Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum–Saunders distributions

Caro-Lopera, Francisco J.; Leiva, Víctor; Balakrishnan, N.

doi:10.1016/j.jmva.2011.07.004

Cited by 42 publications

(24 citation statements)

References 19 publications

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“…We have already detailed the univariate and multivariate GBS distributions, and they are natural extensions of the univariate and multivariate BS distributions. Along the same lines, Caro‐Lopera et al defined the matrix‐variate GBS distribution using an elliptic random matrix. First, we introduce a matrix‐variate elliptic distribution.…”

Section: Gbs Distributionmentioning

confidence: 99%

“…Definition Let

bold-italicZ = false(Z_{i j} false) \sim {EC}_{n \times k} false(bold0, I_{k}, bold-italicI; sans-serifg false)

and T =( T i j ), where

T_{i j} = β_{i j} {[\frac{α_{i j} Z_{i j}}{2} + \sqrt{{\{\frac{α_{i j} Z_{i j}}{2}\}}^{2} + 1}]}^{2}, α_{i j} > 0, β_{i j} > 0, i = 1, …, n, j = 1, …, k .

Then, the random matrix T is said to have a generalized matrix‐variate BS distribution, denoted by

{GBS}_{n \times k} false(bold-italicA, bold-italicB, sans-serifg false)

, where A =( α i j ) and B =( β i j ). It has been shown in the work of Caro‐Lopera et al that if

T \sim {GBS}_{n \times k} false(bold-italicA, bold-italicB, sans-serifg false)

, then the PDF of T is given by

f_{bold-italicT} (T) = \frac{c}{2^{n + k}} g (\sum_{i = 1}^{n} \sum_{j = 1}^{k} \frac{1}{α_{i j}^{2}} [\frac{T_{i j}}{β_{i j}} + \frac{β_{i j}}{T_{i j}} - 2]) \prod

…”

Section: Gbs Distributionmentioning

confidence: 99%

“…A comprehensive discussion on the GBS distributions has been provided by Sanhueza et al Kundu et al introduced the bivariate BS distribution and studied some of its properties and characteristics. The multivariate GBS distribution has been introduced, by replacing the normal kernel by an elliptically symmetric kernel, by Kundu et al Caro‐Lopera et al developed subsequently the generalized matrix‐variate BS distribution and discussed its properties.…”

Section: Introductionmentioning

confidence: 99%

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

Balakrishnan

Kundu

2018

Appl Stoch Models Bus & Ind

Self Cite

110

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Birnbaum and Saunders introduced a two-parameter lifetime distribution to model the fatigue life of a metal, subject to cyclic stress. Since then, extensive work has been done on this model providing different interpretations, constructions, generalizations, inferential methods, and extensions to bivariate, multivariate, and matrix-variate cases. More than 200 papers and one research monograph have already appeared describing all these aspects and developments. In this paper, we provide a detailed review of all these developments and, at the same time, indicate several open problems that could be considered for further research. KEYWORDSBayes estimators, EM algorithm, Fisher information matrix, hazard function, length-biased distribution, moments and inverse moments, probability density function, stochastic orderings, TP 2 property 4

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Section: Gbs Distributionmentioning

confidence: 99%

“…Definition Let

bold-italicZ = false(Z_{i j} false) \sim {EC}_{n \times k} false(bold0, I_{k}, bold-italicI; sans-serifg false)

and T =( T i j ), where

T_{i j} = β_{i j} {[\frac{α_{i j} Z_{i j}}{2} + \sqrt{{\{\frac{α_{i j} Z_{i j}}{2}\}}^{2} + 1}]}^{2}, α_{i j} > 0, β_{i j} > 0, i = 1, …, n, j = 1, …, k .

