Yili Hong

On computing the distribution function for the Poisson binomial distribution

The Poisson binomial distribution is the distribution of the sum of independent and non-identically distributed random indicators. Each indicator follows a Bernoulli distribution and the individual probabilities of success vary. When all success probabilities are equal, the Poisson binomial distribution is a binomial distribution. The Poisson binomial distribution has many applications in different areas such as reliability, actuarial science, survey sampling, econometrics, etc. The computing of the cumulative distribution function (cdf) of the Poisson binomial distribution, however, is not straightforward. Approximation methods such as the Poisson approximation and normal approximations have been used in literature. Recursive formulae also have been used to compute the cdf in some areas. In this paper, we present a simple derivation for an exact formula with a closed-form expression for the cdf of the Poisson binomial distribution. The derivation uses the discrete Fourier transform of the characteristic function of the distribution. We develop an algorithm that efficiently implements the exact formula. Numerical studies were conducted to study the accuracy of the developed algorithm and approximation methods. We also studied the computational efficiency of different methods. The paper is concluded with a discussion on the use of different methods in practice and some suggestions for practitioners.

Reliability Meets Big Data: Opportunities and Challenges

ABSTRACT Reliability field data such as that obtained from warranty claims and maintenance records have been used traditionally for such purposes as generating predictions for warranty costs and optimizing the cost of system operation and maintenance. In the current (and future) generation of many products, the nature of field reliability data is changing dramatically. In particular, products can be outfitted with sensors that can be used to capture information about how and when and under what environmental and operating conditions products are being used. Today some of that information is being used to monitor system health and interest is building to develop prognostic information systems. There are, however, many other potential applications for using such data. In this article we review some applications where field reliability data are used and explore some of the opportunities to use modern reliability data to provide stronger statistical methods to operate and predict the performance of systems in the field. We also provide some examples of recent technical developments designed to be used in such applications and outline remaining challenges.

Prediction of remaining life of power transformers based on left truncated and right censored lifetime data

Prediction of the remaining life of high-voltage power transformers is an important issue for energy companies because of the need for planning maintenance and capital expenditures. Lifetime data for such transformers are complicated because transformer lifetimes can extend over many decades and transformer designs and manufacturing practices have evolved. We were asked to develop statisticallybased predictions for the lifetimes of an energy company’s fleet of high-voltage transmission and distribution transformers. The company’s data records begin in 1980, providing information on installation and failure dates of transformers. Although the dataset contains many units that were installed before 1980, there is no information about units that were installed and failed before 1980. Thus, the data are left truncated and right censored. We use a parametric lifetime model to describe the lifetime distribution of individual transformers. We develop a statistical procedure, based on age-adjusted life distributions, for computing a prediction interval for remaining life for individual transformers now in service. We then extend these ideas to provide predictions and prediction intervals for the cumulative number of failures, over a range of time, for the overall fleet of transformers.

Using Accelerated Life Tests Results to Predict Product Field Reliability

Accelerated life tests (ALTs) provide timely assessments of the reliability of materials, components, and subsystems. ALTs can be run at any of these levels or at the full-system level. Sometimes ALTs generate multiple failure modes. A frequently asked question near the end of an ALT program is “what do these test results say about field performance?” ALTs are carefully controlled, whereas the field environment is highly variable. For example, products in the field have different average use rates across the product population. With good characterization of field use conditions, it may be possible to use ALT results to predict the failure time distribution in the field. When such information is not available but both life test data and field data (from, e.g., warranty returns) are available, it may be possible to find a model to relate the two data sets. Under a reasonable set of practical assumptions, this model then can be used to predict the failure time distribution for a future component or product operating in the same use environment. This paper describes a model and methods for such situations. The methods are illustrated by an example to predict the failure time distribution of a newly designed product with two failure modes. Supplemental material for this article is available online at the Technometrics website.

Field-Failure and Warranty Prediction Based on Auxiliary Use-Rate Information

Assessment of risk due to product failure is important both for purposes of finance (e.g., warranty costs) and safety (e.g., potential loss of human life). In many applications a prediction of the number of future failures is an important input to such an assessment. Usually the field-data response used to make predictions of future failures is the number of weeks (or another unit of real time) in service. Use-rate information usually is not available (automobile warranty data are an exception, where both weeks in service and number of miles driven are available for units returned for warranty repair). With new technology, however, sensors and smart chips are being installed in many modern products ranging from computers and printers to automobiles and aircraft engines. Thus the coming generations of field data for many products will provide information on how the product was used and the environment in which it was used. This article was motivated by the need to predict warranty returns for a product with multiple failure modes. For this product, cycles-to-failure/use-rate information was available for those units that were connected to the network. We show how to use a cycles-to-failure model to compute predictions and prediction intervals for the number of warranty returns. We also present prediction methods for units not connected to the network. To provide insight into the reasons that use-rate models provide better predictions, we also present a comparison of asymptotic variances comparing the cycles-to-failure and time-to-failure models. This article has supplementary material online.

