Weibull tolerance intervals associated with moderate to small survival proportions for use in a new formulation of lanchester combat theory

Given herein is an easily implemented method for obtaining, from complete or censored data, approximate tolerance intervals associated with the upper tail of a Weibull distribution. These approximate intervals are based on point estimators that make essentially most efficient use of sample data. They agree extremely well with exact intervals (obtained by Monte Carlo simulation procedures) for sample sizes of about 10 or larger when specified survival proportions are sufficiently small. Ranges over which the error in the approximation is within 2 percent are determined. The motivation for investigation of the methodology for obtaining the approximate tolerance intervals was provided by the new formulation of Lanchester Combat Theory by Grubbs and Shuford [3], which suggests a Weibull assumption for time-to-incapacitation of key targets. With the procedures investigated herein, one can use (censored) data from battle simulations to obtain confidence intervals on battle times associated with given low survivor proportions of key targets belonging to either specified side in a future battle. It is also possible to calculate confidence intervals on a survival proportion of key targets corresponding to a given battle duration time.     

Optimum quantiles for the linear estimation of the parameters of the extreme value distribution in complete and censored samples

The present study is concerned with the determination of a few observations from a sufficiently large complete or censored sample from the extreme value distribution with location and scale parameters μ and σ, respectively, such that the asymptotically best linear unbiased estimators (ABLUE) of the parameters in Ref. [24] yield high efficiencies among other choices of the same number of observations. (All efficiencies considered are relative to the Cramér-Rao lower bounds for regular unbiased estimators.) The study is on the asymptotic theory and under Type II censoring scheme. For the estimation of μ when σ is known, it has been proved that there exists a unique optimum spacing whether the sample is complete, right censored, left censored, or doubly censored. Several tables are prepared to aid in the numerical computation of the estimates as well as to furnish their efficiencies. For the estimation of σ when μ is known, it has been observed that there does not exist a unique optimum spacing. Accordingly we have obtained a spacing based on a complete sample which yields high efficiency. A similar table as above is prepared. When both μ and σ are unknown, we have considered four different spacings based on a complete sample and chosen the one yielding highest efficiency. A table of the efficiencies is also prepared. Finally we apply the above results for the estimation of the scale and/or shape parameters of the Weibull distribution.     

广义逆指数分布元件的可靠性分析

在II型混合截尾样本下,得到了广义逆指数分布未知参数的最大似然估计。利用最大似然估计的渐近正态性构造了参数的渐近置信区间,运用Lindley's逼近方法和TierneyKadane's逼近方法计算出了参数的Bayes估计。最后,运用Monte-Carlo方法对上述估计方法结果作了模拟比较。     

Tabular aids for fitting weibull moment estimates

Moment estimators for the parameters of the Weibull distribution are considered in the context of analysis of field data. The data available are aggregated, with individual failure times not recorded. In this case, the complexity of the likelihood function argues against the use of maximum-likelihood estimation, particularly for relatively large sets of data, and moment estimators are a reasonable alternative. In this article, we derive the asymptotic covariance matrix of the moment estimators, and provide listings for BASIC computer programs which generate tables useful for calculation of the estimates as well as for estimating the asymptotic covariance matrix using aggregated data.     

Statistical analysis for masked system life data from Marshall‐Olkin Weibull distribution under progressive hybrid censoring

This paper considers the statistical analysis of masked data in a series system, where the components are assumed to have Marshall‐Olkin Weibull distribution. Based on type‐I progressive hybrid censored and masked data, we derive the maximum likelihood estimates, approximate confidence intervals, and bootstrap confidence intervals of unknown parameters. As the maximum likelihood estimate does not exist for small sample size, Gibbs sampling is used to obtain the Bayesian estimates and Monte Carlo method is employed to construct the credible intervals based on Jefferys prior with partial information. Numerical simulations are performed to compare the performances of the proposed methods and one data set is analyzed.     

