Modelling Accelerated Degradation Data Using Wiener Diffusion With A Time Scale Transformation

Engineering degradation tests allow industry to assess the potential life span of long-life products that do not fail readily under accelerated conditions in life tests. A general statistical model is presented here for performance degradation of an item of equipment. The degradation process in the model is taken to be a Wiener diffusion process with a time scale transformation. The model incorporates Arrhenius extrapolation for high stress testing. The lifetime of an item is defined as the time until performance deteriorates to a specified failure threshold. The model can be used to predict the lifetime of an item or the extent of degradation of an item at a specified future time. Inference methods for the model parameters, based on accelerated degradation test data, are presented. The model and inference methods are illustrated with a case application involving self-regulating heating cables. The paper also discusses a number of practical issues encountered in applications. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bayesian Semiparametric Modelling in Quantile Regression

Abstract. We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non‐parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non‐parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.     

Sequential design for nonparametric inference

The performance of nonparametric function estimates often depends on the choice of design points. Based on the mean integrated squared error criterion, we propose a sequential design procedure that updates the model knowledge and optimal design density sequentially. The methodology is developed under a general framework covering a wide range of nonparametric inference problems, such as conditional mean and variance functions, the conditional distribution function, the conditional quantile function in quantile regression, functional coefficients in varying coefficient models and semiparametric inferences. Based on our empirical studies, nonparametric inference based on the proposed sequential design is more efficient than the uniform design and its performance is close to the true but unknown optimal design. The Canadian Journal of Statistics 40: 362–377; 2012 © 2012 Statistical Society of Canada     

Automatic Bayesian quantile regression curve fitting

Quantile regression, including median regression, as a more completed statistical model than mean regression, is now well known with its wide spread applications. Bayesian inference on quantile regression or Bayesian quantile regression has attracted much interest recently. Most of the existing researches in Bayesian quantile regression focus on parametric quantile regression, though there are discussions on different ways of modeling the model error by a parametric distribution named asymmetric Laplace distribution or by a nonparametric alternative named scale mixture asymmetric Laplace distribution. This paper discusses Bayesian inference for nonparametric quantile regression. This general approach fits quantile regression curves using piecewise polynomial functions with an unknown number of knots at unknown locations, all treated as parameters to be inferred through reversible jump Markov chain Monte Carlo (RJMCMC) of Green (Biometrika 82:711–732, 1995). Instead of drawing samples from the posterior, we use regression quantiles to create Markov chains for the estimation of the quantile curves. We also use approximate Bayesian factor in the inference. This method extends the work in automatic Bayesian mean curve fitting to quantile regression. Numerical results show that this Bayesian quantile smoothing technique is competitive with quantile regression/smoothing splines of He and Ng (Comput. Stat. 14:315–337, 1999) and P-splines (penalized splines) of Eilers and de Menezes (Bioinformatics 21(7):1146–1153, 2005).     

Bayesian bridge quantile regression

Regularization methods for simultaneous variable selection and coefficient estimation have been shown to be effective in quantile regression in improving the prediction accuracy. In this article, we propose the Bayesian bridge for variable selection and coefficient estimation in quantile regression. A simple and efficient Gibbs sampling algorithm was developed for posterior inference using a scale mixture of uniform representation of the Bayesian bridge prior. This is the first work to discuss regularized quantile regression with the bridge penalty. Both simulated and real data examples show that the proposed method often outperforms quantile regression without regularization, lasso quantile regression, and Bayesian lasso quantile regression.     

Common threshold in quantile regressions with an application to pricing for reputation

The paper develops a systematic estimation and inference procedure for quantile regression models where there may exist a common threshold effect across different quantile indices. We first propose a sup-Wald test for the existence of a threshold effect, and then study the asymptotic properties of the estimators in a threshold quantile regression model under the shrinking threshold effect framework. We consider several tests for the presence of a common threshold value across different quantile indices and obtain their limiting distributions. We apply our methodology to study the pricing strategy for reputation through the use of a data set from Taobao.com. In our economic model, an online seller maximizes the sum of the profit from current sales and the possible future gain from a targeted higher reputation level. We show that the model can predict a jump in optimal pricing behavior, which is considered as “reputation effect” in this paper. The use of threshold quantile regression model allows us to identify and explore the reputation effect and its heterogeneity in data. We find both reputation effects and common thresholds for a range of quantile indices in seller’s pricing strategy in our application.     

