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Information about the age distribution and survival of wild populations is of much interest in ecology and biodemography, but is hard to obtain. Established schemes such as capture-recapture often are not feasible. In the proposed residual demography paradigm, individuals are randomly sampled from the wild population at unknown ages and the resulting captive cohort is reared out in the laboratory until death. Under some basic assumptions one obtains a demographic convolution equation that involves the unknown age distribution of the wild population, the observed survival function of the captive cohort, and the observed survival function of a reference cohort that is independently raised in the laboratory from birth. We adopt a statistical penalized least squares method for the deconvolution of this equation, aiming at extracting the age distribution of the wild population under suitable constraints. Under stationarity of the population, the age density is proportional to the survival function of the wild population and can thus be inferred. Several extensions are discussed. Residual demography is demonstrated for data on fruit flies Bactrocera oleae.

The use of principal component methods to analyze functional data is appropriate in a wide range of different settings. In studies of ``functional data analysis,'' it has often been assumed that a sample of random functions is observed precisely, in the continuum and without noise. While this has been the traditional setting for functional data analysis, in the context of longitudinal data analysis a random function typically represents a patient, or subject, who is observed at only a small number of randomly distributed points, with nonnegligible measurement error. Nevertheless, essentially the same methods can be used in both these cases, as well as in the vast number of settings that lie between them. How is performance affected by the sampling plan? In this paper we answer that question. We show that if there is a sample of $n$ functions, or subjects, then estimation of eigenvalues is a semiparametric problem, with root-$n$ consistent estimators, even if only a few observations are made of each function, and if each observation is encumbered by noise. However, estimation of eigenfunctions becomes a nonparametric problem when observations are sparse. The optimal convergence rates in this case are those which pertain to more familiar function-estimation settings. We also describe the effects of sampling at regularly spaced points, as opposed to random points. In particular, it is shown that there are often advantages in sampling randomly. However, even in the case of noisy data there is a threshold sampling rate (depending on the number of functions treated) above which the rate of sampling (either randomly or regularly) has negligible impact on estimator performance, no matter whether eigenfunctions or eigenvectors are being estimated., Published at http://dx.doi.org/10.1214/009053606000000272 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of $n$ subjects are investigated under mild conditions, as $n\to \infty$, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard $R^2$ to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index., Published at http://dx.doi.org/10.1214/009053605000000660 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Patterns of behavior were recorded every 10 min during a 2-h period each day from eclosion to death for individual Drosophila melanogaster (both sexes) and Ceratitis capitata (males-only) including walking, preening, feeding, flying, and resting for the former species, and walking, calling (signaling), supine (upside-down), and resting in the latter. Results reveal that, with the exception of preening in D. melanogaster, behavioral patterns are age-specific and the frequency of several behaviors (e.g. supine in medfly; walking and resting in D. melanogaster) are correlated with time-to-death. This is the first set of studies to report the age patterns over a range of behavioral categories throughout the lives of individuals and thus the first that systematically documents the behavior of individuals at advanced ages. We suggest that the new and unique behaviors (e.g. supine) that emerge from the aging process be referred to as degenerative behaviors, not only to distinguish them from the conventional behavioral classifications (innate, learned), but also to reflect their emergent nature.

Age of sexual maturity, daily and lifetime reproductive rates, and life span were recorded in a laboratory cohort of Mexican fruit flies consisting of over 1100 females maintained individually. The results revealed that, relative to the medfly, the Mexfly is slower maturing (14 vs 17 days), more fecund (1400 vs 650–1100 eggs/female), and longer lived (50 vs 35 days). The results reinforced the generality of several earlier findings on the medfly including the deceleration of mortality at older ages and the weakness of the correlation between the rate of egg laying at early ages and both subsequent reproduction and remaining longevity. Discussion includes perspectives on the role of artificial selection in shaping the demographic traits of the mass-reared strain of Mexfly used in this study, as well as the overall significance of large scale biodemographic studies in understanding aging and longevity.

