Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.     

Reliability analysis of wind turbine subassemblies based on the 3-P Weibull model via an ergodic artificial bee colony algorithm

The Weibull distribution is the most widely used model for the reliability evaluation of wind turbine subassemblies. Considering the important role of the location parameter in the three-parameter (3-P) Weibull model and its rare application in wind turbines, this study conducted a reliability analysis of wind turbine subassemblies based on field data that obeyed the 3-P Weibull distribution model via maximum likelihood estimation (MLE). An improved ergodic artificial bee colony algorithm (ErgoABC) was proposed by introducing the chaos search theory, global best solution, and Lévy flights strategy into the classical artificial bee colony (ABC) algorithm to determine the maximum likelihood estimates of the Weibull distribution parameters. This was validated against simulation calculations and proved to be efficient for high-dimensional function optimization and parameter estimation of the 3-P Weibull distribution. Finally, reliability analyses of the wind turbine subassemblies based on different types of field failure data were conducted using ErgoABC. The results show that the 3-P Weibull model can reasonably evaluate the lifetime distribution of critical wind turbine subassemblies, such as generator slip rings and main shafts, on which the location parameter has a significant effect.     

Reliability modeling involving two Weibull distributions

In this paper, we study three different models, each involving two Weibull distributions, to model failure data. We carry out a characterization of the plots on Weibull plotting paper (WPP). This allows one to decide on the appropriateness of a particular model formulation to model a given data set. For each of the three model formulations, we discuss parameter estimation based on the plots and illustrate their application using real data.     

Weibull and inverse Weibull mixture models allowing negative weights

This article presents the finding that when components of a system follow the Weibull or inverse Weibull distribution with a common shape parameter, then the system can be represented by a Weibull or inverse Weibull mixture model allowing negative weights. We also use an example to illustrate that the proposed mixture model can be used to approximate the reliability behaviours of the consecutive-k-out-of-n systems. The example also shows data analysis procedures when the parameters of the component life distributions are either known or unknown.     

Inference for step-stress plan with Khamis-Higgins model under type-II censored Weibull data

The accelerated life testing (ALT) is frequently used in examining the component reliability and acceptance testing. The ALT is carried out by exposing the unit to higher stress levels in order to observe data faster than those are producing under the normal conditions. The simple step-stress model based on type-II censoring Weibull lifetimes is studied here. In addition, the lifetimes satisfy Khamis-Higgins model assumption. In this paper, Bayesian approaches are developed for estimating the model parameters and predicting times to failure of future censored of the simple step-stress model from Weibull distribution using Khamis-Higgins model. The main goal of this work consists of two parts. First, the Bayesian estimation of the unknown parameters involved in the model is considered by adopting Devroye method to generate log-concave densities within sampling-based algorithm under different loss functions. The Bayes and highest posterior density credible intervals are then established. Second, the estimation of the posterior predictive density of the future lifetimes are discussed to obtain the point and prediction intervals with a given coverage probability. Monte Carlo simulation is performed to check the efficiency of the developed procedures and analyze a real data set for illustrative purposes.     

Statistics of single‐fibre tensile strength incorporating spatial distribution of microcracks

The random distribution of single‐fibre tensile strength has been commonly characterized by the two‐parameter Weibull statistics. However, the calibrated Weibull model from one set of strength data at a given gauge length cannot accurately predicts the strength variation of the fibre at different gauge lengths. Instead of presuming the two‐parameter Weibull distribution or any other specific statistical distribution for the single‐fibre strength to begin with, this work proposes an approach to incorporating the appropriate spatial flaw distribution within a fibre and synchronizing multiple sets of tensile strength data to evaluate the single‐fibre strength distribution. The approach is examined and validated by published single‐fibre strength data sets of glass, ceramic and synthetic and natural carbon fibres. It is shown that the single‐fibre strength statistics does not necessarily always follow the two‐parameter Weibull distribution.     

