Saddlepoint Approximation Method Research Articles

Semiparametric evaluation of first‐passage distribution for step‐stress accelerated degradation tests

AbstractIn reliability engineering, different types of accelerated degradation tests have been used to obtain reliability information for evaluating highly reliable or expensive products. The step‐stress accelerated degradation test (SSADT) is one of the useful experimental schemes that can be used to save the resources of an experiment. Motivated by the SSADT data for operational amplifiers collected in Xi'an Microelectronic Technology Institute, in which the underlying degradation mechanism of the operational amplifiers is unknown, we propose a semiparametric approach for SSADT data analysis that does not require strict distributional assumptions. Specifically, the empirical saddlepoint approximation method is proposed to estimate the items' lifetime (first‐passage time) distribution at both stress levels included and not included in the SSADT experiment. Monte Carlo simulation studies are used to evaluate the performance and illustrate the advantages of the proposed approach. Finally, the proposed semiparametric approach is applied to analyze the motivating data set.

Saddlepoint p-values for a class of location-scale tests under randomized block design

This paper deals with a class of nonparametric two-sample location-scale tests. The purpose of this paper is to approximate the exact p-value of the considered class under a randomized block design. The exact p-value of the considered class is approximated by the saddlepoint approximation method, also by the traditional method which is the normal approximation method. The saddlepoint approximation method is more accurate than the normal approximation method in approximating the exact p-value, and does not take a lot of time like the simulation method. This accuracy is proved by applying the mentioned methods to two real data sets and a simulation study.

Time-dependent fatigue reliability estimation and optimal design of scraper chains under corrosive wear effect

Fatigue damage is one of the main failures of the scraper chain. Moreover, corrosive wear has a detrimental effect on fatigue crack propagation. It is so essential to study the fatigue reliability of the scraper chain considering the interaction of corrosive wear and fatigue. This paper is intended to propose a reliability-based estimation and optimal design of the scraper chain considering the performance degradation of the scraper chain under the interaction of corrosive wear and fatigue. First, the dynamic model of the scraper conveyor is established based on the finite element method. Second, Paris Low is used to empirically describe the occurrence of cracking with corrosion effects. Furthermore, the time-dependent reliability of scraper chains under the interaction of the corrosive wear and fatigue crack propagation is addressed by the fourth-moment saddlepoint approximation (FMSA) method, and the reliability-based design optimization (RBDO) of the scraper chain is then carried out. Finally, the applicability of the proposed method is verified by the SGZ1000/1400 scraper conveyor.

Saddlepoint Approximation Method in Reliability Analysis: A Review

Saddlepoint Approximation Method in Reliability Analysis: A Review

Study on reliability of emergency braking performance of high-speed and heavy-load monorail crane

The reliability of monorail crane braking system has an important influence on braking safety, high-speed and heavy-load operation, which poses great challenges to braking safety, and it is necessary to evaluate its braking reliability accurately and efficiently. Firstly, the dynamic performance and thermal-mechanical coupling characteristics of high-speed and heavy-load monorail crane under different braking parameters were analyzed. Secondly, the random response model of braking distance and braking temperature was established by combining the design of experiment method (DoE) and Dendrite Net (DD). Finally, the high-order moment saddlepoint approximation (SPA) method was used to evaluate the emergency braking reliability of the monorail crane. The results can provide a reference for the selection of key parameters and the evaluation of braking safety of the monorail crane braking system under high-speed and heavy-load conditions.