Nonparametric approaches have recently emerged as a flexible way to model longitudinal data. This entry reviews some of the common nonparametric approaches to incorporate time and other covariate effects for longitudinally observed response data. Smoothing procedures are invoked to estimate the associated nonparametric functions, but the choice of smoothers can vary and is often subjective. Both fixed and random effects may be included for vector or longitudinal covariates. A closely related type of data is functional data, where the prevailing approaches to model random effects are through functional principal components analysis and B-splines. Related semiparametric regression models also play an increasingly important role. Keywords: functional data analysis; scatter-plot smoother; mean curve; fixed effects; random effects; principal components analysis; semiparametric regression

Summary Lifespan in individually housed medflies (virgins of both sexes) and daily reproduction for females were studied following one of 12 dietary restriction (DR) treatments in which the availability of high-quality food (yeast–sugar mixture) for each fly was based on a Markov chain feeding scheme – a stochastic dietary regime which specifies that the future dietary state depends only on the present dietary state and not on the path by which the present state was achieved. The stochastic treatments consisted of a combination of one of four values of a ‘discovery’ parameter and one of three values of a ‘persistence’ parameter. The results supported the hypotheses that: (i) longevity is extended in most medfly cohorts subject to stochastic DR; and (ii) longevity is more affected by the patch discovery than the patch persistence parameter. One of the main conclusions of the study is that, in combination with the results of earlier dietary restriction studies on the medfly, the results reinforce the concept that the details of the dietary restriction protocols have a profound impact on the sign and magnitude of the longevity extension relative to ad libitum cohorts and that a deeper understanding of the effect of food restriction on longevity is not possible without an understanding of its effect on reproduction.

The purpose of this study was to investigate medfly longevity and reproduction across a broad spectrum of diet restriction using a protocol similar to those applied in most rodent studies. Age-specific reproduction and age of death were monitored for 1200 adult males and 1200 females, each individually maintained on one of 12 diets from ad libitum to 30% of ad libitum. Diet was provided in a fixed volume of solution that was fully consumed each day, ensuring control of total nutrient consumption for every fly. Contrary to expectation and precedence, increased longevity was not observed at any level of diet restriction. Among females, reproduction continued across all diet levels despite the cost in terms of increased mortality. Among males, life expectancy exceeded that of females at most diet levels. However, in both sexes, mortality increased more sharply and the pattern of survival changed abruptly once the diet level fell to 50% of ad libitum or below, even though the energetic demands of egg production has no obvious counterpart in males. We believe that a more complete picture of the life table response to dietary restriction will emerge when studies are conducted on a wider range of species and include both sexes, more levels of diet, and the opportunity for mating and reproduction.

Purpose To determine a possible relationship between donor epithelial status on the first postoperative day after keratoplasty and the eventual health of the corneal surface. Methods We analyzed 91 patients who underwent penetrating corneal transplantation between January 1998 and January 2000, monitoring the epithelial status of the corneas with fluorescein staining using slit-lamp biomicroscopy. Recipient pre- and postoperative variables and donor characteristics were recorded. Macroepithelial defects were classified into three groups according to the extent of the epithelial defect. The results on the first postoperative day were compared with the first and third operative month. Donor and recipient variables were compared with the epithelial status on the first and third month as well. Results On the first postoperative day, 64.84% of the patients had epithelial defects, 10.99% had defects at the 1-month postoperative visit, and none had defects at the third month. Graft recipients with macroepithelial defects in the first postoperative month were older, had a higher prevalence of blepharitis, higher prevalence of inadequate eye hydration, and slightly increased corneal sensation compared with the group without epithelial defects; however, none of these trends were statistically significant. Patients with macroepithelial defects in the first postoperative month received older donor tissue, and the average preservation-to-surgery time was longer. These donor variables, however, were not significant statistically (p value >0.10) in determining outcome of the epithelial status at the first or third months. Conclusions Our results suggest that the epithelial status on the first postoperative day is not predictive of surface integrity at 1-month postoperative (p value is 0.2676 for the likehood ratio test). The epithelial status on the first postoperative day is not predictive of the status of the third month after keratoplasty, because none of the 91 patients had epithelial defects after 3 months.