A good simple percentile estimator of the weibull shape parameter for use when all three parameters are unknown

In this paper we consider a simple three-order-statistic asymptotically unbiased estimator of the Weibull shape parameter c for the case in which all three parameters are unknown. Optimal quantiles that minimize the asymptotic variance of this estimator, c? are determined and shown to depend only on the true (unknown) shape parameter value c and in a rather insensitive way. Monte Carlo studies further verified that, in practice where the true shape parameter c is unknown, using always c? with the optimal quantities that correspond to c = 2.0 produces estimates, c?, remarkably close to the theoretical optimal. A second stage estimation procedure, namely recalculating c? based on the optimal quantiles corresponding to c?, was not worth the additional effort. Benchmark simulation comparisons were also made with the best percentile estimator of Zanakis [20] and with a new estimator of Wyckoff, Bain and Engelhardt [18], one that appears to be the best of proposed closed-form estimators but uses all sample observations. The proposed estimator, c?, should be of interest to practitioners having limited resources and to researchers as a starting point for more accurate iterative estimation procedures. Its form is independent of all three Weibull parameters and, for not too large sample sizes, it requires the first, last and only one other (early) ordered observation. Practical guidelines are provided for choosing the best anticipated estimator of shape for a three-parameter Weibull distribution under different circumstances.     

Planning accelerated life tests for censored two‐parameter exponential distributions

We consider optimal test plans involving life distributions with failure‐free life, i.e., where there is an unknown threshold parameter below which no failure will occur. These distributions do not satisfy the regularity conditions and thus the usual approach of using the Fisher information matrix to obtain an optimal accelerated life testing (ALT) plan cannot be applied. In this paper, we assume that lifetime follows a two‐parameter exponential distribution and the stress‐life relationship is given by the inverse power law model. Near‐optimal test plans for constant‐stress ALT under both failure‐censoring and time‐censoring are obtained. We first obtain unbiased estimates for the parameters and give the approximate variance of these estimates for both failure‐censored and time‐censored data. Using these results, the variance for the approximate unbiased estimate of a percentile at a design stress is computed and then minimized to produce the near‐optimal plan. Finally, a numerical example is presented together with simulation results to study the accuracy of the approximate variance given by the proposed plan and show that it outperforms the equal‐allocation plan. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 169–186, 1999     

Selecting among weibull,lognormal and gamma distributions using complete and censored smaples

In a recent paper, Kent and Quesenberry [19] considered using certain optimal invariant statistics to select the best fitting member of a collection of probability distributions using complete samples of life data. In the present work extensions of this approach in two directions are given. First, selection for complete samples based on scale and shape invariant statistics is considered. Next, the selection problem for type I censored samples is considered, and both scale invariant and maximum likelihood selection procedures are studied. The two-parameter (scale and shape) Weibull, lognormal, and gamma distributions are considered and applications to real data are given. Results from a (small) comparative simulation study are presented.     

Nonparametric bayesian lifetime data analysis using dirichlet process lognormal mixture model

We propose a nonparametric Bayesian lifetime data analysis method using the Dirichlet process mixture model with a lognormal kernel. A simulation‐based algorithm that implements the Gibbs sampling is developed to fit the Dirichlet process lognormal mixture (DPLNM) model using rightly censored failure time data. Five examples are used to illustrate the proposed method, and the DPLNM model is compared to the Dirichlet process Weibull mixture (DPWM) model. Results indicate that the DPLNM model is capable of estimating different lifetime distributions. The DPLNM model outperforms the DPWM model in all the examples, and the DPLNM model shows promising potential to be applied to analyze failure time data when an appropriate parametric model for the data cannot be specified. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Estimating failure time distribution and its parameters based on intermediate data from a Wiener degradation model

