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Author: Shyamalee Kumary Serasinghe
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Quantiles are useful in describing distributions of component lifetimes. Data, consisting of the lifetimes of sample units, used to estimate quantiles are often censored. Right censoring, the setting investigated here, occurs, for example, when some test units may still be functioning when the experiment is terminated. This study investigated and compared the performance of parametric and nonparametric estimators of quantiles from right censored data generated from Weibull and Lognormal distributions, models which are commonly used in analyzing lifetime data. Parametric quantile estimators based on these assumed models were compared via simulation to each other and to quantile estimators obtained from the nonparametric Kaplan- Meier Estimator of the survival function. Various combinations of quantiles, censoring proportion, sample size, and distributions were considered. Our simulation show that the larger the sample size and the lower the censoring rate the better the performance of the estimates of the 5th percentile of Weibull data. The lognormal data are very sensitive to the censoring rate and we observed that for higher censoring rates the incorrect parametric estimates perform the best. If you do not know the underlying distribution of the data, it is risky to use parametric estimates of quantiles close to one. A limitation in using the nonparametric estimator of large quantiles is their instability when the censoring rate is high and the largest observations are censored. Key Words: Quantiles, Right Censoring, Kaplan-Meier estimator.
Author: Shyamalee Kumary Serasinghe
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Quantiles are useful in describing distributions of component lifetimes. Data, consisting of the lifetimes of sample units, used to estimate quantiles are often censored. Right censoring, the setting investigated here, occurs, for example, when some test units may still be functioning when the experiment is terminated. This study investigated and compared the performance of parametric and nonparametric estimators of quantiles from right censored data generated from Weibull and Lognormal distributions, models which are commonly used in analyzing lifetime data. Parametric quantile estimators based on these assumed models were compared via simulation to each other and to quantile estimators obtained from the nonparametric Kaplan- Meier Estimator of the survival function. Various combinations of quantiles, censoring proportion, sample size, and distributions were considered. Our simulation show that the larger the sample size and the lower the censoring rate the better the performance of the estimates of the 5th percentile of Weibull data. The lognormal data are very sensitive to the censoring rate and we observed that for higher censoring rates the incorrect parametric estimates perform the best. If you do not know the underlying distribution of the data, it is risky to use parametric estimates of quantiles close to one. A limitation in using the nonparametric estimator of large quantiles is their instability when the censoring rate is high and the largest observations are censored. Key Words: Quantiles, Right Censoring, Kaplan-Meier estimator.
Author: W. J. Padgett
Publisher:
ISBN:
Category :
Languages : en
Pages : 25
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Book Description
Based on randomly right-censored data, a smooth nonparametric estimator of the quantile function of the lifetime distribution is studied. The estimator is defined to be the solution x sub n (p) to F sub n (p)) = O, where F sub n is the distribution function corresponding to a kernel estimator of the lifetime density. The strong consistency and asymptotic normality of x sub n (p) are shown. Some simulation results comparing this estimator with the product of the bandwidth required for computing F sub n is investigated using bootstrap methods. Illustrative examples are given. (Author).
Author: W. J. Padgett
Publisher:
ISBN:
Category :
Languages : en
Pages : 13
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Book Description
Major results have been obtained in the areas of nonparametric estimation of quantiles and of density functions under censoring, discrete failure models, and multiple comparisons. In particular, smooth nonparametric estimators of quantile functions from censored data were developed which give better estimates of percentiles of the lifetime distribution than the usual product-limit quantile function. Also, smooth density estimators from censored data were investigated using maximum penalized likelihood procedures. Several parametric models were proposed for the case of discrete failure data. These models provide a better fit to such data than some previously used discrete models. Finally, new methods of constructing simultaneous confidence intervals for pairwise differences of means of normal populations were developed, and the problem of selecting an asymptotically optimal design for comparing several new treatments with a control was solved. Work is continuing on the study of properties of kernel type quantile function estimators and development of goodness-of-fit tests for the model assumptions in accelerated life testing. Keywords: Nonparametric quantile estimation; Density estimation; Right-censored data; Discrete failure models; Multiple comparisons; Accelerated life testing.
Author: W. J. Padgett
Publisher:
ISBN:
Category :
Languages : en
Pages : 29
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The objectives of this paper are two-fold. One is to report results of extensive Monte Carlo simulations which demonstrate the behavior of the mean squared error of the kernel estimator with respect to bandwidth. These simulations provide a method of choosing an optimal bandwidth when the form of the lifetime and censoring distributions are known. Also, they compare the kernel-type estimator with the product-limit qauntile estimator. Five commonly used parametric lifetime distributions, two censoring mechanisms, and four different kernel functions are considered in this study, which is an extension of the brief simulations for exponential distributions reported by Padgett (1986). The second objective is to present a nonparametric method for bandwidth selection based on the bootstrap for right-censored data. This data-based procedure used the bootstrap to estimate mean squared error, and is both an extension and modification of the methods proposed by Padgett. Bandwidth selection using the bootstrap is important for small and moderately large samples since no exact expressions exist for the mean squared error of the kernel-type quantile estimator.
