On the hazard rate of the lognormal distribution

It is argued that plots of the hazard rate for the lognormal random variable which have appeared in some recent literature are incorrect and/or misleading; the hazard rate always begins at zero, rises to a maximum, then decreases very slowly to zero. An equation for the location of the maximum of the hazard rate is derived. The maximum lies in a finite interval for all positive values of the standard deviation of the associated normal random variable. Approximations that can be used to compute the hazard rate for parameter values outside of the usual range in the tables associated with the normal (Gaussian) random variable are presented     

Confidence limits for steady state availability of systems with lognormal operating time and inverse Gaussian repair time

A 100 p% confidence interval for the steady state availability of a system is derived, when the operating time distribution is lognormal and the repair time distribution is Inverse Gaussian (IG). It is assumed that one of the parameters of lognormal distribution and also the ratio of parameters of IG distribution are known.     

Confidence limits for steady-state availability of systems with a mixture of exponential and gamma operating time and lognormal repair time

A 100% confidence interval for the steady-state availability of a system is derived, when the operating time distribution is a mixture of exponential and gamma and the repair time distribution is lognormal. It is assumed that one of the parameters of the operating time distribution and also one of the parameters of the repair time distribution are known.     

Confidence regions having different shapes for the failure distribution function

A failure distribution represents an attempt to describe mathematically the length of life of a device.Most often the possibility remains that the analysts may be hesitant or unwilling to entertain any well known theoretical failure distribution.Therefore, one must rely on actual observations of the time to failure to construct an empirical cumulative failure distribution function. We present some useful results in constructing statistical confidence regions for the entire failure cumulative distribution function (cdf), F(x) from which a random sample has been drawn. The bandwidth of these regions becomes narrower in some parts of the distribution where we may want to have more precise information about failure cdf than is afforded by the ordinary Kolmogorov-Smirnov (K−S) confidence region. The problem of constructing the best, having minimum risk, confidence region from a decision theoretic approach is also considered.Illustrative numerical examples are presented.     

Predictive densities for the lognormal distribution and their applications

Maximum likelihood predictive densities (MLPDs) for a future lognormal observation are obtained and their applications to reliability and life testing are considered. When applied to reliability and failure rate estimations, they give estimators that can be much less biased and less variable than the usual maximum likelihood estimations (MLEs) obtained by replacing the unknown parameters in the density function by their MLEs. When applied to lifetime predictions, they give prediction intervals that are shorter than the usual frequentist intervals. Using the MLPDs, it is also rather convenient to construct the shortest prediction intervals. Extensive simulations are performed for comparisons. A numerical example is given for illustration.     

对数正态分布下的无线光通信系统误码率分析

对开关键控(OOK)强度调制直接检测(IM/DD)方式的无线光通信接收发射系统的光电信号进行分析,并使用最大后验概率(MAP)方法确定判决阈值。将对数正态分布作为湍流信道上光强闪烁模型,建立了系统误码率与湍流强度、光源相干参数、激光发射器光功率等系统参数之间的定量关系。采用部分相干光作为信号光能降低光源相干度,有效抑制湍流效应。计算结果表明：在一定条件下,湍流强度改变0.5个量级系统误码率相差6-8个量级,并且最优源相干参数可以通过计算得到,其对应的部分相干光最有利于提高通信系统性能。     

A comprehensive review of the lognormal failure distribution with application to LED reliability

Life-test data for many semiconductor devices obey the lognormal law of failure. In this paper we offer a survey of the key statistical properties of the lognormal distribution which are relevant in device engineering. The statistical developments are illustrated throughout by specific examples drawn from life-test data for GaP red LEDs which show lognormal failure behavior. Additional theoretical justification for accepting the lognormal distribution in the case of these devices is provided by the statistical recasting of the diffusion theory of red LED degradation.The treatment of the fundamental concepts of reliability in terms of cumulative failure function, instantaneous failure rate, mean time for failure (MTTF), etc., although concise, is self-contained. Moreover, the estimation of the two characteristic parameters (median life, standard deviation) of the lognormal distribution from sample data by means of a number of alternative procedures (graphical, least-square, maximum likelihood) is discussed in detail. The efficient determination of the MTTF and failure rate is facilitated by convenient charts developed here which relate these quantities to the parameters median life and standard deviation at 5, 10, 20 and 40 years of device service life. In order to assess the uncertainty associated with the estimates, the confidence limits on the median life and standard deviation of the population as a function of confidence level and sample size are given in novel graphical forms. Finally, error bounds for the MTTF and failure rate of the population are also presented.     

