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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 22

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Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 22 studies the combination of various methods of designing for reliability, availability, maintainability and safety, as well as the latest techniques in probability and possibility modelling, mathematical algorithmic modelling, evolutionary algorithmic modelling, symbolic logic modelling, artificial intelligence modelling, and object-oriented computer modelling, in a logically structured approach to determining the integrity of engineering design. . | 3.3 Analytic Development of Reliability and Performance in Engineering Design 193 where aj and xj regression parameters and covariates 3 and J. the shape and scale parameters. It is often more convenient to define an additional covariate xo 1 in order to allow the Weibull scale parameter to be included in the vector of regression coefficients and the proportional hazards model expressed solely by the beta shape parameter together with the regression parameters and covariates. The PH failure rate can then be written as k t X 3 t P 1exp m s aJxJ 0 3.149 The PH reliability function is thus given by the expression t R t X exp y h u du R t X exp -J k u X du 0 R t X exp -tP exp m s aJxJ J 0 3.150 0 The probability density function p.d.f. can be obtained by taking the partial derivative with respect to time of the reliability function given by Eq. 3.150 . The PH probability density functionis given by the expression f t X k t X R t X . The total number of unknowns to solve in this model is m 2 i.e. 3 P- a1 a2 a3 . am . The maximum likelihood estimation method can be used to determine these parameters. Solving for the parameters that maximise the maximum likelihood estimation will yield the parameters for the PH Weibull model. For 3 1 the equation then becomes the likelihood function for the PH exponential model which is similar to the original form of the proportional hazards model proposed by Cox 1972 . c Maximum Likelihood Estimation MLE Parameter Estimation The idea behind maximum likelihood parameter estimation is to determine the parameters that maximise the probability likelihood of the sample data. From a statistical point of view the method of maximum likelihood is considered to be more robust with some exceptions and yields estimators with good statistical properties. In other words MLE methods are versatile and apply to most models and to different types of data. In addition they provide efficient methods for quantifying uncertainty through confidence bounds. .