A Smooth Nonparametric Quantile Estimator from Right-Censored Data PDF Download
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Author: W. J. Padgett
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ISBN:
Category :
Languages : en
Pages : 25
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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 : 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: Yuhlong Lio
Publisher:
ISBN:
Category : Nonparametric statistics
Languages : en
Pages : 162
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Author: W. J. Padgett
Publisher:
ISBN:
Category :
Languages : en
Pages : 29
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Book Description
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: Katsuhiro Uechi
Publisher:
ISBN:
Category : Mathematical statistics
Languages : en
Pages :
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The development of an estimator of a quantile function Q(p) is discussed. The smooth nonparametric estimator Qn(p) of a quantile function Q(p) is defined as the solution to Fn(Qn(p)) = p, where Fn is a smooth Kaplan-Meier estimator of an unknown continuous distribution function F(x). The asymptotic properties of the smooth quantile process, n(Qn(p) - Q(p)) , based on right censored lifetimes are studied. The asymptotic properties of the bootstrap quantile process, n(Q n(p) - Q(p)) are also investigated and shown to have the same limiting distribution as the smooth quantile process. The bootstrap method to approximate the sampling distribution of the smooth quantile process is used to construct simultaneous confidence bands for a quantile function and the difference of two quantile functions. A Monte Carlo simulation is conducted to assess the performance of these confidence bands by computing the lengths and coverage probabilities of the bands. The optimum bandwidth is also investigated.
Author: Y. L. Lio
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ISBN:
Category :
Languages : en
Pages : 26
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Based on right-censored data from a lifetime distribution F sub 0, a smooth alternative to the product-limit estimator as a nonparametric quantile estimator of a population quantile is proposed. The estimator is a generalized product-limit quantile obtained by averaging appropriate subsample product-limit quantiles over all subsamples of a fixed size. Under the random censorship model and some conditions of F sub 0, it is shown that the estimator is consistent and has the same asymptotic normal distribution as the product-limit quantile estimator performs better than the product-limit quantile estimator in the sense of estimated mean squared errors.
Author: K. J. Padgett
Publisher:
ISBN:
Category :
Languages : en
Pages : 15
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Book Description
The main objectives of this research have been the development of smooth nonparametric estimators of quantile functions from right-censored data and the further study of smooth density estimators from censored observations. In particular, kernel-type quantile estimators have been obtained under censoring which give better estimates of percentiles of the lifetime distribution than the usual product-limit quantile estimator. During the past year, asymptotic properties of these kernel quantile estimators have been developed, including asymptotic normality, consistency, and mean square convergence. In addition, a data-based procedure for selecting the bandwidth has been investigated using the bootstrap, and approximate confidence for the true quantile have been proposed using bootstrap estimates of the sampling distribution. Theoretical results on the optimal bandwidth selection for kernel density estimators under random right censorship have also been obtained. New results in several other problem areas were also developed. These included the study of linear empirical Bayes estimators, prediction intervals for the inverse Gaussian distribution, nonparametric hazard rate estimation under censoring, nonparametric inference for step-stress accelerated life tests under censoring, discrete failure models, simultaneous confidence intervals for pairwise differences of normal means, and optimal designs for comparing treatments with a control.
Author: Shyamalee Kumary Serasinghe
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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 : 21
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Book Description
Right-censored data arise very naturally in life testing and reliability studies. For such data, it is important to be able to obtain good nonparametric estimates of various characteristics of the unknown lifetime distribution. This report concerns the computational procedure for a kernel-type nonparametric estimator of the quantile function of the lifetimne distribution from right-censored data. This estimator was suggested by Padgett (1986), extending the complete sample results of Yang (1985). The large sample properties of the estimator, such as asymptotic normality and mean square convergence, were studied by Lio, Padgett and Yu (1986) and by Lio and Padgett (1985). In this report, a procedure for calculation of the kernel-type quantile estimate from right-censored data is described, and a listing of a computer program in FORTRAN code is provided.
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 : 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.
