Weibull-Related Distributions for the Modelling of Bathtub Shaped Failure Rate Functions

Xie, Min; Lai, C. D.; Murthy, D. N. P.

doi:10.1142/9789812795250_0019

Cited by 11 publications

(4 citation statements)

References 32 publications

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“…Recall that the case p<−1/( −1) leads to a non-valid generalized mixture. The monotonicity of h p (t) at t = 0 can be determined from Theorem 3.1, A(0) = ( 0 − 1 / ) 2 are the lifetimes of the components. Then, the distribution function F T of T can be written as F T = s 1 F 1:2 +s 2 F 2:2 (12) where F 1:2 and F 2:2 are the distribution functions of X 1:2 and X 2:2 , respectively.…”

Section: Lemma 36mentioning

confidence: 99%

“…Weibull or gamma), piecewise distributions, stochastic processes, etc. For example, several authors consider the introduction of new parameters in classical models such as Weibull [2][3][4][5][6]. Thus, one can choose a model 325 reliability function through the following inversion formula: (1) for t 0 and hence it can be used to characterize or construct probability models.…”

Section: Introductionmentioning

confidence: 99%

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Generalized mixtures in reliability modelling: Applications to the construction of bathtub shaped hazard models and the study of systems

Navarro¹,

Guillamón²,

Ruiz³

2008

Appl Stoch Models Bus & Ind

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In this paper, we obtain and discuss some general properties of hazard rate (HR) functions constructed via generalized mixtures of two members. These results are applied to determine the shape of generalized mixtures of an increasing hazard rate (IHR) model and an exponential model. In addition, we note that these kind of generalized mixtures can be used to construct bathtub-shaped HR models. As examples, we study in detail two cases: when the IHR model chosen is a linear HR function and when the IHR model is the extended exponential-geometric distribution. Finally, we apply the results and show the utility of generalized mixtures in determining the shape of the HR function of different systems, such as mixed systems or consecutive k-out-of-n systems. whose hazard rate (HR) function includes Weibull HR and BHR depending on the values of some parameters. Another popular option is to consider models obtained from mixtures [7][8][9][10][11]. For example, BHR models are obtained in [11] from the mixtures of an increasing hazard rate (IHR) gamma distribution and a decreasing hazard rate (DHR) continuous mixture.Stochastic mixtures may be used to represent heterogeneous populations, or to represent coherent systems (see [12][13][14]). BHR models constructed via mixtures reflect properties of the heterogeneous population mixands, such as for example, when one studies the lifetimes of electronic devices and there exist electronic devices with and without manufacturing defects. Hence, although BHR models have three typical phases, a mixture model with two components may be sufficient.Generalized mixtures are mixtures that represent the distribution function as a linear combination of the distribution functions of components in the mixture, and the coefficients may be any real number. By contrast, the coefficients in usual mixtures are proportions that lie between 0 and 1. The generalized mixtures appear in characterization problems, in inference procedures and in some representations of systems (see [15][16][17][18][19][20][21][22] and the references therein). For example, the distribution function of the lifetime of a parallel system can be written as a generalized mixture of the distributions of series system lifetimes (see, e.g.[20]). Some properties and applications of generalized mixtures were given in [3,16,[23][24][25].The main advantage of mixtures is that there exists a wide literature on how to manage them in practice. However, despite their simplicity, it is not easy to determine the shape of ageing functions such as HR or mean residual life functions of mixtures (see [7,8,15,21]). The same holds for generalized mixtures, which are more flexible models with similar properties that appear in different practical situations (coherent systems, inference procedures, etc.). The main disadvantage of generalized mixtures is that we need to study in general whether they define a proper distribution function, except in some situations, for example, when they are used to represent coherent systems.In this paper, we obtain...

