Birnbaum‐Saunders distribution: A review of models, analysis, and applications

Balakrishnan, N.; Kundu, Debasis

doi:10.1002/asmb.2348

Cited by 110 publications

(76 citation statements)

References 247 publications

(443 reference statements)

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“…Consider T = (T 1 , T 2 ) ⊤ as a two-component system having a GBVBS distribution with equal shapes 1 (2) ) and F (i) (t; , h (2) ) be the PDF and CDF of (2) ) be the corresponding RF of T (i) . Theorem 1.…”

Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

“…where (t, ) = √ t∕ − √ ∕t and (X (1) , X (2) ) ⊤ is the order statistic of (X 1 , X 2 ) ⊤ . We also know that, for each symmetric elliptically distribution, F EC 1 (0; ′ , h (1) ) = 1∕2.…”

Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

“…Remark 2. Using Theorem 1 and the result of Remark 1, the mean of order statistics T (1) and T (2) can be expressed by…”

Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

“…It is also interesting to note that the hazard function of T (i) is (2) ) , t > 0. Figure 1 displays the density and hazard function curves of T (1) and T (2) for the density generator of the SUN and SUT models. It is clearly seen that T (1) and T (2) can produce decreasing and both unimodal and bimodal PDFs.…”

Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

“…This motivates us to study the distribution and statistical properties of order statistics when T = ( T 1 , … , T p ) ⊤ is followed by the GMBS distribution. It is evident that the lifetime of parallel (r = p) and series (r = 1) systems are T (p) = max(T 1 , … , T p ) and T (1) = min(T 1 , … , T p ), respectively. In this note, we present a mixture representation for the reliability function (RF) of an order statistic from a GMBS distribution in terms of RFs of a new extension of the BS distribution based on the unified skew-elliptical (SUE) model.…”

Section: Introductionmentioning

confidence: 99%

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Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications”

Jamalizadeh

Hashemi

Naderi

2018

Appl Stoch Models Bus & Ind

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Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

“…Remark 2. Using Theorem 1 and the result of Remark 1, the mean of order statistics T (1) and T (2) can be expressed by…”

Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

Section: The Rfs Of Series and Parallel Gbvbs Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications”

Jamalizadeh

Hashemi

Naderi

2018

Appl Stoch Models Bus & Ind

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Top‐Down Fabrication of Atomic Patterns in Twisted Bilayer Graphene

et al. 2023

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Atomic‐scale engineering typically involves bottom‐up approaches, leveraging parameters such as temperature, partial pressures, and chemical affinity to promote spontaneous arrangement of atoms. These parameters are applied globally, resulting in atomic‐scale features scattered probabilistically throughout the material. In a top‐down approach, different regions of the material are exposed to different parameters, resulting in structural changes varying on the scale of the resolution. In this work, the application of global and local parameters is combined in an aberration‐corrected scanning transmission electron microscope (STEM) to demonstrate atomic‐scale precision patterning of atoms in twisted bilayer graphene. The focused electron beam is used to define attachment points for foreign atoms through the controlled ejection of carbon atoms from the graphene lattice. The sample environment is staged with nearby source materials such that the sample temperature can induce migration of the source atoms across the sample surface. Under these conditions, the electron‐beam (top‐down) enables carbon atoms in the graphene to be replaced spontaneously by diffusing adatoms (bottom‐up). Using image‐based feedback control, arbitrary patterns of atoms and atom clusters are attached to the twisted bilayer graphene with limited human interaction. The role of substrate temperature on adatom and vacancy diffusion is explored by first‐principles simulations.

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Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications” and a novel financial extreme value data analytics from natural disasters

Leiva

Lillo

Gomes

et al. 2018

Appl Stoch Models Bus & Ind

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The univariate Birnbaum-Saunders (BS) distribution was first postulated to model failure times in material science (see the work of Birnbaum and Saunders 1 ). In this modeling, a cumulative damage exceeds a threshold to produce the failure (see the work of Leiva and Saunders 2 ). The univariate BS distribution is unimodal, positively skewed (although close to a symmetric distribution in some cases), supported over a positive range of values and possessor of diverse and attractive properties (see the book of Johnson et al 3 ). The univariate BS distribution has been extensively studied and applied (see the works of Leiva et al 4,5 ). Most of its mathematical and statistical results until 2016 were published in the book of Leiva. 6 Multivariate BS distributions were derived as a natural extension to the univariate case, based on mathematical methods, with no fatigue theoretical arguments, different to the univariate BS distribution. Aykroyd et al 7 published recently a review on multivariate BS distributions with some applications. In addition, cumulative damage models and their relation to times of occurrence were recently modeled in a multivariate setting for multicomponent systems by Fierro et al. 8 Balakrishnan and Kundu 9 conducted a complete and interesting review of the BS distribution, which considered physical justifications, mathematical and statistical issues, shape analysis and links to other models, as well as formulations and generalizations for the univariate case. Furthermore, extensions to multivariate and matrix-variate versions of the BS distribution were also included. This review provides a full and updated list of references on the topic. However, in the book of Leiva 6 and in the review of Balakrishnan and Kundu, 9 no attention was paid to an extreme value BS (EVBS) distribution, which has several attractive properties and a different conception with respect to its standard version. Indeed, the standard BS distribution cannot be obtained as a particular case of the EVBS distribution, as it happens with other generalizations and extensions of the BS distribution.There are several reasons to justify the use of the BS distribution in the modeling of data with heavy tails. In particular, the EVBS distribution developed by Ferreira et al 10 provides some of these justifications. The EVBS distribution is based on the generalized extreme value (GEV) distribution. As mentioned, the standard BS distribution was derived to solve a problem of material fatigue, but the BS distribution has been also used to solve problems in areas as diverse as environment, finance, and medicine, where heavy-tailed phenomena are often detected. This kind of phenomena is frequently described by extreme value models. Then, the EVBS distribution can be a good alternative to BS distributions because of its good properties to model heavy-tailed phenomena.We discuss some aspects of the BS distribution reviewed by Balakrishnan and Kundu 9 and delve into the EVBS distribution, considering its features, modeling properties, and ne...

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Birnbaum‐Saunders distribution: A review of models, analysis, and applications

Cited by 110 publications

References 247 publications

Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications”

Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications”

Top‐Down Fabrication of Atomic Patterns in Twisted Bilayer Graphene

Discussion of “Birnbaum‐Saunders distribution: A review of models, analysis, and applications” and a novel financial extreme value data analytics from natural disasters

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