Generalized mixtures of Weibull components

Weibull mixtures have been used extensively in reliability and survival analysis, and they have also been generalized by allowing negative mixing weights, which arise naturally under the formation of some structures of reliability systems. These models provide flexible distributions for modeling dependent lifetimes from heterogeneous populations. In this paper, we study conditions on the mixing weights and the parameters of the Weibull components under which the considered generalized mixture is a well-defined distribution. Specially, we characterize the generalized mixture of two Weibull components. In addition, some reliability properties are established for these generalized two-component Weibull mixture models. One real data set is also analyzed for illustrating the usefulness of the studied model.     

Conditional mean- and median-based cumulative sum control charts for Weibull data

This article deals with the monitoring of censored data using the cumulative sum (CUSUM) control charts for Weibull lifetimes under type-I censoring. To develop an efficient CUSUM structure for censored data, we use the conditional expected value (CEV) and conditional median (CM) approaches. In particular, we focus on the detection of shifts in the mean of Weibull lifetimes assuming censored data. In addition to fixed/known parameter values, the effect of estimation is assessed on the detection power of control charts. The performance of the proposed charts is evaluated by the average run length (ARL). Furthermore, the ARL performance of CUSUM charts is compared with CEV- and CM-based exponentially weighted moving average (EWMA) control charts. Besides an extensive simulation study, the significance of the current work is illustrated by a data set on the response time of a thermostat experiment.     

一类基于改进Weibull分布模型的 电力电缆寿命评估方法

对电缆的剩余寿命进行评估是电缆寿命管理的关键环节,以往电缆寿命评估的方法中有Arrhenius方法和Weibull分布模型。对这两种方法分别进行分析,特别是确定Arrhenius模型中激活能的计算,以及应用Weibull分布模型对电缆寿命进行评估。最后以实际电缆的状态监测数据,对简化的Arrhenius模型进行仿真,说明所提出的方法是有效的。     

Parameter estimation for bivariate Weibull distribution using generalized moment method for reliability evaluation

Bivariate Weibull distribution can address the life of a system exhibiting 2‐dimensional characteristics in risk and reliability engineering. The applicability of bivariate Weibull distribution has been hindered by its difficulty with parameter estimation, as the number of parameters in bivariate Weibull distribution is more than those in univariate Weibull distribution. Considering a particular structure of a bivariate Weibull distribution model, this paper proposes a generalized moment method (GMM) for parameter estimation. This GMM method is simple, and it has proved to be efficient. The GMM can guarantee the existence and the uniqueness of the solution. A confidence interval for each estimator is derived from the moments of the bivariate distribution. The paper presents a simulation case and 2 real cases to demonstrate the proposed methods.     

Modeling the brittle–ductile transition in ferritic steels. Part II: analysis of scatter in fracture toughness

A dislocation simulation model has been proposed to predict the brittle–ductile transition in ferritic steels in Part I. Here we extend the model to address the problem of inherent scatter in fracture toughness measurements. We carried out a series of Monte Carlo simulations using distributions of microcracks situated on the plane of a main macrocrack. Detailed statistical analysis of the simulation results showed the following: (a) fracture is initiated at one of the microcracks whose size is at the tail of the size distribution function, and (b) the inherent scatter arises from the distribution in the size of the critical microcrack that initiates the fracture and not from the variation of the location of the critical microcrack. Utilizing the weakest-link theory, Weibull analysis shows good agreement with the Weibull modulus values obtained from fracture toughness measurements.     

Prediction of fibre strength at the critical length: a simulation theory and experimental verification for bimodally distributed carbon fibre strengths

A computer simulation model of fragment distribution with respect to the fibre strength in a single-filament composite test is developed using the bimodal Weibull statistics. The predictions of the theory are examined with experimental results for AU carbon fibres coated by zirconium-n-propoxide or a zircoaluminate complex. Weibull analysis reveals a bimodal distribution of fibre strengths, in which the fractions of low- and high-strength populations vary with gauge length. It is seen that the simulation results are in good agreement with experimental data if the best fit model of strength distribution is applied. Thus, the use of a bimodal distribution term in the simulation theory yields a predicted strength at the critical length which is in good agreement with the results of extrapolation of experimental data, while the unimodal distribution term leads to overestimation of the strength.     