A Multi-Fidelity Integration Method for Reliability Analysis of Industrial Robots

Abstract A multi-fidelity integration method is proposed to analyze the reliability of multiple performance indicators (MPI) for industrial robots. In order to high-fidelity mapping the performance of industrial robots, a unified multi-domain model (UMDM) is first established. The contribution-degree analysis is then used to classify the input random variables into interacting and non-interacting ones. Thus, the high-dimensional integration of reliability analysis is separated into a low-dimensional integration and multiple one-dimensional integrations in an additive form. Here, the low-dimensional integration consisting of the interacting variables is calculated using the high-precision mixed-degree cubature formula (MDCF), and the computational results are treated as high-fidelity data. The one-dimensional integration consisting of non-interacting variables is then computed by the highly efficient five-point Gaussian Hermite quadrature (FGHQ), and the computational results are named low-fidelity data. A multi-fidelity integration method is constructed by fusing the high-fidelity data and the low-fidelity data to obtain the statistical moments of the MPI. Subsequently, the probability density function and the failure probability of the MPI are estimated using the saddlepoint approximation method. Finally, some representative methods are performed to verify the superiority of the proposed method.

Linear rank tests for left-truncated data using randomized block design: saddlepoint p-values and confidence intervals

Survival analysis is most typically characterized by left truncation; left-truncated data are composed of two recorded times, the time of occurrence of an event before the event of interest, and the time of the interested event. In this study, p-values of the permutation distribution of the linear rank tests are computed using saddlepoint approximation method for left-truncated data using a randomized block design. Simulation studies demonstrate the saddlepoint approximation method’s accuracy in approximating p-value of the linear rank tests. Due to the efficiency of the proposed method over the normal approximation, 95% confidence intervals for the treatment impact are performed.

Time-dependent reliability analysis under random and interval uncertainties based on Kriging modeling and saddlepoint approximation

Time-dependent reliability analysis is capable of evaluating the reliability of the system over its full life cycle, which is of highly concerned for designers. In practice, some uncertainties cannot be simply described by a deterministic distribution due to lack of knowledge or information, and generally, their upper and lower bounds can be obtained. Thus, this paper develops a novel computational method for time-dependent reliability under random and interval uncertainties. The first step of the proposed method is to construct the Kriging surrogating model of the most probable point trajectory, with which the bound of the limit-state function can be efficiently recast as an equivalent Gaussian process. In this case, the computation of the bound of time-dependent failure probability (TDFP) is converted to a high-dimensional Gaussian integral problem. To solve this problem, the expansion optimal linear estimation, sparse-grid Gaussian quadrature technique and saddlepoint approximation method are integrated so that the above integral can be efficiently computed. Finally, four engineering examples are studied to demonstrate the effectiveness of the proposed method by comparison with previous methods. The results show this novel method can provide accurate and efficient time-dependent reliability analysis in terms of random and interval uncertainties.

Approximation for the distribution function of non-central complex Wilks statistic

In this article, we use the saddlepoint approximation method to compute the cumulative distribution function of non-central complex Wilks statistic. This approximation does not require conditions on the dimension, degrees of freedom, and non-centrality matrix. We also compare our results with known approximations to show the accuracy of our approximation. For applications, we compute the power functions of the Wilks statistic in the complex multivariate analysis of variance and the probability of detection of the Wilks detector in signal detection in noise model.

A novel geometric AR(1) model and its estimation

This paper proposes a first-order geometric autoregressive model based on a new thinning operator. The parameters of the model are estimated by the method of Two-Stage Conditional Least Squares and Saddlepoint Approximation. Simulation studies demonstrate that the saddlepoint approximation estimator performs well, even in the case of small sample sizes. To illustrate the significance of the model, a real data analysis of the number of downloads of a TeX editor is presented.

Saddlepoint approximations to score test statistics in logistic regression for analyzing genome-wide association studies.

We investigate saddlepoint approximations of tail probabilities of the score test statistic in logistic regression for genome-wide association studies. The inaccuracy in the normal approximation of the score test statistic increases with increasing imbalance in the response and with decreasing minor allele counts. Applying saddlepoint approximation methods greatly improve the accuracy, even far out in the tails of the distribution. By using exact results for a simple logistic regression model, as well as simulations for models with nuisance parameters, we compare double saddlepoint methods for computing two-sided -values and mid- -values. These methods are also compared to a recent single saddlepoint procedure. We investigate the methods further on data from UK Biobank with skin and soft tissue infections as phenotype, using both common and rare variants.

