A Log-Linear Model for the Birnbaum—Saunders Distribution

Rieck, James R.; Nedelman, Jerry

doi:10.1080/00401706.1991.10484769

Cited by 82 publications

(122 citation statements)

References 14 publications

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“…If we consider the model given in and if ϵ ∼ BS( α ,1), then by using Property (C3), a log‐linear regression model of the form T = γ x − η ϵ ∼ BS( α , γ x − η ) can be derived. Now, applying logarithm, we obtain the linear model

normallog (T) MathClass-rel= normallog (γ) MathClass-bin- η normallog (x) MathClass-bin+ normallog (ϵ) MathClass-punc,

where the term log( ϵ ) follows a logarithmic version of the BS distribution . Thus, estimates of γ and η can be obtained by the least square method.…”

Section: Birnbaum–saunders Accelerated Life Modelsmentioning

confidence: 99%

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Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data

Leiva

Rojas

Galea

et al. 2012

Appl Stoch Models Bus & Ind

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In industrial statistics, there is great interest in predicting with precision lifetimes of specimens that operate under stress. For example, a bad estimation of the lower percentiles of a life distribution can produce significant monetary losses to organizations due to an excessive amount of warranty claims. The Birnbaum–Saunders distribution is useful for modeling lifetime data. This is because such a distribution allows us to relate the total time until the failure occurs to some type of cumulative damage produced by stress. In this paper, we propose a methodology for detecting influence of atypical data in accelerated life models on the basis of the Birnbaum–Saunders distribution. The methodology developed in this study should be considered in the design of structures and in the prediction of warranty claims. We conclude this work with an application of the proposed methodology on the basis of real fatigue life data, which illustrates its importance in a warranty claim problem. Copyright © 2012 John Wiley & Sons, Ltd.

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normallog (T) MathClass-rel= normallog (γ) MathClass-bin- η normallog (x) MathClass-bin+ normallog (ϵ) MathClass-punc,

where the term log( ϵ ) follows a logarithmic version of the BS distribution . Thus, estimates of γ and η can be obtained by the least square method.…”

Section: Birnbaum–saunders Accelerated Life Modelsmentioning

confidence: 99%

“…where the term log. "/ follows a logarithmic version of the BS distribution [52]. Thus, estimates of and Á can be obtained by the least square method.…”

Section: Birnbaum-saunders Accelerated Life Modelsmentioning

confidence: 99%

Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data

Leiva

Rojas

Galea

et al. 2012

Appl Stoch Models Bus & Ind

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“…The error term ln X has the sinh-normal distribution, and properties of this distribution are given in Ref. 11. If instead the power-law acceleration model (1) was used for β in model (2), as given by model (3), the log-linear model would then be represented by ln T = ln γ − η ln V + ln X , so that regression estimates for γ and η could still be obtained (note that this is simply a log transform of the acceleration variable V from the model of Rieck and Nedelman 11 ).…”

Section: Modelmentioning

confidence: 99%

Power-Law Accelerated Birnbaum–saunders Life Models

Owen

Padgett

2000

Int. J. Rel. Qual. Saf. Eng.

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An accelerated life testing procedure can reduce the lifetime of a material by observing the material's behavior under higher levels of stress than what is normally encountered. Useful inference hinges on the selection of an appropriate lifetime distribution and the substitution of an acceleration model for a distribution parameter, such as the mean or scale. The (inverse) power-law model is one such acceleration model that has applications to fatigue studies in metals, where failure tends to be crack-induced. The Birnbaum-Saunders distribution was developed to model fatigue in materials where the failure of a specimen is due to the propagation of a dominant crack. This paper will compare two Birnbaum-Saunders type models from the literature (that have power-law accelerated features) with a new but distinctive model proposed here. The new model is an accelerated life model for a reparameterization of the baseline distribution. Comparison of the three models will be via the aluminum coupon data set from Birnbaum and Saunders 5 and issues of accelerated testing will be discussed.

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“…This explicit form is useful to obtain the PDF and the unconditional SF and HR. It is flexible in terms of bimodality when the logarithm of a BS RV is taken into account. Note that (i)If

U \sim BS (1, δ)

Y = log false(U false) \sim log-BS (\sqrt{2 / δ}, log false(δ / (δ + 1) false))

; see Rieck and Nedelman (); (ii)If

U \sim LN (1, σ^{2})

Y = log false(U false) \sim N false(1, σ^{2} false)

; see Crow and Shimizu (); (iii)If

U \sim IG (1, σ^{2})

Y = log false(U false) \sim log-IG false(σ^{2}, 0 false)

; see Kotz et al. (); (iv)If

U \sim GA (1 / ζ, 1 / ζ)

Y = log (U) \sim log-GA (1 / ζ, 1 / ζ)

; see Johnson et al.…”

Section: Introductionmentioning

confidence: 99%

“…().Some properties of the log‐BS distribution are as follows. If

Y \sim \log - BS (\sqrt{2 / δ}, log false(δ μ / (δ + 1) false))

, then (a)

U = exp (Y) \sim BS (μ, δ)

; (b)

E (Y) = log (δ μ / (δ + 1))

; (c) there is no closed form for the variance of Y , but based upon an asymptotic approximation for the log‐BS moment generating function, it follows that, as

δ \to \infty

Var false(Y false) = 2 / δ - 1 / δ^{2}

, whereas that, in contrast, as

δ \to 0

Var false(Y false) = 4 ({prefixlog}^{2} false(2 / \sqrt{δ} false) + 2 - 2 log false(2 / \sqrt{δ} false))

; and (d) the distribution of Y is symmetric around μ, unimodal for

δ \geq 0.5

, and bimodal for

δ < 0.5

; see Rieck and Nedelman () and Leiva (). Figure shows some shapes for the PDF of

Y = log (U)

in each aforementioned distribution.…”

Section: Introductionmentioning

confidence: 99%

Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data

et al. 2017

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In survival models, some covariates affecting the lifetime could not be observed or measured. These covariates may correspond to environmental or genetic factors and be considered as a random effect related to a frailty of the individuals explaining their survival times. We propose a methodology based on a Birnbaum-Saunders frailty regression model, which can be applied to censored or uncensored data. Maximum-likelihood methods are used to estimate the model parameters and to derive local influence techniques. Diagnostic tools are important in regression to detect anomalies, as departures from error assumptions and presence of outliers and influential cases. Normal curvatures for local influence under different perturbations are computed and two types of residuals are introduced. Two examples with uncensored and censored real-world data illustrate the proposed methodology. Comparison with classical frailty models is carried out in these examples, which shows the superiority of the proposed model.

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A Log-Linear Model for the Birnbaum—Saunders Distribution

Cited by 82 publications

References 14 publications

Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data

Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data

Power-Law Accelerated Birnbaum–saunders Life Models

Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data

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