How do heterogeneities in operating environments affect field failure predictions and test planning

The main objective of accelerated life tests (ALTs) is to predict fraction failings of products in the field. However, there are often discrepancies between the predicted fraction failing from the lab testing data and that from the field failure data, due to the yet unobserved heterogeneities in usage and operating conditions. Most previous research on ALT planning and data analysis ignores the discrepancies, resulting in inferior test plans and biased predictions. In this paper we model the heterogeneous environments together with their effects on the product failures as a frailty term to link the lab failure time distribution and field failure time distribution of a product. We show that in the presence of the heterogeneous operating conditions, the hazard rate function of the field failure time distribution exhibits a range of shapes. Statistical inference procedure for the frailty models is developed when both the ALT data and the field failure data are available. Based on the frailty models, optimal ALT plans aimed at predicting the field failure time distribution are obtained. The developed methods are demonstrated through a real life example.

Field-Failure Predictions Based on Failure-Time Data With Dynamic Covariate Information.

Modern technological developments, such as smart chips, sensors, and wireless networks, have changed many data-collection processes. For example, there are more and more products being produced with automatic data-collecting devices that track how and under which environments the products are being used. Although there is a tremendous amount of dynamic data being collected, there has been little research on using such data to provide more accurate reliability information for products and systems. Motivated by a warranty-prediction application, this article focuses on using failure-time data with dynamic covariate information to make field-failure predictions. We provide a general method for prediction using failure-time data with dynamic covariate information. The dynamic covariate information is incorporated into the failure-time distribution through a cumulative exposure model. We develop a procedure to predict field-failure returns up to a specified future time. This procedure accounts for unit-to-unit variability in the covariate process. We also define a metric to quantify the improvements in prediction accuracy obtained by using dynamic information. We conduct simulations to study the effect of different sources of covariate process variability on predictions. We also provide some discussion of future opportunities for using dynamic data. This article has supplementary material online.

Reliability Analysis of Repairable Systems With Dependent Component Failures Under Partially Perfect Repair

Existing reliability models for repairable systems with a single component can be well applied for a range of repair actions from perfect repair to minimal repair. Establishing reliability models for multi-component repairable systems, however, is still a challenge problem when considering the dependency of component failures. This paper focuses on a special repair assumption, called partially perfect repair, for repairable systems with dependent component failures, where only the failed component is repaired to as good as new condition. A parametric reliability model is proposed to capture the statistical dependency among different component failures, in which the joint distribution of the latent component failure time is established using copula functions. The model parameters are estimated by using the maximum likelihood method, and the maximum likelihood function is calculated based on the conditional probability. Based on the proposed reliability model, statistical hypothesis testing procedures are developed to determine the dependency of component failures. The developed methods are illustrated with an application in a cylinder head assembling cell that consists of multiple stations.

A Tool for Evaluating Time-Varying-Stress Accelerated Life Test Plans With Log-Location-Scale Distributions

Accelerated life tests (ALTs) are often used to make timely assessments of the lifetime distribution of materials and components. The goal of many ALTs is the estimation of a quantile of a log-location-scale failure time distribution. Much of the previous work on planning accelerated life tests has focused on deriving test-planning methods under a specific log-location-scale distribution. This paper presents a new approach for computing approximate large-sample variances of maximum likelihood estimators of a quantile of a general log-location-scale distribution with censoring, and time-varying stress. The approach is based on a cumulative exposure model. Using sample data from a published paper describing optimum ramp-stress test plans, we show that our approach and the one used in the previous work give the same variance-covariance matrix of the quantile estimator from the two different approaches. Then, as an application of this approach, we extend the previous work to a new optimum ramp-stress test plan obtained by simultaneously adjusting the ramp rate with the lower start level of stress. We find that the new optimum test plan can have a smaller variance than that of the optimum ramp-stress test plan previously obtained by adjusting only the ramp rate. We compare optimum ramp-stress test plans with the more commonly used constant-stress accelerated life test plans. We also conduct simulations to provide insight, and to check the adequacy of the large-sample approximate results obtained by the approach.

Statistical Methods for Degradation Data With Dynamic Covariates Information and an Application to Outdoor Weathering Data

Degradation data provide a useful resource for obtaining reliability information for some highly reliable products and systems. In addition to product/system degradation measurements, it is common nowadays to dynamically record product/system usage as well as other life-affecting environmental variables, such as load, amount of use, temperature, and humidity. We refer to these variables as dynamic covariate information. In this article, we introduce a class of models for analyzing degradation data with dynamic covariate information. We use a general path model with individual random effects to describe degradation paths and a vector time series model to describe the covariate process. Shape-restricted splines are used to estimate the effects of dynamic covariates on the degradation process. The unknown parameters in the degradation data model and the covariate process model are estimated by using maximum likelihood. We also describe algorithms for computing an estimate of the lifetime distribution induced by the proposed degradation path model. The proposed methods are illustrated with an application for predicting the life of an organic coating in a complicated dynamic environment (i.e., changing UV spectrum and intensity, temperature, and humidity). This article has supplementary material online.