Statistical Inference of the Class of Nonparametric Tests for the Panel Count and Current Status Data from the Perspective of the Saddlepoint Approximation

Many statisticians resort to using the asymptotic normal approximation method to carry out statistical inference for many statistical tests, especially nonparametric ones. In this article, the saddlepoint approximation method is proposed as an alternative to the asymptotic normal approximation method to carry out statistical inference for a number of nonparametric tests for an important type of data that appears frequently in many clinical studies such as cancer and tumorigenicity studies. In clinical trials, there are many strategies through which treatments are assigned to patients. Equal allocation of both treatments is a largely prevalent approach in clinical trials to eliminate experimental bias and increase power. Accordingly, the statistical analysis is carried out based on the truncated binomial design, which is one of the designs that achieve a perfect balance between the two treatments. To clarify the accuracy of the proposed approximation method, two sets of real data are analyzed, and for the same purpose, a comprehensive simulation study is carried out.

Saddlepoint approximation p-values of weighted log-rank tests based on censored clustered data under block Efron's biased-coin design.

Clustered survival data frequently occurs in biomedical research fields and clinical trials. The log-rank tests are used for two independent samples of clustered data tests. We use the block Efron's biased-coin randomization (design) to assign patients to treatment groups in a clinical trial by forcing a sequential experiment to be balanced. In this article, the -values of the null permutation distribution of log-rank tests for clustered data are approximated via the double saddlepoint approximation method. Comprehensive numerical studies are carried out to assess the accuracy of the saddlepoint approximation. This approximation demonstrates great accuracy over the asymptotic normal approximation.

Saddlepoint p-values for a class of nonparametric tests for the current status and panel count data under generalized permuted block design

<abstract><p>Current status and panel count data appear in many applied fields, including medicine, clinical trials, epidemiology, econometrics, demography, engineering and public health. Therefore, in this article, we use the saddlepoint approximation method to approximate the exact p-value of a number of nonparametric tests for the current status and panel count data under a generalized permuted block design. The saddlepoint approximation is referred to as higher-order approximation and it is more accurate than the methods that lead to approximations that are accurate to the first order, such as the asymptotic normal approximation method. To verify the accuracy and efficiency of the saddlepoint approximation method, a simulation study is conducted. The simulation study results confirm that the saddlepoint approximation method is more powerful than the existing approximation method. Furthermore, number of real current status and panel count data sets are analyzed and displayed as illustrative examples.</p></abstract>

An Uncertainty Analysis and Reliability-Based Multidisciplinary Design Optimization Method Using Fourth-Moment Saddlepoint Approximation

In uncertainty analysis and reliability-based multidisciplinary design and optimization (RBMDO) of engineering structures, the saddlepoint approximation (SA) method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables involved in SA should be easy to handle. Additionally, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization, a fourth-moment saddlepoint approximation (FMSA) method is introduced into RBMDO. In FMSA, the approximate cumulative generating functions are constructed based on the first four moments of the limit state function. The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment, which is based on the performance measure approach strategy. Two engineering examples are introduced to verify the effectiveness of proposed method.

Kinematic trajectory accuracy reliability analysis for industrial robots considering intercorrelations among multi-point positioning errors

The kinematic trajectory accuracy reliability is an essential index to evaluate the service performance of industrial robots. This study proposes a new kinematic trajectory accuracy reliability analysis method for industrial robots by integrating the sparse grid technique, the saddlepoint approximation method and copula functions. The novelty of this study lies in comprehending the intercorrelations among single point three-coordinate directional positioning errors and multiple points positioning errors on the kinematic trajectory. To start with, the statistical moments of the positioning errors at an arbitrary point on the trajectory of industrial robots are calculated by the extended sparse grid (SPGR) technique. The single-point positioning accuracy reliability is then evaluated by the saddlepoint approximation (SPA) method. The joint failure probability of positioning accuracy between any two points and the matrix of correlation coefficients of positioning errors among multiple points are subsequently calculated by copula functions. Two methods, namely the boundary theory and the multivariate normal distribution theory, are employed in the current study to calculate the failure probability of kinematic trajectory accuracy. Three examples are implemented to demonstrate the proficiency of the currently proposed methods.