The relation between fertility and postmenopausal longevity is investigated for a sample of 1635 women from a historical (17th to 18th century) French-Canadian cohort who lived past the age of 50 years. We find that increased fertility is linked to increased rather than decreased postreproductive survival. Postreproductive life expectancy extension is found to be tied to late births. This finding sheds new light on the cost of reproduction and may be viewed as supporting a new paradigm that states that reproductive potential drives remaining longevity. The emerging reproductive potential concept complements the well-established cost of reproduction hypothesis. Alternative explanations for the observed association are also explored. A specific finding is that the degree to which mortality increases for 50-year-old mothers as a result of senescence is closely tied to the logarithm of the age of their youngest child. For example, 50-year-old mothers experience a mortality decrease of 38% and an increase of remaining lifetime of 3.93 years for every 10-fold decrease in the age of their youngest child. This amount of gain in remaining life expectancy would apply to a mother with a two-year-old child as compared with a mother with a 20-year-old offspring. We also find evidence for the existence of vulnerable periods in human life history that are characterized by phases of heightened mortality and are found to be tied to reproduction and senescence.

Mortality rates were measured over the lifetime of 65,000 female Mediterranean fruit flies, Ceratitis capitata , maintained in either all-female (virgin) cages or cages with equal initial numbers of males, to determine the effect of sexual activity and mating on the mortality trajectory of females at older ages. Although a greater fraction of females maintained in all-female (virgin) cages survived to older ages, the life expectancy of the surviving virgins was less than the life expectancy of surviving non-virgins at older ages. This was due to a mortality crossover where virgin flies experience lower mortality than mated flies from eclosion to Day 20 but higher mortality thereafter. These results suggest that there are two consequences of mating—a short-term mortality increase (cost) and a longer term mortality decrease (benefit).

Sex-specific mortality rates of medflies were monitored in cages containing individuals of both sexes and with food (either sugar-only or full diet) removed every 2nd, 3rd, or 4th day (plus ad libitum controls). The general finding is that periodic starvation led to marked oscillations in raw mortality rates. The specific findings are as follows: (i) female medflies live longer than male medflies when they are subjected to periodic starvation; (ii) male medflies maintained on a full diet experience a catastrophic increase in mortality (40%) on the first day food is removed. This mortality surge was not observed for females on either diet or for males maintained on a sugar-only diet; (ii) life expectancy is inversely related to the amplitude of mortality oscillations caused by food deprivation; and (iv) the large perturbations in mortality at younger ages caused by periodic starvation has little effect on the amplitude of mortality at older ages. In general, our data shed new light on the complexity of the mortality response of medflies to both the type and availability of food and thus provide a complimentary perspective to findings from dietary restriction studies on both vertebrate and invertebrate systems.

Without parametric assumptions, high-dimensional regression analysis is already complex. This is made even harder when data are subject to censoring. In this article, we seek ways of reducing the dimensionality of the regressor before applying nonparametric smoothing techniques. If the censoring time is independent of the lifetime, then the method of sliced inverse regression can be applied directly. Otherwise, modification is needed to adjust for the censoring bias. A key identity leading to the bias correction is derived and the root-n consistency of the modified estimate is established. Patterns of censoring can also be studied under a similar dimension reduction framework. Some simulation results and an application to a real data set are reported. 1. Introduction. Survival data are often subject to censoring. When this occurs, the incompleteness of the observed data may induce a substantial bias in the sample. Several approaches have been suggested to overcome the associated difficulties in regression, including the accelerated failure time model, censored linear regression, the Cox proportional hazard model and many others. Survival analysis becomes even more intricate when the dimension of the regressor increases. To apply any of the aforementioned methods, users are required to specify a functional form which relates the outcome variables to the input ones. However, in reality, knowledge needed for an appropriate model specification is often inadequate. As a matter of fact, the acquisition of such information may well turn out to be one of the primary goals of the study itself. Under such circumstances, it seems preferable to have exploratory tools that rely less on such model specification. This is the issue to be addressed in this article. The dimension reduction approach of Li Ž. 1991 will be extended to settings which allow for censoring in the data. We shall offer methods of finding low-dimensional projections of the data for visually examining the censoring pattern. We shall show how censored regression data can still be analyzed without assuming the functional form a priori.