Instead of measuring a Wiener degradation or performance process at predetermined time points to track degradation or performance of a product for estimating its lifetime, we propose to obtain the first‐passage times of the process over certain nonfailure thresholds. Based on only these intermediate data, we obtain the uniformly minimum variance unbiased estimator and uniformly most accurate confidence interval for the mean lifetime. For estimating the lifetime distribution function, we propose a modified maximum likelihood estimator and a new estimator and prove that, by increasing the sample size of the intermediate data, these estimators and the above‐mentioned estimator of the mean lifetime can achieve the same levels of accuracy as the estimators assuming one has failure times. Thus, our method of using only intermediate data is useful for highly reliable products when their failure times are difficult to obtain. Furthermore, we show that the proposed new estimator of the lifetime distribution function is more accurate than the standard and modified maximum likelihood estimators. We also obtain approximate confidence intervals for the lifetime distribution function and its percentiles. Finally, we use light‐emitting diodes as an example to illustrate our method and demonstrate how to validate the Wiener assumption during the testing. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Linear estimation of the location and scale parameters from type 11 censored samples from symmetric unimodal distributions

Estimates of the location and scale parameters, linear in the order statistics of a Type II censored or complete sample, from a continuous symmetric unimodal distribution satisfying certain conditions are obtained. Their coefficients are explicit functions of the expectations of the order statistics or population quantiles from the known parameter-free standardized distribution. Linear estimates with simpler coefficients are also obtained. The theorems state the complete sample case, and the singly and doubly censored cases. The more general case, the multiple censoring, is an extension of these cases and is indicated. All the estimates obtained are asymptotically efficient in the strict sense.     

混合指数分布模型的Bayes分析

针对截尾试验数据的情况,给出了二元混合指数分布模型的平均寿命和可靠性函数的严格的Ｂａｙｅｓ点估计,并运用最大熵准则给出了可靠性函数的近似的Ｂａｙｅｓ置信下限估计。     

Progressive Type‐II censoring and coherent systems

A new connection between the distribution of component failure times of a coherent system and (adaptive) progressively Type‐II censored order statistics is established. Utilizing this property, we develop inferential procedures when the data is given by all component failures until system failure in two scenarios: In the case of complete information, we assume that the failed component is also observed whereas in the case of incomplete information, we have only information about the failure times but not about the components which have failed. In the first setting, we show that inferential methods for adaptive progressively Type‐II censored data can directly be applied to the problem. For incomplete information, we face the problem that the corresponding censoring plan is not observed and that the available inferential procedures depend on the knowledge of the used censoring plan. To get estimates for distributional parameters, we propose maximum likelihood estimators which can be obtained by solving the likelihood equations directly or via an Expectation‐Maximization‐algorithm type procedure. For an exponential distribution, we discuss also a linear estimator to estimate the mean. Moreover, we establish exact distributions for some estimators in the exponential case which can be used, for example, to construct exact confidence intervals. The results are illustrated by a five component bridge system. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 512–530, 2015     

Signal‐to‐interference‐plus‐noise ratio estimation for wireless communication systems: Methods and analysis

The Signal‐to‐Interference‐plus‐Noise Ratio (SINR) is an important metric of wireless communication link quality. SINR estimates have several important applications. These include optimizing the transmit power level for a target quality of service, assisting with handoff decisions and dynamically adapting the data rate for wireless Internet applications. Accurate SINR estimation provides for both a more efficient system and a higher user‐perceived quality of service. In this paper, we develop new SINR estimators and compare their mean squared error (MSE) performance. We show that our new estimators dominate estimators that have previously appeared in the literature with respect to MSE. The sequence of transmitted bits in wireless communication systems consists of both pilot bits (which are known both to the transmitter and receiver) and user bits (which are known only by the transmitter). The SINR estimators we consider alternatively depend exclusively on pilot bits, exclusively on user bits, or simultaneously use both pilot and user bits. In addition, we consider estimators that utilize smoothing and feedback mechanisms. Smoothed estimators are motivated by the fact that the interference component of the SINR changes relatively slowly with time, typically with the addition or departure of a user to the system. Feedback estimators are motivated by the fact that receivers typically decode bits correctly with a very high probability, and therefore user bits can be thought of as quasipilot bits. For each estimator discussed, we derive an exact or approximate formula for its MSE. Satterthwaite approximations, noncentral F distributions (singly and doubly) and distribution theory of quadratic forms are the key statistical tools used in developing the MSE formulas. In the case of approximate MSE formulas, we validate their accuracy using simulation techniques. The approximate MSE formulas, of interest in their own right for comparing the quality of the estimators, are also used for optimally combining estimators. In particular, we derive optimal weights for linearly combining an estimator based on pilot bits with an estimator based on user bits. The optimal weights depend on the MSE of the two estimators being combined, and thus the accurate approximate MSE formulas can conveniently be used. The optimal weights also depend on the unknown SINR, and therefore need to be estimated in order to construct a useable combined estimator. The impact on the MSE of the combined estimator due to estimating the weights is examined. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

