Semiparametric simultaneous confidence bands for the difference of survival functions

In the analysis of censored survival data, simultaneous confidence bands are useful devices to help determine the efficacy of a treatment over a control. Semiparametric confidence bands are developed for the difference of two survival curves using empirical likelihood and compared with the nonparametric counterpart. Simulation studies are presented to show that the proposed semiparametric approach is superior, with the new confidence bands giving empirical coverage closer to the nominal level. Further comparisons reveal that the semiparametric confidence bands are tighter and, hence, more informative. For censoring rates between 10 and 40 %, the semiparametric confidence bands provide a relative reduction in enclosed area amounting to between 2 and 10 % over their nonparametric bands, with increased reduction attained for higher censoring rates. The methods are illustrated using an University of Massachusetts AIDS data set.     

Families of smooth confidence bands for the survival function under the general random censorship model

Randomly right censored data often arise in industrial life testing and clinical trials. Several authors have proposed asymptotic confidence bands for the survival function when data are randomly censored on the right. All of these bands are based on the empirical estimator of the survival function. In this paper, families of asymptotic (1-)100% level confidence bands are developed from the smoothed estimate of the survival function under the general random censorship model. The new bands are compared to empirical bands, and it is shown that for small sample sizes, the smooth bands have a higher coverage probability than the empirical counterparts.     

Optimal distribution-free confidence bands for a distribution function

Distribution-free confidence bands for a distribution function are typically obtained by inverting a distribution-free hypothesis test. We propose an alternate strategy in which the upper and lower bounds of the confidence band are chosen to minimize a narrowness criterion. We derive necessary and sufficient conditions for optimality with respect to such a criterion, and we use these conditions to construct an algorithm for finding optimal bands. We also derive uniqueness results, with the Brunn–Minkowski Inequality from the theory of convex bodies playing a key role in this work. We illustrate the optimal confidence bands using some galaxy velocity data, and we also show that the optimal bands compare favorably to other bands both in terms of power and in terms of area enclosed.     

An improved survival estimator for censored medical costs with a kernel approach

Cost assessment serves as an essential part in economic evaluation of medical interventions. In many studies, costs as well as survival data are frequently censored. Standard survival analysis techniques are often invalid for censored costs, due to the induced dependent censoring problem. Owing to high skewness in many cost data, it is desirable to estimate the median costs, which will be available with estimated survival function of costs. We propose a kernel-based survival estimator for costs, which is monotone, consistent, and more efficient than several existing estimators. We conduct numerical studies to examine the finite-sample performance of the proposed estimator.     

A nonparametric test for diagnosis of the proportionality assumption

We propose a nonparametric test for diagnosis of the proportionality assumption between hazard functions based on a functional equation. Because of involvement of censoring distribution, we consider the test procedure in an asymptotic manner and obtain the asymptotic normality for the proposed test statistic. Then we discuss the rationale of use of the functional equation for the initial effect model. Finally we compare our test with others using an example.     

A Bayesian nonparametric estimator of a multivariate survival function

Using reinforced processes related to beta-Stacy process and generalized Pólya urn scheme jointly with a structure assumption about dependence, a Bayesian nonparametric prior and a predictive estimator for a multivariate survival function are provided. This estimator can be computed through an easy implementation of a Gibbs sampler algorithm. Moreover consistency of the estimator is studied.     

Familles of confidence bands for the survival function under the general random censorship model and the Koziol-Green model

Familles of asymptotic 100(1 – α)% level confidence bands for the survival function under the general random right-censorship (GRC) model and the proportional-hazards model of random right-censorship, also known as the Koziol-Green (KG) model, are developed. The family of bands under the GRC model is based on the well-known product-limit estimator (PLE), and this family is rich in that it contains as special cases the bands of Hall and Wellner (1980) and Gillespie and Fisher (1979), and more generally, the GF-type and HW-type bands of Csörg? and Horváth (1986), as well as new bands not previously studied. The familles of bands under the KG model are based on the maximum-likelihood estimator of F under this particular model. We compare the PLE-based bands and the MLE-based bands under the KG model. This enables us to study the loss in efficiency of the former bands when used in a setting where they are not optimal. The notion of asymptotic relative width efficiency (ARWE), defined to be the limiting ratio of the sample sizes needed by the bands to achieve the same asymptotic widths, is employed to compare two bands. Through this efficiency measure it is shown that if the censoring parameter β is known, then the PLE-based bands are highly inefficient relative to the MLE-based bands when β is large. When β is not known, the MLE-based bands are asymptotically conservative. Despite their conservatism, they still dominate the PLE-based bands when β is not too small or equivalently when the degree of censoring is not too light. We also compare the various PLE-based bands under the GRC model. The resulting information is valuable for evaluating competing PLE-based bands. We illustrate the confidence bands by utilizing the well-known Channing House data.     

Bivariate frailty model for the analysis of multivariate survival time

Because of limitations of the univariate frailty model in analysis of multivariate survival data, a bivariate frailty model is introduced for the analysis of bivariate survival data. This provides tremendous flexibility especially in allowing negative associations between subjects within the same cluster. The approach involves incorporating into the model two possibly correlated frailties for each cluster. The bivariate lognormal distribution is used as the frailty distribution. The model is then generalized to multivariate survival data with two distinguished groups and also to alternating process data. A modified EM algorithm is developed with no requirement of specification of the baseline hazards. The estimators are generalized maximum likelihood estimators with subject-specific interpretation. The model is applied to a mental health study on evaluation of health policy effects for inpatient psychiatric care.     

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.     

Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter

In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.     

Comparison of renewal function estimators for censored data

The computation of the renewal function when the distribution function is completely known has received much attention in the literature. However, in many cases the form of the distribution function is unknown and has to be estimated nonparametrically. A nonparametric estimator for the renewal function for complete data was suggested by Frees (1986). In many cases, however, censoring of the lifetime might occur. We shall present parametric and nonparametric estimators of the renewal function based on censored data. In a simulation study we compare the nonparametric estimators with parametric estimators for the Weibull and lognormal distribution. The study suggests that the nonparametric estimator is a viable alternative to the parametric estimators when the lifetime distribution is unknown. Also, the nonparametric estimator is computationally simpler than the parametric estimator.     

Deficiency of the mle of a smooth survival function under the proportional hazard model

The problem of estimating a smooth distribution function F at a point t is treated under the proportional hazard model of random censorship. It is shown that a certain class of properly chosen kernel type estimator of F asymptotically perform better than the maximum likelihood estimator. It is shown that the relative deficiency of the maximum likelihood estimator of F under the proportional hazard model with respect to the properly chosen kernel type estimator tends to infinity as the sample size tends to infinity.     

On the distribution of a test for exponentiality based on progressively type-II right censored spacings

A test for exponentiality based on progressively Type-II right censored spacings has been proposed recently by Balakrishnan et al. (2002). They derived the asymptotic null distribution of the test statistic. In this work, we utilize the algorithm of Huffer and Lin (2001) to evaluate the exact null probabilities and the exact critical values of this test statistic.     

Confidence bands for the bias of linear estimators of a function of one variable

The choice of a parametric family of curves used in fitting a response in dependence of an exploratory variable is always debatable, since the fitted curve is merely an approximation of the true response curve. The present paper discusses the computation of (simultaneous) confidence bands for the bias of a fitted curve. Our development is based on integral expressions for the bias using derivatives of the unknown response curve. Such formulas can be used to convert confidence bands on the derivative(s) into confidence band(s) for the bias.