This paper provides a data analysis and some methodological advances which contribute to an ongoing scientific debate about the patterns of aging. One of the problems we address is how to estimate a hazard function when only aggregated information on the lifetimes in the form of a lifetable is available. This problem affects the the estimation of oldest-old mortality which in turn plays an important role in the quantification of biological lifespan and longevity. We illustrate these issues with an analysis of mortality data obtained from cohorts of nematodes. The methods involve data transformation with the aim of bias reduction when estimating the hazard function. We provide rigorous asymptotic results for the smoothing of lifetables and show that the transformation approach is supported by both asymptotic and simulation results. We also demonstrate how the information contained in many samples of lifetables, as typically obtained in aging experiments, can be summarized in a two-dimensional hazard surface.

We discuss various methods which can be employed for the comparative analysis of samples of response curves. In the application discussed here, these curves are hazard functions, each generated by the survival data obtained for a cohort of experimental subjects which are fed a specific diet. It is demonstrated how comparisons of the effects of different diets on survival can be carried out by employing statistical techniques for inference on samples of curves. The methods are illustrated with data on the survival of large cohorts of male and female Mediterranean fruit flies under full diet and under protein deprivation. These statistical methods allow one to investigate differences between the samples of hazard functions generated by the four groups defined by combinations of sex and diet.

Experiments based on over 400,000 medflies revealed that females maintained on a normal diet (sucrose plus protein) have a higher life expectancy than males maintained on a normal diet, with a difference of 1.30 ± 0.27 days in favor of females. However, this sex differential reverses under protein deprivation, with a difference of 2.24 ± 0.18 days in favor of males. The reversal of the male–female life expectancy differential is caused by a sustained surge in early female mortality under protein deprivation that is tied to egg-laying and physiological processes. In contrast, male mortality and life expectancy are only mildly affected by protein deprivation. The surge in early mortality for female medfly cohorts is an instance of a vulnerable period. These vulnerable periods are linked with patterns in hazard rates.

Pearson’s original chi-square test for goodness-of-fit has been extended and generalized in various ways to test the composite hypothesis of a certain parametric family {F (x; θ): θ ∈ Θ}. We illustrate in this paper that, for lifetime data that are subject to incomplete observation, a unified approach is available to derive general chi-square tests for parametric families, regardless of the sampling plan for such incomplete data. Let be a nonparametric estimate of the target distribution function F, and let be any parametric estimate of the true parameter under the null hypothesis. The general test statistic is a quadratic form of the estimated sample process evaluated at r points (called cell boundaries) of the x′s. A unified large sample theory is available for such test statistics, and one tractable test is recommended. Several applications are discussed involving various choices of .

A review is presented of parametric and nonparametric models and corresponding estimation procedures for change-points in hazard functions where the data are possibly subject to random censoring. In particular, we discuss nonparametric models and the application of nonparametric smoothing techniques for change-point estimation and estimation of a hazard function when a changepoint is present. Preliminary theoretical results are mentioned and a simulation study provides further insight.

We discuss the estimation of hazard rates under random censoring with the kernel method. Two practically relevant problems that occur when applying unmodified kernel estimators are boundary effects near the endpoints of the support of the hazard rate, and a substantial increase in the variance from left to right over the range of abscissae where the hazard rate is estimated. A new class of boundary kernels is proposed for the first problem. Explicit formulas for these kernels are developed, and it is shown that this boundary correction works well in practice. A data-adaptive varying bandwidth selection procedure is proposed for the second problem. This procedure generally will lead to increasing bandwidths near the left endpoint and toward the right endpoint, and will lead to smaller integrated mean squared error of the hazard rate estimator as compared to a fixed bandwidth method. A practically feasible method incorporating the new boundary kernels and local bandwidth choices is implemented and illustrated with survival data from a leukemia study.