A general regression model for lifetime evaluation and prediction

A particular form of a probabilistic model for materials under fatigue which embodies Weibull features and the size effect in a weakest-link framework is derived. The parametric and functional form of the model arises from a certain set of assumptions, as the weakest-link principle, stability, limit behavior, limited range and compatibility, which can be justified as being consistent with experimental features of fatigue (mainly of highly drawn steel wires) and the mathematics of extreme value theory. These assumptions, which are discussed, can be used to rule out other possible forms as being fundamentally inconsistent. The authors also discuss estimation procedures for the parameters based on two steps: a non-linear regression step, in which the threshold lifetime and stress range values are determined, and a second step in which the Weibull parameters are estimated by pooling data from different stress levels and using a probability-weighted moments approach or the Castillo-Hadi estimators. Next, the damage accumulation problem is dealt with and two different proposals for the damage index are given. The model, originally developed to handle a fixed load parameter (such as the stress range in cyclic fatigue), is extended to handle a block load sequence involving many load levels, as well as random load programs. Some formulas for calculating the accumulated damage index for constant, block and random loading are given. Finally, the model and methods are applied to a particular fatigue program on concrete to illustrate all concepts and the practical use of formulas.     

Investigation of the strength of surface-treated carbon fibres embedded in resin, by means of model composite tests

The strength of single fibres of carbon embedded in epoxy resin has been estimated by continuous monitoring of fragmentation tests on single embedded fibres. These fibres had previously been treated to enhance their adhesion to the resin, and the effect of this treatment on the fibre strength is studied. The data generated have been analysed by using an adapted Weibull model which includes the influence of the ‘ineffective’ length at each fibre break. Weibull parameters determined from this analysis have been compared with those from tests carried out on fibres in free air. Similar Weibull moduli are obtained and, although absolute values of strength differ, similar trends are observed and a peak in strength is observed at an intermediate level of treatment in both sets of results.     

Reliability Data Analysis for Life Test Designed Experiments with Sub‐Sampling

The ability to model lifetime data from life test experiments is of paramount importance to all manufacturers, engineers and consumers. The Weibull distribution is commonly used to model the data from life tests. Standard Weibull analysis assume completely randomized designs. However, not all life test experiments come from completely randomized designs. Experiments involving sub‐sampling require a method for properly modeling the data. We provide a Weibull nonlinear mixed models (NLLMs) methodology for incorporating random effects in the analysis. We apply this methodology to a reliability life test on glass capacitors. We compare the NLLMs methodology to other available methods for incorporating random effects in reliability analysis. A simulation study reveals the method proposed in this paper is robust to both model misspecification and increasing levels of variance on the random effect. Copyright © 2012 John Wiley & Sons, Ltd.     

Life Prediction Model of Machine Tool based on Deep Learning

In view of the shortage of traditional life prediction methods for machine tools, such as low accuracy of life prediction and few samples basis attributes, a life prediction model of machine tools combined with machine tool attributes is proposed. The life prediction model of machine tool adopts KL dispersion distribution theory, uses modal superposition method to carry out machine tool life analysis, calculates the theoretical life of machine tool, and then carries on the simulation, obtains the machine tool life prediction value. Compared with the traditional method of machine tool life prediction, the model is based on the application life fatigue damage model, which superimposes the service times and maintenance cycle of the machine tool, derives the influence factor of machine tool life, and obtains the linear relationship between the influence factor of machine tool life and the life of machine tool. The influence factor of machine tool life is introduced as the life prediction parameter of machine tool. The data transformation relationship of HT300 parts is constructed. The original part data is enhanced. The effective training set is obtained. The life prediction model of machine tool based on deep learning is completed. The quantitative analysis of machine tool life is carried out. The experiment of machine tool life prediction using training data set proves the validity of the model. Regression test was carried out on the training data set to reflect the robustness of the model. The prediction accuracy of the model is further verified by Weibull test.     

