Two Probability Plots of the Three-Parameter Lognormal Distribution

The two-parameter lognormal distribution is a variant of the normal distribution and the three-parameter lognormal distribution is an extension of the two-parameter lognormal distribution by introducing a location parameter. The Q-Q plot of the three-parameter lognormal distribution is widely used. To obtain the Q-Q plot one needs to iteratively try different values of the shape parameter and subjectively judge the linearity of the Q-Q plot. In this paper,a mathematical method was proposed to determine the value of the shape parameter so as to simplify the generation of the Q-Q plot. Then a new probability plot was proposed,which was more easily obtained and provided more accurate parameter estimates than the Q-Q plot. These are illustrated by three realworld examples.

Hypothesis testing for reliability with a three-parameter Weibull distribution using minimum weighted relative entropy norm and bootstrap

With the help of relative entropy theory,norm theory,and bootstrap methodology,a new hypothesis testing method is proposed to verify reliability with a three-parameter Weibull distribution.Based on the relative difference information of the experimental value vector to the theoretical value vector of reliability,six criteria of the minimum weighted relative entropy norm are established to extract the optimal information vector of the Weibull parameters in the reliability experiment of product lifetime.The rejection region used in the hypothesis testing is deduced via the area of intersection set of the estimated truth-value function and its confidence interval function of the three-parameter Weibull distribution.The case studies of simulation lifetime,helicopter component failure,and ceramic material failure indicate that the proposed method is able to reflect the practical situation of the reliability experiment.

Dynamic Multi-Attribute Decision Analysis on Three-Parameter Interval Grey Number of Upper(Lower)Deviation Quasi-Normal Distribution

Aiming at the dynamic multi-attribute decision making problem where the weight of each decision stage and attribute weight are completely unknown and the attribute value is unknown distributed three-parameter interval grey number,a threeparameter interval grey number dynamic multiattribute grey target decision making method with attribute value following quasi-normal distribution is proposed.Firstly,the position relationship between the“center of gravity”point and the kernel of the threeparameter interval grey number is discussed.According to the characteristicthat the attribute value obeys the quasi-normal distribution,anew weight isgiventothe“centerof gravity”point,and a new distance measure formula of the three-parameter interval grey number is defined.Secondly,according to the principle of maximum entropy,the objective programming model is constructed to determine the stage weight and attributeweight.Then,the schemes aresorted according to thesize of the comprehensive bull's-eye distance Finally an example is given to illustrate the effectiveness of the decision model.

Lithium-ion cell inconsistency analysis based on three-parameter Weibull probability model

The inconsistency of lithium-ion cells degrades battery performance,lifetime and even safety.The complexity of the cell reaction mechanism causes an irregular asymmetrical distribution of various cell parameters,such as capacity and internal resistance,among others.In this study,the Newman electrochemical model was used to simulate the 1 C discharge curves of 100 LiMn2 O4 pouch cells with parameter variations typically produced in manufacturing processes,and the three-parameter Weibull probability model was used to analyze the dispersion and symmetry of the resulting discharge voltage distributions.The results showed that the dispersion of the voltage distribution was related to the rate of decrease in the discharge voltage,and the symmetry was related to the change in the rate of voltage decrease.The effect of the cells’capacity dominated the voltage distribution thermodynamically during discharge,and the phase transformation process significantly skewed the voltage distribution.The effects of the ohmic drop and polarization voltage on the voltage distribution were primarily kinetic.The presence of current returned the right-skewed voltage distribution caused by phase transformation to a more symmetrical distribution.Thus,the Weibull parameters elucidated the electrochemical behavior during the discharge process,and this method can guide the prediction and control of cell inconsistency,as well as detection and control strategies for cell management systems.

Estimation of load and resistance factors using the thirdmoment method based on the 3P-Iognormal distribution

Load and resistance factors are generally obtained using the first order reliability method(FORM)in which the design point should be determined and derivative-based iterations used.In this article,the thirdmoment reliability index,based on the three-parameter lognormal(3P-lognormal)distribution,is investigated.A simple method based on the third-moment method for estimating load and resistance factors is then proposed,and a simple formula for the target mean resistance is also presented to avoid iterative computations.Unlike the currently used method,the proposed method can be used to determine load and resistance factors,even when the probability density functions(PDFs)of the basic random variables are not available.Moreover,the proposed method does not require the iterative computation of derivatives or any design points.Thus,the method provides a more convenient and effective way to estimate load and resistance factors in practical engineering applications.Numerical examples are presented to demonstrate the advantages of the proposed third moment method for determining load and resistance factors.

Characterizing the Relationship between Weibull Location Parameter and the Minimal Observation in a Small Size of Sample Based on Stochastic Simulation

Three-parameter Weibull distribution is one of the preferable distribution models to describe product life. However, it is difficult to estimate its location parameter in the situation of a small size of sample. This paper presents a stochastic simulation method to estimate the Weibull location parameters according to a small size of sample of product life observations and a large amount of statistically simulated life date. Big data technique is applied to find the relationship between the minimal observation in a product life sample of size n (n ≥ 3) and the Weibull location parameter. An example is presented to demonstrate the applicability and the value of the big data based stochastic simulation method. Comparing with other methods, the stochastic simulation method can be applied to very small size of sample such as the sample size of three, and it is easy to apply.