Author: Sneh Gulati
Publisher: Springer Science & Business Media
ISBN: 0387215492
Category : Mathematics
Languages : en
Pages : 123
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Book Description
By providing a comprehensive look at statistical inference from record-breaking data in both parametric and nonparametric settings, this book treats the area of nonparametric function estimation from such data in detail. Its main purpose is to fill this void on general inference from record values. Statisticians, mathematicians, and engineers will find the book useful as a research reference. It can also serve as part of a graduate-level statistics or mathematics course.
Author: W. J. Padgett
Publisher:
ISBN:
Category :
Languages : en
Pages : 25
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Book Description
Arbitrarily right-censored data arise naturally in industrial life testing and medical follow-up studies. In these situations it is important to be able to obtain nonparametric estimates of various characteristics of the survival function S. Based on such right-censored data, Kaplan and Meier gave the nonparametric maximum likelihood estimator of S, called the product-limit estimator, and, among others, Reid has proposed methods of estimating the median survival time from the product-limit estimator. Recently, Nair studied the problem of confidence bands for the survival function obtained from the product-limit estimator. Also, Padgett and McNichols and McNichols and Padgett have discussed estimation of a density for the survival distribution based on right-censored data. One characteristic of the survival distribution that is of interest is the quantile function, which is useful in reliability and medical studies. The quantile function of the product-limit estimator is a step function with jumps corresponding to the uncensored observations. The purpose of this paper is to present a smoothed nonparametric estimator of the quantile function from arbitrarily right-censored data based on the kernel method. It will be shown that under general conditions this estimator, mentioned briefly by Parzen is strongly consistent, and based on the results of a small Monte-Carol simulation study, performs better than quantile function of the product-limit estimator in the sense of smaller mean squared error. In particular, better estimates of the median survival time are obtainable. In addition, an approximation to the kernel estimator will be shown to be almost surely asymptotically equivalent to it under certain conditions.
Author: Y. L. Lio
Publisher:
ISBN:
Category :
Languages : en
Pages : 14
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For randomly right-censored data, new asymptotic expressions for the mean squared errors of the product-limit quantile estimator and a kernel-type quantile estimator are presented in this paper. From these results a comparison of the two quantile estimators with respect to their mean squared errors is given. (Author).
Author: Mei-Chu Tang
Publisher:
ISBN:
Category :
Languages : en
Pages : 130
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Author: Yuhlong Lio
Publisher:
ISBN:
Category : Nonparametric statistics
Languages : en
Pages : 162
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Author: Clifford Isaac Anderson-Bergman
Publisher:
ISBN: 9781303810138
Category :
Languages : en
Pages : 213
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Interval censoring occurs when event times are known to have occurred within an interval, rather than observing the exact time of event. This includes observations that are right censored, left censored and contained in intervals such that the left side is greater than the origin and the right side is finite (i.e. neither right censored or left censored). For interval censored data, the most common survival estimator used is the non-parametric maximum likelihood estimator (NPMLE), a generalization of the Kaplan-Meier curve which does not require any uncensored event times. The popularity of this estimator is due in part to the fact that assessing model fit for interval censored data can be very difficult. However, the extreme flexibility of the estimator comes at the cost of high variance, often providing an n^(1/3) convergence rate rather than the more typical n^(1/2). In a compromise between a highly constrained parametric estimator and the overly flexible NPMLE, we apply the popular log-concave density constraint to the NPMLE. By constraining a non-parametric estimator to have a log-concave density, an inves- tigator can improve the performance without needing to select a parametric family or smoothing parameter. We describe a fast algorithm we have developed for finding the log-concave NPMLE for interval censored data. We demonstrate that using the constraint significantly reduces the variance of the survival estimates in comparison to the unconstrained NPMLE via simulations. Next, we present three inference methods for our new estimator. This includes a goodness of fit test, two methods of confidence interval construction and a Cox PH model which incorporates a baseline log-concave distribution. We evaluate the power of the goodness of fit test and compare the other inference methods with the unconstrained counterparts via simulation. We apply these methods to a study on the effects of different environments on the rates of lung cancer among mice and another study investigating age at menopause. While our work demonstrates that the application of the shape constraints can be very helpful in the context of interval censored data, in some situations the log- concave constraint may not allow for as heavy tailed distributions as the investigator would like. To address this, we propose a new, more flexible "inverse convex" shape constraint, examine its behavior via simulation and show that it provides a better fit than the log-concave estimator when applied to real income data, which is well known to be heavy tailed. We are very optimistic about applying this new estimator to censored data, although we have yet to implement an algorithm to do so. We end this work with an algorithm for finding the (unconstrained) bivariate NPMLE for interval censored data. The bivariate NPMLE is used when each subject has two censored outcomes and the investigator is interested in modeling the relation between the two outcomes. Quickly finding the NPMLE has proven to be a challenging computational problem, as the number of parameters to consider is of order O(n^2). We present an efficient EM algorithm to find the bivariate NPMLE. We note that this is not related to shape constrained estimation.
