Mixtures of distributions with increasing linear failure rates

Block, Henry W.; Savits, Thomas H.; Wondmagegnehu, Eshetu T.

doi:10.1017/s0021900200019446

Cited by 16 publications

(11 citation statements)

References 9 publications

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“…We must note that this definition for the BHR class does not coincide with the definitions given by other authors (see, e.g. [8]) and that with this definition the IHR models are not BHR. In addition, note that t 1 can be equal to t 2 .…”

Section: Construction Of Bhr Modelsmentioning

confidence: 75%

“…In this case, the results given in [10] can be obtained now using the results given in this section. Another interesting particular case is the mixture of an exponential and an LHR model studied in [8] for 0< p<1. In this case, we can also consider p-generalized mixtures with p<0 obtaining the following general result.…”

Section: Theorem 31mentioning

confidence: 99%

“…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.).…”

mentioning

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: Construction Of Bhr Modelsmentioning

confidence: 75%

Section: Theorem 31mentioning

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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“…One notable success with a direct calculation was in Block et al (2003b) for the case of two subpopulations, each having an increasing linear failure rate. Thus, a direct analysis of its failure rate is often a difficult task.…”

Section: Review Of Previous Literature Discussion and Concluding Rmentioning

confidence: 99%

A Glaser Twist: Focus on the Mixture Parameters

Block¹,

Langberg²,

Savits³

2012

J. Appl. Probab.

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“…(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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Mixtures of distributions with increasing linear failure rates

Cited by 16 publications

References 9 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

A Glaser Twist: Focus on the Mixture Parameters

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

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