Bounds on the distribution of a sum of independent lognormal randomvariables

The distribution function of a sum of lognormal random variables (RVs) appears in several communication problems. Approximations are usually used for such distribution as no closed form nor bounds exist. Bounds can be very useful in assessing the performance of any given system. In this letter, we derive upper and lower bounds on the distribution function of a sum of independent lognormal RVs. These bounds are given in a closed form and can be used in studying the performance of cellular radio and broadcasting systems     

Confidence limits for steady state availability of a parallel system

100 (1 − α)% confidence limits for the steady state availability of a parallel system, when the failure rate of an operable unit is a constant and the repair time of a failed unit is an Erlangian distribution are obtained.     

Shortest bayes credibility intervals for the lognormal failure model

The lognormal distribution with density function

相似文献   

A note on constructing the confidence bounds of the failure rate for the inverse Gaussian distribution

In this paper, a Bonferroni confidence interval for the failure rate of the inverse Gaussian distribution is derived. It can be solved by formulating the problem as a nonlinear programming. A numerical example on repair time data is provided to illustrate the procedure.     

Confidence interval for the mean of the exponential distribution,based on grouped data

When the available data from an exponential distribution are grouped, the maximum likelihood estimator (MLE) for the mean and several modified MLE have been discussed in literature. However, little work has been done on interval estimators based on such grouped data. This paper derives the asymptotic property of a statistic which is used to construct an approximate confidence interval for the mean. The width of this approximate confidence interval, based on grouped data, is compared with those based on complete samples, and samples with type-I and type-II censoring. The limits of the ratios of these widths are derived when the sample size approaches infinity. The approximate confidence interval from grouped data is wider than those from complete and censored samples. However, Monte Carlo simulation indicates that the proposed method based on grouped data is adequate, considering the restricted information in this case     

Reliability estimation and tolerance limits for Laplace distribution based on censored samples

In this paper, we first present an estimator for the reliability function based on the best linear unbiased estimators (BLUEs) of the location and scale parameters, μ and σ, for the Laplace distribution based on Type-II censored samples. We show that this estimator is almost unbiased at varying levels of reliability. Next, we determine through Monte Carlo simulations the values of the tolerance factor t_γ that are necessary for the construction of lower and upper tolerance limits for the distribution. We also illustrate how these tables for tolerance limits could be used to determine lower confidence limits for the reliability. Finally, we present an example to illustrate the methods of inference developed in this paper.     

Partially coherent imaging in two dimensions and the theoretical limits of projection printing in microfabrication

Optical projection printing is quantitatively studied, using the diffracted image of two bars of equal width and space and an aspect ratio of three. Hopkins' four-dimensional integral for partially coherent imaging is used for the numerical analysis. Diffraction-limited lenses with a circular pupil are used. Constant intensity profiles in an absorptionless photoresist and nonreflective substrate are evaluated. Then the tolerances ot defocus and intensity are derived using a linewidth tolerance of ±2.5 percent of the mask linewidth which varies from 0.5, 0.75, 1, 1.5, 2, to 2.5 µm. The illuminations used have σ = 0, 0.5, 0.7, 0.78, 1, 1.3, and infinity.     

Estimating the IC upset/failure threshold of dose rate for a radiation pulse sequence

The conditions are analyzed for upsets and failures of ICs exposed to a sequence of arbitrarily shaped radiation pulses. Within an asymptotic model, a procedure is proposed for converting the upset/failure threshold due to a single rectangular pulse into that due to a pulse sequence.     

A maintenance inspection model for a single machine with general failure distribution

In this paper we develop a mathematical model for determining a periodic inspection schedule in a preventive maintenance program for a single machine subject to random failure. We formulate the problem as a profit maximization model with general failure time distribution. We show that under certain conditions on the probability density function of failure, a unique optimal inspection interval can be obtained. When the failure times are exponentially distributed, we propose alternative optimal and heuristic procedures to find exact and approximate inspection intervals. Our heuristic solution method is shown numerically to be more efficient than an earlier published heuristic procedure. We also investigated the sensitivity of the optimal inspection interval and expected profit per unit of time with respect to the changes in the two parameters of the Weibull time to failure distribution.     

Lower Confidence Limits and a Test of Hypotheses for System Availability

Any estimate of system availability calculated from time-to-failure and time-to-repair test data will be subject to some degree of uncertainty due to the uncertainty associated with the sample estimates of MTTF and MTTR. Although decisions about the true availability of the system should take this uncertainty into account, a point estimate of availability is usually the only statistic calculated. This paper derives techniques for placing a lower confidence limit on system availability and for deciding if the true system availability differs significantly from a specified value when MTTF and MTTR are estimated from test data. These techniques could be used to analyze existing test data or to design a test program for demonstrating system availability and/or detecting significant deviations from specified values of system availability. To facilitate utilization of these techniques, curves of the lower 0.90 and 0.95 confidence limits and power curves for the test of hypotheses at the 0.05 and 0.10 levels of significance are presented. Examples illustrating the use of the curves are given.