Micro and macro analysis of a thermonuclear fusion plant for reliability and performance

The Next European Torus (NET) must be operated with appropriate reliability and availability levels if it is to attain its targets. A basic simulation model, the principles and potentials of which had been previously demonstrated, has been considerably extended to provide a more accurate representation of the operations of NET and to facilitate its study under various operating and reliability considerations. The latest simulation model of NET incorporates the physics phase as well as the technology phase. It encompasses complex features such as time and event dependent failure rates and independent failures. For ease of manipulating the uncertainties associated with the reliability data, the model has been furnished with the normal, exponential, lognormal and Weibull distribution modules. The level of subsystem breakdown is easily adjustable to match the needs of the design that is growing in detail. A goal-seeking procedure has been developed for use in conjunction with the simulation model. The application of this procedure culminates in operating and reliability strategies which will enable NET to achieve its intended targets. This paper describes the operating and reliability features of NET together with the ways in which they have been captured by the NET simulation model. The underlying philosophy and application of the goal seeking procedure is also demonstrated through illustrative results.     

The validity of Weibull estimators

The parameters of the two-parametric Weibull distribution, the Weibull modulus and the scale parameter, were estimated by using not only analytical means but also Monte-Carlo simulations. The precision of the measurement of both parameters, i.e. their variation coefficient, has been calculated. It is shown that the variation coefficient of the scale parameter is dependent on the number of experiments, M, which were performed, and on the Weibull modulus itself, whereas the variation coefficient of the Weibull modulus is only dependent on M. Furthermore, the correctly interpreted results show that each single measurement gives the statistically correct Weibull parameters and the biasing arises only from the method of adding single measurements to obtain a mean value. Thus, in practice, when only one set of experiments for further evaluation is available, there is no need for adjustment factors.     

Reliability analysis for wind turbines with incomplete failure data collected from after the date of initial installation

Reliability has an impact on wind energy project costs and benefits. Both life test data and field failure data can be used for reliability analysis. In wind energy industry, wind farm operators have greater interest in recording wind turbine operating data. However, field failure data may be tainted or incomplete, and therefore it needs a more general mathematical model and algorithms to solve the model. The aim of this paper is to provide a solution to this problem. A three-parameter Weibull failure rate function is discussed for wind turbines and the parameters are estimated by maximum likelihood and least squares. Two populations of German and Danish wind turbines are analyzed. The traditional Weibull failure rate function is also employed for comparison. Analysis shows that the three-parameter Weibull function can obtain more accuracy on reliability growth of wind turbines. This work will be helpful in the understanding of the reliability growth of wind energy systems as wind energy technologies evolving. The proposed three-parameter Weibull function is also applicable to the life test of the components that have been used for a period of time, not only in wind energy but also in other industries.     

A probabilistic analysis on thickness effect in fracture toughness

Fracture toughness values are often influenced by specimen thickness and they indicate generally decreasing toughness with increasing thickness. In the present paper, a probabilistic analysis has been carried out by using various kinds of toughness data in order to clarify the applicability of the weakest link model to thickness effect in fracture toughness. Moreover, a new statistical method is proposed for determining fracture toughness distribution, which is necessary for the above analysis, with taking the temperature dependency of fracture toughness into account. Thickness effect in fracture toughness is brought about by its statistical nature and the weakest link model can be applied to evaluate the thickness effect for both steel plate and its welds with heterogeneity in toughness. This thickness effect is considerably affected by Weibull shape parameter and the probability of cleavage fracture for the material. The statistical method proposed newly in this paper is sufficiently applicable and superior to the conventional method. By using this new method, Weibull parameters at a temperature of interest can be determined with much the same reliability as in the conventional method, and also Weibull parameters at lower and higher temperatures can be obtained with a certain confidence depending on the number of specimens tested.