A performance comparison of empirical Bayes and Bayes estimators of the Weibull and gamma scale parameters

A mean-squared error comparison of smooth empirical Bayes and Bayes estimators for the Weibull and gamma scale parameters is studied based on a computer simulation. The smooth empirical Bayes estimators are determined as functions of up to 15 past estimates of the parameter of interest. Results indicate that at best the mean-squared errors of the empirical Bayes estimators are about 20–40% larger than those of the corresponding squared-error optimal Bayes estimators.     

An empirical bayes estimator for the scale parameter of the two-parameter weibull distribution

An empirical Bayes estimator is given for the scale parameter in the two-parameter Weibull distribution. The scale parameter is assumed to vary randomly throughout a sequence of experiments according to a common, but unknown, prior distribution. The shape parameter is assumed to be known, however, it may be different in each experiment. The estimator is obtained by means of a continuous approximation to the unknown prior density function. Results from Monte Carlo simulation are reported which show that the estimator has smaller mean-squared errors than the usual maximum-likelihood estimator.     

Optimum step‐stress accelerated life test plans for log‐location‐scale distributions

This article presents new tools and methods for finding optimum step‐stress accelerated life test plans. First, we present an approach to calculate the large‐sample approximate variance of the maximum likelihood estimator of a quantile of the failure time distribution at use conditions from a step‐stress accelerated life test. The approach allows for multistep stress changes and censoring for general log‐location‐scale distributions based on a cumulative exposure model. As an application of this approach, the optimum variance is studied as a function of shape parameter for both Weibull and lognormal distributions. Graphical comparisons among test plans using step‐up, step‐down, and constant‐stress patterns are also presented. The results show that depending on the values of the model parameters and quantile of interest, each of the three test plans can be preferable in terms of optimum variance. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

低轨预警自由段弹道估计的多项式逼近算法

低轨单星对自由段弹道的估计是天基预警系统需解决的关键技术之一.建立了低轨预警卫星对自由段弹道的观测模型,针对极大似然估计批处理算法的大运算量问题,给出了一种多项式逼近算法,由观测数据的逼近多项式在一些特定采样点的值形成伪观测数据,以伪观测数据代替原观测数据进行弹道估计.仿真表明,精度与极大似然估计相当,运算量显著降低.     

Test for the extreme value and weibull distributions based on normalized spacings

Discussed in this article are tests for the extreme-value distribution, or, equivalently, for the two-parameter Weibull distribution when parameters are unknown and the sample may be censored. The three tests investigated are based on the median, the mean, and the Anderson-Darling A² statistic calculated from a set zⁱ of values derived from the spacings of the sample. The median and the mean have previously been discussed by Mann, Scheuer, and Fertig [10] and by Tiku and Singh [14]. Asymptotic distributions and points are given for the test statistics, based on recently developed theory, and power studies are conducted to compare them with each other and with two other statistics suitable for the test. Of the normalized spacings tests, A² is recommended overall; the mean also gives good power in many situations, but can be nonconsistent.