Mixture inverse Gaussian distributions and its transformations, moments and applications

Leiva, Víctor; Sanhueza, Antonio; Cabrera, Enrique

doi:10.1080/02331880701829948

Cited by 78 publications

(32 citation statements)

References 15 publications

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“…First, we consider the BS, IG and LN distributions. The reasons for doing this are the following: (i) as mentioned, the LN distribution has been frequently used in environmental sciences; (ii) the BS distribution is the basic model for the new EBS one and, in addition, it has also been used for describing environmental data; and (iii) the IG distribution is a model that has received great attention in the last decades and it is adequately implemented; see Balakrishnan et al (2009a). In addition, the BS, IG and LN distributions have \-shaped HRs, such as the TTT curves of both data sets indicate.…”

Section: Model Checking and Goodness-of-fitmentioning

confidence: 99%

“…In addition to the scale and reciprocation transformations given in (P1) and (P2), respectively, logarithmic and raising to a power transformations of variates are of interest; for more details, see Balakrishnan et al (2009a) and references therein.…”

Section: Transformationsmentioning

confidence: 99%

“…There may also be several very unequal contamination sources, each contributing to the overall distribution. Then, it is useful to consider models that are built as mixtures, where the BS distribution is one of these; see Balakrishnan et al (2009a). Thus, the BS model is a good candidate for describing environmental data.…”

Section: Introductionmentioning

confidence: 99%

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An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile

Vilca

Sanhueza

Leiva

et al. 2010

Stoch Environ Res Risk Assess

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In this article, we introduce, characterize and apply an extended version of the Birnbaum-Saunders model based on the Mudolkar-Hutson skew distribution. This model is appropriated for describing phenomena involving accumulation of some type, as is the case of environmental contamination. Specifically, we find the density, distribution function, and moments of the new model. In addition, we derive several properties and transformations related to this distribution. Furthermore, we propose an estimation method for the parameters of the model. Moreover, we conduct a study of its hazard rate focuses in environmental analysis. A computational implementation in R language of the obtained results is discussed. Finally, we present two examples with real data from environmental quality in Chile that illustrate the proposed methodology.

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Section: Model Checking and Goodness-of-fitmentioning

confidence: 99%

Section: Transformationsmentioning

confidence: 99%

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An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile

Vilca

Sanhueza

Leiva

et al. 2010

Stoch Environ Res Risk Assess

Self Cite

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“…The values of ℓ, AIC, A * , W * and K-S statistic are provided in Table 4. Leiva, Sanhuzea, & Cabrera (2009), which was originally given by Chhikara & Folks (1989). The actual observations are listed below.…”

Section: Data Setmentioning

confidence: 99%

The Topp-Leone Generated Weibull Distribution: Regression Model, Characterizations and Applications

Aryal¹,

Ortega²,

Hamedani³

et al. 2016

IJSP

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This paper introduces a new four-parameter lifetime model called the Topp Leone Generated Weibull (TLGW) distribution. This distribution is a generalization of the two parameter Weibull distribution using the genesis of Topp-Leone distribution. We derive many of its structural properties including ordinary and incomplete moments, quantile and generating functions and order statistics. Parameter estimation using maximum likelihood method and simulation results to assess effectiveness of the distribution are discussed. Also, for the first time, we introduce a regression model based on the new distribution. We prove empirically the importance and flexibility of the new model in modeling various types of real data sets.

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“…This has the advantage that both the inverse Gaussian and the Birnbaum-Saunders distributions appear almost naturally under simple integrate-and-fire neurons. Additionally, they have a close relationship (Leiva et al 2015;Desmond 1986;Balakrishnan et al 2009) where its distributions become closer as PðV 0 VðkÞÞ becomes higher, for any k [ 0, and thus, inverse Gaussian and Birnbaum-Saunders families can be considered as belonging to the family of fatigue life distributions. When such probability decreases the goodness-of-fit of the Birnbaum-Saunders distribution remains high but one of its parameters, the median, losses its interpretation (Leiva et al 2015).…”

mentioning

confidence: 99%

The membrane potential process of a single neuron seen as a cumulative damage process

Tejo

Niklitschek-Soto

2016

Cogn Neurodyn

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A simple integrate-and-fire mechanism of a single neuron can be compared with a cumulative damage process, where the spiking process is analogous to rupture sequences of a material under cycles of stress. Although in some cases lognormal-like patterns can be recognized in the inter-spike times under a simple integrate-and-fire mechanism, fatigue life models as the inverse Gaussian distribution and the Birnbaum-Saunders distribution (which was recently introduced in the neural activity framework) provide theoretical arguments that make them more suitable for the modeling of the resulting inter-spike times.Keywords Lognormal Á Inverse Gaussian and BirnbaumSaunders distributions Á Integrate-and-fire model Á Interspike distribution In a recent communication, Kish et al. (2015) highlight lognormal-like features in the inter-spike distribution of a single neuron under a simple integrate-and-fire scheme with an additive noise with no long-tail but exponential cutoff. To be more precise, let V(k) be the membrane potential of a typical neuron at discrete time k 2 N. Its dynamics is given by:with initial condition Vð0Þ ¼ V 0 . The injected current is described through the process fnðkÞg k2N , which represents a white noise process (typically normally or uniformly distributed), and the parameters d and D, which represent the drift velocity and the diffusion coefficient, respectively. Here, the condition V 0 VðkÞ is enforced during the whole motion (i.e., V 0 is a reflecting boundary). The process, then, stops at time j ¼ inffk : VðkÞ [ V th g, where V th represents a constant (membrane potential) threshold at which the neuron spikes. Once it occurs, the process resets at V 0 . Although this model is too simple to capture a great part of reality, it is still a reasonable approximation when the input rate is high in comparison with the membrane capacitance. A lognormal-like feature in the inter-spike distribution appears whenActually, this is not a theoretical result but a graphical similarity. Indeed, if we assume that fnðkÞg k2N in (1) is a standard normal process, when no condition on the motion is imposed, it is well-known that the resulting inter-spike times are inverse Gaussian distributed (which is already mentioned in Kish et al. 2015). The fact that the inverse Gaussian and the lognormal distributions are graphically similar under some specific values of its parameters is just an akin of the pictures of its probability density functions.In Takagi et al. (1997)

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Mixture inverse Gaussian distributions and its transformations, moments and applications

Cited by 78 publications

References 15 publications

An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile

An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile

The Topp-Leone Generated Weibull Distribution: Regression Model, Characterizations and Applications

The membrane potential process of a single neuron seen as a cumulative damage process

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