A flexible distribution and its application in reliability engineering

Abstract Probability distributions of random variables are necessary for reliability evaluation. Generally, probability distributions are determined using one or two parameters evaluated from the mean and standard deviation of statistical data. However, these distributions are not sufficiently flexible to represent the skewness and kurtosis of data. This study therefore proposes a probability distribution based on the cubic normal transformation, whose parameters are determined using the skewness and kurtosis, as well as the mean and standard deviation of available data. This distribution is categorized into six different types based on different combinations of skewness and kurtosis. The boundaries of each type are identified, and the completeness of each type is proved. The cubic normal distribution is demonstrated to provide significant flexibility, and its applicable range covers a large area in the skewness–kurtosis plane, thus enabling it to approximate well-known distributions. The distribution is then applied in reliability engineering: simulating distributions of statistical data, calculating fourth-moment reliability index, finding optimal inspection intervals for condition-based maintenance system, and assessing the influence of input uncertainties on the whole output of a system. Several examples are presented to demonstrate the accuracy and efficacy of the distribution in the above-mentioned reliability engineering practices.

[1]  K. Pearson.  Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material , 1895 .

[2]  Hanfeng Wang,et al.  Effects of oncoming flow conditions on the aerodynamic forces on a cantilevered square cylinder , 2017 .

[3]  M. Grigoriu.  Existence and construction of translation models for stationary non-Gaussian processes , 2009 .

[4]  Wilson H. Tang,et al.  Probability concepts in engineering planning and design , 1984 .

[5]  Ahmed A. Soliman,et al.  Modified Weibull model: A Bayes study using MCMC approach based on progressive censoring data , 2012, Reliab. Eng. Syst. Saf..

[6]  Zhao-Hui Lu,et al.  4P-LAMBDA DISTRIBUTION AND ITS APPLICATIONS TO STRUCTURAL RELIABILITY ASSESSMENT , 2006 .

[7]  Yan-Gang Zhao,et al.  Moment methods for structural reliability , 2001 .

[8]  Emanuele Borgonovo,et al.  Global sensitivity measures from given data , 2013, Eur. J. Oper. Res..

[9]  A. Kareem,et al.  Peak Factors for Non-Gaussian Load Effects Revisited , 2011 .

[10]  D. Boos.  Introduction to the Bootstrap World , 2003 .

[11]  Yan-Gang Zhao,et al.  Complete monotonic expression of the fourth-moment normal transformation for structural reliability , 2018 .

[12]  Allen I. Fleishman.  A method for simulating non-normal distributions , 1978 .

[13]  Emanuele Borgonovo,et al.  A new uncertainty importance measure , 2007, Reliab. Eng. Syst. Saf..

[14]  Eberhard O. Voit,et al.  The S‐Distribution A Tool for Approximation and Classification of Univariate, Unimodal Probability Distributions , 1992 .

[15]  N. Okasha.  An improved weighted average simulation approach for solving reliability-based analysis and design optimization problems , 2016 .

[16]  Mahesh D. Pandey,et al.  The probability distribution of maintenance cost of a system affected by the gamma process of degradation: Finite time solution , 2012, Reliab. Eng. Syst. Saf..

[17]  M D Pandey,et al.  Finite-time maintenance cost analysis of engineering systems affected by stochastic degradation , 2011 .

[18]  Yan-Gang Zhao,et al.  Fourth-Moment Standardization for Structural Reliability Assessment , 2007 .

[19]  Xiang Li,et al.  A hybrid approach combining uniform design and support vector machine to probabilistic tunnel stability assessment , 2016 .

[20]  Saralees Nadarajah,et al.  Modifications of the Weibull distribution: A review , 2014, Reliab. Eng. Syst. Saf..

[21]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[22]  E. Cornish,et al.  The Percentile Points of Distributions Having Known Cumulants , 1960 .

[23]  S. Winterstein.  Nonlinear Vibration Models for Extremes and Fatigue , 1988 .

[24]  M. O'Callaghan.  Modelling non-Gaussian random fields using transformations and Hermite polynomials , 2001 .

[25]  Min Xie,et al.  Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function , 1996 .

[26]  Yan-Gang Zhao,et al.  A THREE-PARAMETER DISTRIBUTION USED FOR STRUCTURAL RELIABILITY EVALUATION , 2001 .

[27]  Yan-Gang Zhao,et al.  New Point Estimates for Probability Moments , 2000 .

[28]  B. Keshtegar.  Stability iterative method for structural reliability analysis using a chaotic conjugate map , 2016 .

[29]  Ronald L. Iman,et al.  A Matrix-Based Approach to Uncertainty and Sensitivity Analysis for Fault Trees1 , 1987 .

[30]  Ying Min Low,et al.  A new distribution for fitting four moments and its applications to reliability analysis , 2013 .

[31]  S. Kotz,et al.  Statistical Size Distributions in Economics and Actuarial Sciences , 2003 .

[32]  Bruce W. Schmeiser,et al.  An approximate method for generating symmetric random variables , 1972, CACM.

[33]  Feng Xing,et al.  Time-dependent probability assessment for chloride induced corrosion of RC structures using the third-moment method , 2015 .

[34]  D. L. Wallace.  Asymptotic Approximations to Distributions , 1958 .

[35]  Zhenzhou Lu,et al.  A new method for evaluating Borgonovo moment-independent importance measure with its application in an aircraft structure , 2014, Reliab. Eng. Syst. Saf..