Then, the random matrix T is said to have a generalized matrix‐variate BS distribution, denoted by

{GBS}_{n \times k} false(bold-italicA, bold-italicB, sans-serifg false)

, where A =( α i j ) and B =( β i j ). It has been shown in the work of Caro‐Lopera et al that if

T \sim {GBS}_{n \times k} false(bold-italicA, bold-italicB, sans-serifg false)

, then the PDF of T is given by

f_{bold-italicT} (T) = \frac{c}{2^{n + k}} g (\sum_{i = 1}^{n} \sum_{j = 1}^{k} \frac{1}{α_{i j}^{2}} [\frac{T_{i j}}{β_{i j}} + \frac{β_{i j}}{T_{i j}} - 2]) \prod

…”

Section: Gbs Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Balakrishnan

Kundu

2018

Appl Stoch Models Bus & Ind

Self Cite

110

View full text Add to dashboard Cite

show abstract

“…Pan and Balakrishnan (2011) used this distribution for a race model constructed from the first hitting times of two correlated gamma processes. Multivariate versions of the Birnbaum Saunders distribution have been proposed by Kundu, Balakrishnan, and Jamalizadeh (2013), Caro-Lopera, Leiva, and Balakrishnan (2001) and Lemonte, Martinez-Florez, and Moreno-Arenas (2013).…”

Section: The Bivariate Birnbaum Saunders Distributionmentioning

confidence: 99%

A Race Model for Responses and Response Times in Tests

2014

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Latent trait models for responses and response times in tests are often pure statistical models without a close connection to features of the assumed response process. In the present paper, a new model is presented that is more closely related to assumptions about the response process. The model is based on two increasing stochastic processes. Each stochastic process represents the accumulation of knowledge with respect to one of two response options, the correct and incorrect response. Both accumulators compete and the accumulator that first exceeds a critical level determines the response. General assumptions about the accumulators result in a race between two response times that follow a bivariate Birnbaum Saunders distribution. The model can be calibrated with marginal maximum likelihood estimation. Feasibility of the estimation approach is demonstrated in a simulation study. Additionally, a test of model fit is proposed. Finally, the model will be used for the analysis of an empirical data set.

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“…Bivariate BS distributions were proposed by Kundu et al (2010) and Vilca et al (2014), being then extended to the multivariate case by Kundu et al (2013), with a matrix version introduced by Caro-Lopera et al (2012); see also Sánchez et al (2015). Other work related to the multivariate BS distribution include Jamalizadeh and Kundu (2015); Khosravi et al (2015); Kundu (2015b); Lemonte et al (2015); Marchant et al ( , 2018; Garcia-Papani et al (2017).…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

Multivariate Birnbaum-Saunders Distributions: Modelling and Applications

Aykroyd

Leiva

Marchant

2018

Risks

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Abstract:Since its origins and numerous applications in material science, the Birnbaum-Saunders family of distributions has now found widespread uses in some areas of the applied sciences such as agriculture, environment and medicine, as well as in quality control, among others. It is able to model varied data behaviour and hence provides a flexible alternative to the most usual distributions. The family includes Birnbaum-Saunders and log-Birnbaum-Saunders distributions in univariate and multivariate versions. There are now well-developed methods for estimation and diagnostics that allow in-depth analyses. This paper gives a detailed review of existing methods and of relevant literature, introducing properties and theoretical results in a systematic way. To emphasise the range of suitable applications, full analyses are included of examples based on regression and diagnostics in material science, spatial data modelling in agricultural engineering and control charts for environmental monitoring. However, potential future uses in new areas such as business, economics, finance and insurance are also discussed. This work is presented to provide a full tool-kit of novel statistical models and methods to encourage other researchers to implement them in these new areas. It is expected that the methods will have the same positive impact in the new areas as they have had elsewhere.

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Connection between the Hadamard and matrix products with an application to matrix-variate Birnbaum–Saunders distributions

Cited by 42 publications

References 19 publications

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

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

A Race Model for Responses and Response Times in Tests

Multivariate Birnbaum-Saunders Distributions: Modelling and Applications

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