An adaptive hybrid dimension-reduction model and its application for statistical moments estimation of stochastic systems and reliability analysis of structures

From the perspective of accuracy, efficiency and versatility, calculating statistical moments for different forms of performance functions in a unified way remains a challenge, and their accuracy also considerably affects the results of subsequent reliability analysis. In this article, a new dimension-reduction model, termed as adaptive hybrid dimension-reduction model (AH-DRM), is derived to approximate the performance functions with different characteristics relatively accurately. Based on the AH-DRM, a highly flexible point estimation method (PEM), which has good versatility for the various forms of performance function, is developed for statistical moments estimation of stochastic systems. Subsequently, to illustrate the performance of the proposed method for different performance function forms, three examples, including weakly, moderately and strongly nonlinear benchmark examples, are investigated. The results indicate that the proposed method is versatile for each example and can receive desirable accuracy for evaluating the first-four central moments of the performance functions with high efficiency. Finally, the developed PEM is applied to reconstruct the probability density function of the performance function and calculate the reliability of structures with the aid of the saddle-point approximation (SPA) method, and two practical engineering examples are used to verify the applicability of the proposed method in structural reliability analysis.

Saddlepoint approximation for weighted log-rank tests based on block truncated binomial design

ABSTRACT Clustered data frequently occur in biomedical research fields and clinical trials. The log-rank tests are widely used for two-independent samples of clustered data tests. The randomized block design and truncated binomial design are used for forcing balance in clinical trials and reducing selection bias. In this paper, survival clustered data are randomized by generalized randomized block, and subsequently clustered data in each block are randomized by truncated binomial design. Consequently, the p-values of the null permutation distribution of log-rank tests for clustered data are approximated via the double saddlepoint approximation method. Comprehensive numerical studies are carried out to assess the accuracy of the saddlepoint approximation. This approximation has a great accuracy over the asymptotic normal approximation.

Envelope Method for Time- and Space-Dependent Reliability Prediction

Abstract Reliability can be predicted by a limit-state function, which may vary with time and space. This work extends the envelope method for a time-dependent limit-state function to a time- and space-dependent limit-state function. The proposed method uses the envelope function of time- and space-dependent limit-state function. It at first searches for the most probable point (MPP) of the envelope function using the sequential efficient global optimization in the domain of the space and time under consideration. Then the envelope function is approximated by a quadratic function at the MPP for which analytic gradient and Hessian matrix of the envelope function are derived. Subsequently, the second-order saddlepoint approximation method is employed to estimate the probability of failure. Three examples demonstrate the effectiveness of the proposed method. The method can efficiently produce an accurate reliability prediction when the MPP is within the domain of the space and time under consideration.

Parameter estimation of MIMO FSO systems using saddlepoint approximation

In this paper, the challenges of parameter estimation for multiple-input multiple-out (MIMO) free-space optical (FSO) communication systems employing equal gain combining diversity techniques over the Gamma–Gamma turbulence channels with pointing errors are addressed. According to the goodness-of-fit statistical tests results previously reported by Kolmogorov-Smirnov, the approximate probability density function obtained by a saddlepoint approximation (SAP) method provide a highly efficient approximation of the actual channel models over a wide range of channel conditions. To reveal the importance of the SAP approximation method, we propose a maximum likelihood estimation algorithm to estimate its parameters. The accuracy of the proposed estimator is investigated by using the mean square error (MSE) and normalized mean square error (NMSE); they exhibit satisfactory performance in both noiseless and noisy environments. The bit error rate (BER) performance for MIMO FSO systems with an SAP estimator is also discussed.