A variety of statistical measures have been used to analyze interdistributional income inequality. Two recently introduced measures indicate a much larger secular change in the black–white income differential than the currently used measures. In order to understand this phenomenon both the theoretical properties of and empiric results obtained from five measures are given. It is shown that the new measures are more sensitive to the type of change that actually occured than the usual measures which were designed to detect a shift in location.

In life testing and survival analyses which involve the use of expensive equipment the cost of continuing an experiment until all the items on test have failed can be quite high. In these situations it is reasonable to make a statistical test when a pre-specified percentile, e.g. median of the control group has been observed. This article adapts some existing procedures for complete samples to randomly censored data. The results of Lo and Singh (1985) who extended the Bahadur representation of quantiles to the censored case enable us to use the methods of Gastwirth (1968) and Hettmansperger (1973) which were based on Bahadur's result to extend the procedures of Mathisen (1943), Gart (1963) and Slivka (1970). The large sample efficiency of the control median test is the same as that of Brookmeyer and Crowley's (1982) extension of the usual median test. For the two-sample shift problem with observations following the double-exponential law, the median remains the optimum percentile to use until the censoring becomes quite heavy. On the other hand, in the two-sample scale parameter problem for data from an exponential distribution the percentile (80th in the uncensored case) yielding the asymptotically most powerful test in the family of control percentile tests no longer is optimum. The effect becomes noticeable when 25% or more of the data is censored.

We study the estimation of a density and a hazard rate function based on censored data by the kernel smoothing method. Our technique is facilitated by a recent result of Lo and Singh (1986) which establishes a strong uniform approximation of the Kaplan-Meier estimator by an average of independent random variables. (Note that the approximation is carried out on the original probability space, which should be distinguished from the Hungarian embedding approach.) Pointwise strong consistency and a law of iterated logarithm are derived, as well as the mean squared error expression and asymptotic normality, which is obtain using a more traditional method, as compared with the Hajek projection employed by Tanner and Wong (1983).

A nonparametric estimator G n of a distribution function F with increasing failure rate average (IFRA) is constructed. This estimator G n is so constructed that its hazard function C n is the greatest star-shaped minorant of the sample hazard function. The estimator G n is itself IFRA. Denoting the sample distribution function by F n and the sample hazard function by H n , it follows from Millar (1979) that F n is an asymptotically minimax estimator of F if F is IFRA. We prove, under suitable restrictions of F , and for any fixed λ with F ( λ ) n 1 2 sup x⩽λ ||G n (x)−F n (x)|| and n 1 2 sup x⩽λ ||C n (x)−H n (x)|| both tend to zero in probability. That is, G n and F n are asymptotically n 1 2 - equivalent as estimators of the true distribution functions. Similar results can be obtained for a DFRA distribution.

Unbalanced-size samples arise naturally in equal-employment cases, as the minority fraction of all employees or applicants are invariably less than one half. Motivated by an actual case in which the median test with no power to detect disparate treatment was accepted in court, we develop a symmetrized form of the control median test having the same asymptotic properties as the median test. Since the actual case concerned the relative merits of the median and Wilcoxon test, a Monte Carlo study of the power of the new test and other nonparametric tests is reported. The results show that the new procedure is more powerful than the ordinary median test in small unbalanced samples. When the data come from a normal or double-exponential law, the Wilcoxon test is however usually superior to either of the others. When the data come from a Cauchy distribution, on the other hand, the powers of the procedures typically are reversed. Les echantillons de tailles inegales surviennent naturellement dans les causes de discrimination a l'embauche puisque la fraction minoritaire de tous les employes ou de tous les postulants est invariablement inferieure a un demi. Incites par une cause precise ou le test de la mediane, pourtant sans puissance pour detecter un traitement disparate, fut admis en cour, nous developpons une forme symetrique du test temoin de la mediane qui possede des proprietes asymptotiques identiques a ce dernier. Puisque la cause en question portait sur les merites relatifs des tests de la mediane et de Wilcoxon, nous presentons une etude par methodes de Monte Carlo de la puissance du nouveau test et de d'autres competiteurs non parametriques. Les resultats montrent que la nouvelle procedure est plus puissante que le test ordinaire de la mediane pour de petitis echantillons de tailles inegales. Lorsque les donnees sont regies par une loi normale ou double-exponentielle, le test de Wilcoxon est toutefois generalement superieur aux deux test. Cependant, lorsque les observations sont issues d'une population Cauchy, les puissances des procedures sont typiquement renversees.