Direct estimation of the percentile 'p-value' for the one-sample median test

In this paper we outline and illustrate an easy-to-use inference procedure for directly calculating the approximate bootstrap percentile-type p-value for the one-sample median test, i.e. we calculate the bootstrap p -value without resampling, by using a fractional order statistics based approach. The method parallels earlier work on fractionalorder-statistics-based non-parametric bootstrap percentile-type confidence intervals for quantiles. Monte Carlo simulation studies are performed, which illustrate that the fractional-order-statistics-based approach to the one-sample median test has accurate type I error control for small samples over a wide range of distributions; is easy to calculate; and is preferable to the sign test in terms of type I error control and power. Furthermore, the fractional-order-statistics-based median test is easily generalized to testing that any quantile has some hypothesized value; for example, tests for the upper or lower quartile may be performed using the same framework.     

Quantile inference based on partially rank-ordered set samples

This paper develops statistical inference for population quantiles based on a partially rank-ordered set (PROS) sample design. A PROS sample design is similar to a ranked set sample with some clear differences. This design first creates partially rank-ordered subsets by allowing ties whenever the units in a set cannot be ranked with high confidence. It then selects a unit for full measurement at random from one of these partially rank-ordered subsets. The paper develops a point estimator, confidence interval and hypothesis testing procedure for the population quantile of order p. Exact, as well as asymptotic, distribution of the test statistic is derived. It is shown that the null distribution of the test statistic is distribution-free, and statistical inference is reasonably robust against possible ranking errors in ranking process.     

Uniform Inference on Quantile Effects under Sharp Regression Discontinuity Designs

ABSTRACT

This study develops methods for conducting uniform inference on quantile treatment effects for sharp regression discontinuity designs. We develop a score test for the treatment significance hypothesis and Wald-type tests for the hypotheses related to treatment significance, homogeneity, and unambiguity. The bias from the nonparametric estimation is studied in detail. In particular, we show that under some conditions, the asymptotic distribution of the score test is unaffected by the bias, without under-smoothing. For situations where the conditions can be restrictive, we incorporate a bias correction into the Wald tests and account for the estimation uncertainty. We also provide a procedure for constructing uniform confidence bands for quantile treatment effects. As an empirical application, we use the proposed methods to study the effect of cash-on-hand on unemployment duration. The results reveal pronounced treatment heterogeneity and also emphasize the importance of considering the long-term unemployed.     

Improving extreme quantile estimation via a folding procedure

In many applications (geosciences, insurance, etc.), the peaks-over-thresholds (POT) approach is one of the most widely used methodology for extreme quantile inference. It mainly consists of approximating the distribution of exceedances above a high threshold by a generalized Pareto distribution (GPD). The number of exceedances which is used in the POT inference is often quite small and this leads typically to a high volatility of the estimates. Inspired by perfect sampling techniques used in simulation studies, we define a folding procedure that connects the lower and upper parts of a distribution. A new extreme quantile estimator motivated by this theoretical folding scheme is proposed and studied. Although the asymptotic behaviour of our new estimate is the same as the classical (non-folded) one, our folding procedure reduces significantly the mean squared error of the extreme quantile estimates for small and moderate samples. This is illustrated in the simulation study. We also apply our method to an insurance dataset.     

Jackknife method for intermediate quantiles

Quantile function plays an important role in statistical inference, and intermediate quantile is useful in risk management. It is known that Jackknife method fails for estimating the variance of a sample quantile. By assuming that the underlying distribution satisfies some extreme value conditions, we show that Jackknife variance estimator is inconsistent for an intermediate order statistic. Further we derive the asymptotic limit of the Jackknife-Studentized intermediate order statistic so that a confidence interval for an intermediate quantile can be obtained. A simulation study is conducted to compare this new confidence interval with other existing ones in terms of coverage accuracy.     

Non-parametric quantile estimate for length-biased and right-censored data with competing risks

In this article, we propose a non-parametric quantile inference procedure for cause-specific failure probabilities to estimate the lifetime distribution of length-biased and right-censored data with competing risks. We also derive the asymptotic properties of the proposed estimators of the quantile function. Furthermore, the results are used to construct confidence intervals and bands for the quantile function. Simulation studies are conducted to illustrate the method and theory, and an application to an unemployment data is presented.     

