the expected proportion of time that the system is operating in the interval [0, t]. The present article discusses the nonparametric estimation of the average availability when (i) the data on ‘n’ complete cycles of system operation are available, (ii) the data are subject to right censorship, and (iii) the process is observed upto a specified time ‘T’. In each case, a nonparametric confidence interval for the average availability is also constructed. Simulations are conducted to assess the performance of the estimators.     

A self-consistent estimator of marginal survival functions based on dependent competing risk data and an assumed copula

When the time to death, X, and the time to censoring, Y, are associated some additional information is need to identify the marginal survival functions. A natural function which provides this additional information is the copula of X and Y. Assuming that the copula is known, we use the notion of self consistency to construct an estimator of the marginal survival functions based on dependent competing risk data. Results of a small simulation study are shown to compare this estimator to other estimators of the marginal survival function based on an assumed copula.     

Empirical Likelihood for Nonparametric Models Under Linear Process Errors

In this paper, we study the construction of confidence intervals for a nonparametric regression function under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically distributed. The result is used to obtain EL based confidence intervals for the nonparametric regression function. The finite‐sample performance of the method is evaluated through a simulation study.     

Nonparametric estimation of renewal processes from count data

Abstract: The authors address the problem of estimating an inter‐event distribution on the basis of count data. They derive a nonparametric maximum likelihood estimate of the inter‐event distribution utilizing the EM algorithm both in the case of an ordinary renewal process and in the case of an equilibrium renewal process. In the latter case, the iterative estimation procedure follows the basic scheme proposed by Vardi for estimating an inter‐event distribution on the basis of time‐interval data; it combines the outputs of the E‐step corresponding to the inter‐event distribution and to the length‐biased distribution. The authors also investigate a penalized likelihood approach to provide the proposed estimation procedure with regularization capabilities. They evaluate the practical estimation procedure using simulated count data and apply it to real count data representing the elongation of coffee‐tree leafy axes.     

Nonparametric estimation of the bivariate cdf for arbitrarily censored data

Right, left or interval censored multivariate data can be represented by an intersection graph. Focussing on the bivariate case, the authors relate the structure of such an intersection graph to the support of the nonparametric maximum likelihood estimate (NPMLE) of the cumulative distribution function (CDF) for such data. They distinguish two types of non‐uniqueness of the NPMLE: representational, arising when the likelihood is unaffected by the distribution of the estimated probability mass within regions, and mixture, arising when the masses themselves are not unique. The authors provide a brief overview of estimation techniques and examine three data sets.     

Nonparametric Bayes estimation of gap-time distribution with recurrent event data

Nonparametric Bayes (NPB) estimation of the gap-time survivor function governing the time to occurrence of a recurrent event in the presence of censoring is considered. In our Bayesian approach, the gap-time distribution, denoted by F, has a Dirichlet process prior with parameter α. We derive NPB and nonparametric empirical Bayes (NPEB) estimators of the survivor function F?=1?F and construct point-wise credible intervals. The resulting Bayes estimator of F? extends that based on single-event right-censored data, and the PL-type estimator is a limiting case of this Bayes estimator. Through simulation studies, we demonstrate that the PL-type estimator has smaller biases but higher root-mean-squared errors (RMSEs) than those of the NPB and the NPEB estimators. Even in the case of a mis-specified prior measure parameter α, the NPB and the NPEB estimators have smaller RMSEs than the PL-type estimator, indicating robustness of the NPB and NPEB estimators. In addition, the NPB and NPEB estimators are smoother (in some sense) than the PL-type estimator.     

Nonparametric estimation of the survival distribution in censored data

Two nonparametric estimators o f the survival distributionare discussed. The estimators were proposed by Kaplan and Meier (1958) and Breslow (1972) and are applicable when dealing with censored data. It is known that they are asymptotically unbiased and uniformly strongly consistent, and when properly normalized that they converge weakly to the same Gaussian process. In this paper, the properties of the estimators are carefully inspected in small or moderate samples. The Breslow estimator, a shrinkage version of the Kaplan-Meier, nearly always has the smaller mean square error (MSE) whenever the truesurvival probabilityis at least 0.20, but has considerably larger MSE than the Kaplan-Meier estimator when the survivalprobability is near zero.     