Identification of differentially expressed (DE) genes across two conditions is a common task with microarray. Most existing approaches accomplish this goal by examining each gene separately based on a model and then control the false discovery rate over all genes. We took a different approach that employs a uniform platform to simultaneously depict the dynamics of the gene trajectories for all genes and select differently expressed genes. A new Functional Principal Component (FPC) approach is developed for time-course microarray data to borrow strength across genes. The approach is flexible as the temporal trajectory of the gene expressions is modeled nonparametrically through a set of orthogonal basis functions, and often fewer basis functions are needed to capture the shape of the gene expression trajectory than existing nonparametric methods. These basis functions are estimated from the data reflecting major modes of variation in the data. The correlation structure of the gene expressions over time is also incorporated without any parametric assumptions and estimated from all genes such that the information across other genes can be shared to infer one individual gene. Estimation of the parameters is carried out by an efficient hybrid EM algorithm. The performance of the proposed method across different scenarios was compared favorably in simulation to two-way mixed-effects ANOVA and the EDGE method using B-spline basis function. Application to the real data on C. elegans developmental stages also suggested that FPC analysis combined with hybrid EM algorithm provides a computationally fast and efficient method for identifying DE genes based on time-course microarray data.

We construct nonparametric estimators of the distribution function F and its hazard function in the class of all increasing failure rate (IFR) distribution. Denoting the empirical distribution and empirical hazard function by $F_n$ and $H_n$ , respectively, let $C_n$ be the greatest convex minorant of $H_n$, and $G_n$ the distribution with hazard function $C_n$. The estimator $G_n$ is itself IFR. We prove that under suitable restrictions on F, and for any fixed $\lambda$ with $F(\lambda)

Wald's general analytic conditions that imply strong consistency of the approximate maximum likelihood estimators (AMLEs) have been extended by Le Cam, Kiefer and Wolfowitz, Huber, Bahadur, and Perlman. All these conditions use the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_0)\rbrack$, where $\theta_0$ is the true value of the parameter. However these methods usually fail in the nonparametric case. Thus, in this paper, for each $\theta \neq \theta_0$, we look at the log likelihood ratio of the type $\log\lbrack f(x, \theta)/f(x, \theta_r(\theta))\rbrack$, where $\theta_r(\theta)$ is a certain parameter selected in a neighborhood $V_r$ of $\theta_0$. Some general analytic conditions that imply strong consistency of the AMLE are given. The results are shown to be applicable to several nonparametric families having densities, e.g., concave distributions functions, and increasing failure rate distributions. In particular, they can be applied to several censored data cases.

This document studies the estimation of a hazard rate function based on censored data by the kernel smoothing method. Our technique is facilitated by a recent result of Lo and Singh (1984) which establishes a strong uniform approximation of the Kaplan-Meier estimator by an average of independent random variables. Pointwise and uniform strong consistency are derived, as well as the mean squared error expression and asymptotic normality, which is obtained using a more traditional method, as compared with the Hajek projection employed by Tanner and Wong (1983). (Author)

This paper considers the problem of estimating a survival curve from randomly right censored data when it is known to have increasing Failure Rate Average (IFRA), or to be New Better than Used (NBU). Let F sub n(t) be the product-limit estimator (PL-estimator) of Kaplan and Meier for the life distribution. Since F sub n(t) never has the IFRA property and may not be NBU, we modify F sub n(t) to have the desired IFRA (NBU) properties. The modified estimators are easy to compute and, under mild conditions, are shown to be asymptotically n raised to the 1/2 power-equivalent to F sub n(t) on compact intervals. Thus the modified estimators share the asymptotic properties of the PL-estimator F sub n(t). (Author)

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