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Section: Lemma 36mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized mixtures in reliability modelling: Applications to the construction of bathtub shaped hazard models and the study of systems

Navarro¹,

Guillamón²,

Ruiz³

2008

Appl Stoch Models Bus & Ind

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show abstract

“…These techniques include mixtures, extensions of known models such as Weibull or gamma by adding new parameters, piecewise distributions, stochastic processes, etc. (see [1][2][3][4][5]). In my opinion, the more natural option is to consider models obtained from mixtures (see [6][7][8][9][10] and the references therein).…”

Section: Discussionmentioning

confidence: 99%

‘Understanding the shape of the mixture failure rate’ by Maxim Finkelstein: Discussion 2

Navarro¹

2009

Appl Stoch Models Bus & Ind

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DISCUSSIONFirst of all I would like to say that it is both an honor and a pleasure to join Professor Finkelstein in the discussion of this very interesting and relevant topic.The 'strange' behaviour of the failure (or hazard) rate function of heterogeneous populations has been a key point in the development of Reliability and Survival theories. The failure rate of classical distribution models are usually increasing or decreasing. However, usually, the failure rate functions obtained from real data sets are not monotone. This fact can be explained by a mixture of classical distribution models, that is, by the heterogeneity of the population. The mixture of two decreasing failure rate (DFR) models is also DFR, but the failure rate of a mixture of two increasing failure rate (IFR) models (i.e. models that have ageing) can have several shapes including a decreasing behaviour in some intervals. A well-known example is the case of two exponential distributions that have constant failure rates, but their mixture is strictly DFR. The shape is more complicated if we consider general discrete or continuous mixtures.The paper by Finkelstein shows that this strange behaviour is a simple consequence of the mixture (heterogeneity) and the fact that the weakest populations are dying first (the WPDF principle). Therefore, in the two-component mixture case (Section 2), if the failure rate functions are ordered, then the mixture failure rate moves from one failure rate to the other and, finally (when the time increases), it is close to the failure rate of the strongest population. This fact explains the behaviour of the failure rate of the mixture of two exponential distributions. A similar interpretation is given

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“…Weibull distributions can flexibly represent a wide range of other distributions including exponential, normal, Rayleigh, Poisson and Binomial [12]. However, a single Weibull curve is not able to represent all the three stages namely increasing, decreasing and constant value, of a Bathtub-shaped Failure Rate (BFR) [13]. The US military handbook on reliability prediction of electronic equipment [7] uses two Weibull parameters to model equipment reliability and the prediction failure rate, [failures/10 6 hours] is expressed as:…”

Section: Failure Rate Modelmentioning

confidence: 99%

Experimental Determination of Low-Cost Servomotor Reliability for Small Unmanned Aircraft Applications

Sarson-Lawrence

Sabatini

Clothier

et al. 2014

AMM

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One of the key challenges of designing low-cost Unmanned Aircraft Systems (UAS) is to ensure acceptable and certifiable reliability factors for the adopted Commercial-off-the-Shelf (COTS) components since their reliability is often not quantified. In this paper, the experimental results obtained for quantifying the reliability of mini Unmanned Aircraft (UA) servomotors (by recording their time-to-failure on a defined set of test runs) are presented. The Weibull prediction model is adopted for quantitative analysis and the associated key mathematical models. The methodology adopted for performing the reliability analysis including the test bench setup used for the experiments is described. The results indicate a level of reliability expected for low-cost servos. Such servos could be used for low-risk UAS operations (e.g. small UA operating over sparsely populated regions) and where the economics of the business case permitted higher loss rates.

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Weibull-Related Distributions for the Modelling of Bathtub Shaped Failure Rate Functions

Cited by 11 publications

References 32 publications

Generalized mixtures in reliability modelling: Applications to the construction of bathtub shaped hazard models and the study of systems

Generalized mixtures in reliability modelling: Applications to the construction of bathtub shaped hazard models and the study of systems

‘Understanding the shape of the mixture failure rate’ by Maxim Finkelstein: Discussion 2

Experimental Determination of Low-Cost Servomotor Reliability for Small Unmanned Aircraft Applications

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