Nonparametric estimation of the ROC curve for length-biased and right-censored data

Abstract

ROC curve is a fundamental evaluation tool in medical researches and survival analysis. The estimation of ROC curve has been studied extensively with complete data and right-censored survival data. However, these methods are not suitable to analyze the length-biased and right-censored data. Since this kind of data includes the auxiliary information that truncation time and residual time share the same distribution, the two new estimators for the ROC curve are proposed by taking into account this auxiliary information to improve estimation efficiency. Numerical simulation studies with different assumed cases and real data analysis are conducted.     

Nonparametric Estimation of Bounded Survival Functions with Censored Observations

Stochastic ordering of survival functions is a useful concept in many areas of statistics, especially in nonparametric and order restricted inferences. In this paper we introduce an algorithm to compute maximum likelihood estimates of survival functions where both upper and lower bounds are given. The algorithm allows censored survival data. In a simulation study, we found that the proposed estimates are more efficient than the unrestricted Kaplan-Meier product limit estimates both with and without censored observations.     

Estimation of Distribution Functions Using Data from Different Environments

Suppose that when a unit operatesin a certain environment, its lifetime has distribution G,and when the unit operates in another environment, its lifetimehas a different distribution, say F. Moreover, supposethe unit is operated for a certain period of time in the firstenvironment and is then transferred to the second environment.Thus we observe a censored lifetime in the first environmentand a failure time of a ``used' unit in the second environment.We propose an EM algorithm approach for obtaining a self-consistentestimator of F. Moreover, suppose using observations from both environments.The case where failure times are subject to right censoring isconsidered as well. We also establish the maximum likelihoodestimator of F. Moreover, suppose when the unit is repairable. Applicationand simulation studies are presented to illustrate the methodsderived.     

Nonparametric confidence intervals for Tmax in sequence-stratified crossover studies

Tmax is the time associated with the maximum serum or plasma drug concentration achieved following a dose. While Tmax is continuous in theory, it is usually discrete in practice because it is equated to a nominal sampling time in the noncompartmental pharmacokinetics approach. For a 2-treatment crossover design, a Hodges-Lehmann method exists for a confidence interval on treatment differences. For appropriately designed crossover studies with more than two treatments, a new median-scaling method is proposed to obtain estimates and confidence intervals for treatment effects. A simulation study was done comparing this new method with two previously described rank-based nonparametric methods, a stratified ranks method and a signed ranks method due to Ohrvik. The Normal theory, a nonparametric confidence interval approach without adjustment for periods, and a nonparametric bootstrap method were also compared. Results show that less dense sampling and period effects cause increases in confidence interval length. The Normal theory method can be liberal (i.e. less than nominal coverage) if there is a true treatment effect. The nonparametric methods tend to be conservative with regard to coverage probability and among them the median-scaling method is least conservative and has shortest confidence intervals. The stratified ranks method was the most conservative and had very long confidence intervals. The bootstrap method was generally less conservative than the median-scaling method, but it tended to have longer confidence intervals. Overall, the median-scaling method had the best combination of coverage and confidence interval length. All methods performed adequately with respect to bias.     

Constrained estimation and likelihood intervals for censored data

The authors propose a reduction technique and versions of the EM algorithm and the vertex exchange method to perform constrained nonparametric maximum likelihood estimation of the cumulative distribution function given interval censored data. The constrained vertex exchange method can be used in practice to produce likelihood intervals for the cumulative distribution function. In particular, the authors show how to produce a confidence interval with known asymptotic coverage for the survival function given current status data.     