Bayesian elastic net single index quantile regression

Single index model conditional quantile regression is proposed in order to overcome the dimensionality problem in nonparametric quantile regression. In the proposed method, the Bayesian elastic net is suggested for single index quantile regression for estimation and variables selection. The Gaussian process prior is considered for unknown link function and a Gibbs sampler algorithm is adopted for posterior inference. The results of the simulation studies and numerical example indicate that our propose method, BENSIQReg, offers substantial improvements over two existing methods, SIQReg and BSIQReg. The BENSIQReg has consistently show a good convergent property, has the least value of median of mean absolute deviations and smallest standard deviations, compared to the other two methods.     

Empirical likelihood for quantile regression models with longitudinal data

We develop two empirical likelihood-based inference procedures for longitudinal data under the framework of quantile regression. The proposed methods avoid estimating the unknown error density function and the intra-subject correlation involved in the asymptotic covariance matrix of the quantile estimators. By appropriately smoothing the quantile score function, the empirical likelihood approach is shown to have a higher-order accuracy through the Bartlett correction. The proposed methods exhibit finite-sample advantages over the normal approximation-based and bootstrap methods in a simulation study and the analysis of a longitudinal ophthalmology data set.     

Inference from Accelerated Degradation and Failure Data Based on Gaussian Process Models

An important problem in reliability and survival analysis is that of modeling degradation together with any observed failures in a life test. Here, based on a continuous cumulative damage approach with a Gaussian process describing degradation, a general accelerated test model is presented in which failure times and degradation measures can be combined for inference about system lifetime. Some specific models when the drift of the Gaussian process depends on the acceleration variable are discussed in detail. Illustrative examples using simulated data as well as degradation data observed in carbon-film resistors are presented.     

Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Nonparametric Inference for Time-Varying Coefficient Quantile Regression

The article considers nonparametric inference for quantile regression models with time-varying coefficients. The errors and covariates of the regression are assumed to belong to a general class of locally stationary processes and are allowed to be cross-dependent. Simultaneous confidence tubes (SCTs) and integrated squared difference tests (ISDTs) are proposed for simultaneous nonparametric inference of the latter models with asymptotically correct coverage probabilities and Type I error rates. Our methodologies are shown to possess certain asymptotically optimal properties. Furthermore, we propose an information criterion that performs consistent model selection for nonparametric quantile regression models of nonstationary time series. For implementation, a wild bootstrap procedure is proposed, which is shown to be robust to the dependent and nonstationary data structure. Our method is applied to studying the asymmetric and time-varying dynamic structures of the U.S. unemployment rate since the 1940s. Supplementary materials for this article are available online.     

Optimal design of accelerated degradation tests based on Wiener process models

Optimal accelerated degradation test (ADT) plans are developed assuming that the constant-stress loading method is employed and the degradation characteristic follows a Wiener process. Unlike the previous works on planning ADTs based on stochastic process models, this article determines the test stress levels and the proportion of test units allocated to each stress level such that the asymptotic variance of the maximum-likelihood estimator of the qth quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are also developed for checking the validity of the relationship between the model parameters and the stress variable. Finally, using an example, sensitivity analysis procedures are presented for evaluating the robustness of optimal and compromise plans against the uncertainty in the pre-estimated parameter value, and the importance of optimally determining test stress levels and the proportion of units allocated to each stress level are illustrated.     

Bayesian Methods for a Growth-Curve Degradation Model with Repeated Measures

The increasingreliability of some manufactured products has led to fewer observedfailures in reliability testing. Thus, useful inference on thedistribution of failure times is often not possible using traditionalsurvival analysis methods. Partly as a result of this difficulty,there has been increasing interest in inference from degradationmeasurements made on products prior to failure. In the degradationliterature inference is commonly based on large-sample theoryand, if the degradation path model is nonlinear, their implementationcan be complicated by the need for approximations. In this paperwe review existing methods and then describe a fully Bayesianapproach which allows approximation-free inference. We focuson predicting the failure time distribution of both future unitsand those that are currently under test. The methods are illustratedusing fatigue crack growth data.