Nonparametric estimation of varying-coefficient single-index models

The varying-coefficient single-index model has two distinguishing features: partially linear varying-coefficient functions and a single-index structure. This paper proposes a nonparametric method based on smoothing splines for estimating varying-coefficient functions and an unknown link function. Moreover, the average derivative estimation method is applied to obtain the single-index parameter estimates. For interval inference, Bayesian confidence intervals were obtained based on Bayes models for varying-coefficient functions and the link function. The performance of the proposed method is examined both through simulations and by applying it to Boston housing data.     

Identity for the NPMLE in Censored Data Models

We derive an identity for nonparametric maximum likelihood estimators (NPMLE) and regularized MLEs in censored data models which expresses the standardized maximum likelihood estimator in terms of the standardized empirical process. This identity provides an effective starting point in proving both consistency and efficiency of NPMLE and regularized MLE. The identity and corresponding method for proving efficiency is illustrated for the NPMLE in the univariate right-censored data model, the regularized MLE in the current status data model and for an implicit NPMLE based on a mixture of right-censored and current status data. Furthermore, a general algorithm for estimation of the limiting variance of the NPMLE is provided. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interval estimation of the negative binomial dispersion parameter

Inference concerning the negative binomial dispersion parameter, denoted by c, is important in many biological and biomedical investigations. Properties of the maximum-likelihood estimator of c and its bias-corrected version have been studied extensively, mainly, in terms of bias and efficiency [W.W. Piegorsch, Maximum likelihood estimation for the negative binomial dispersion parameter, Biometrics 46 (1990), pp. 863–867; S.J. Clark and J.N. Perry, Estimation of the negative binomial parameter κ by maximum quasi-likelihood, Biometrics 45 (1989), pp. 309–316; K.K. Saha and S.R. Paul, Bias corrected maximum likelihood estimator of the negative binomial dispersion parameter, Biometrics 61 (2005), pp. 179–185]. However, not much work has been done on the construction of confidence intervals (C.I.s) for c. The purpose of this paper is to study the behaviour of some C.I. procedures for c. We study, by simulations, three Wald type C.I. procedures based on the asymptotic distribution of the method of moments estimate (mme), the maximum-likelihood estimate (mle) and the bias-corrected mle (bcmle) [K.K. Saha and S.R. Paul, Bias corrected maximum likelihood estimator of the negative binomial dispersion parameter, Biometrics 61 (2005), pp. 179–185] of c. All three methods show serious under-coverage. We further study parametric bootstrap procedures based on these estimates of c, which significantly improve the coverage probabilities. The bootstrap C.I.s based on the mle (Boot-MLE method) and the bcmle (Boot-BCM method) have coverages that are significantly better (empirical coverage close to the nominal coverage) than the corresponding bootstrap C.I. based on the mme, especially for small sample size and highly over-dispersed data. However, simulation results on lengths of the C.I.s show evidence that all three bootstrap procedures have larger average coverage lengths. Therefore, for practical data analysis, the bootstrap C.I. Boot-MLE or Boot-BCM should be used, although Boot-MLE method seems to be preferable over the Boot-BCM method in terms of both coverage and length. Furthermore, Boot-MLE needs less computation than Boot-BCM.     

Nonparametric equivalence testing with respect to the median difference

The problem of comparing two independent groups of univariate data in the sense of testing for equivalence is considered for a fully nonparametric setting. The distribution of the data within each group may be a mixture of both a continuous and a discrete component, and no assumptions are made regarding the way in which the distributions of the two groups of data may differ from each other – in particular, the assumption of a shift model is avoided. The proposed equivalence testing procedure for this scenario refers to the median of the independent difference distribution, i.e. to the median of the differences between independent observations from the test group and the reference group, respectively. The procedure provides an asymptotic equivalence test, which is symmetric with respect to the roles of ‘test’ and ‘reference’. It can be described either as a two‐one‐sided‐tests (TOST) approach, or equivalently as a confidence interval inclusion rule. A one‐sided variant of the approach can be applied analogously to non‐inferiority testing problems. The procedure may be generalised to equivalence testing with respect to quantiles other than the median, and is closely related to tolerance interval type inference. Copyright © 2009 John Wiley & Sons, Ltd.