Birnbaum–Saunders statistical modelling: a new approach

Leiva, Víctor; Santos‐Neto, Manoel; Cysneiros, Francisco José A.; Barros, Michelli

doi:10.1177/1471082x13494532

Cited by 72 publications

(59 citation statements)

References 39 publications

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“…All statistical modeling must be evaluated by diagnostic tools. One of the most used tools in regression models is the residual analysis, which is a very useful technique to detect departures from the model assumptions and to identify atypical observations . We consider in this paper two types of residuals for log‐symmetric regression models.…”

Section: Log‐symmetric Regression Models and Information Criteriamentioning

confidence: 99%

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

Ventura

Saulo

Leiva

et al. 2019

Appl Stoch Models Bus & Ind

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This work deals with log‐symmetric regression models, which are particularly useful when the response variable is continuous, strictly positive, and following an asymmetric distribution, with the possibility of modeling atypical observations by means of robust estimation. In these regression models, the distribution of the random errors is a member of the log‐symmetric family, which is composed by the log‐contaminated‐normal, log‐hyperbolic, log‐normal, log‐power‐exponential, log‐slash and log‐Student‐t distributions, among others. One way to select the best family member in log‐symmetric regression models is using information criteria. In this paper, we formulate log‐symmetric regression models and conduct a Monte Carlo simulation study to investigate the accuracy of popular information criteria, as Akaike, Bayesian, and Hannan‐Quinn, and their respective corrected versions to choose adequate log‐symmetric regressions models. As a business application, a movie data set assembled by authors is analyzed to compare and obtain the best possible log‐symmetric regression model for box offices. The results provide relevant information for model selection criteria in log‐symmetric regressions and for the movie industry. Economic implications of our study are discussed after the numerical illustrations.

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Section: Log‐symmetric Regression Models and Information Criteriamentioning

confidence: 99%

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

Ventura

Saulo

Leiva

et al. 2019

Appl Stoch Models Bus & Ind

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“…We use the notation U ∼BS( μ , δ ) and its PDF is given by

f_{U} false(u; μ, δ false) = \frac{\exp false(δ false/ 2 false) \sqrt{δ + 1}}{4 u^{\frac{3}{2}} \sqrt{π μ}} ((), u + \frac{δ μ}{δ + 1}) \exp ((), - \frac{δ}{4} ((), \frac{u false(δ + 1 false)}{δ μ} + \frac{δ μ}{u false(δ + 1 false)})), u > 0 .

Some appealing properties of the BS distribution are the following. If U ∼BS( μ , δ ), then (i) c U ∼BS( c μ , δ ), with c > 0, which means that the BS distribution is closed under scalar multiplication (proportionality); (ii) 1/ U ∼BS( μ ⋆ , δ ), where μ ⋆ = ( δ + 1)/( δ μ ), implying that the BS distribution is closed under reciprocation; (iii) the median of the distribution of U is ( δ /( δ + 1)) μ , which can be directly obtained when q = 0.5 from its quantile function given by

u (q; μ, δ) = F_{U}^{- 1} (q; μ, δ) = (δ μ / (δ + 1)) {(z (q) / \sqrt{2 δ} + \sqrt{{(z (q) / \sqrt{2 δ})}^{2} + 1})}^{2}, 0 < q < 1,

where z ( q ) is the q × 100th percentile (or quantile function) of the standard normal distribution and

F_{U}^{- 1}

is the inverse of the cumulative distribution function (CDF) F U ; (iv) the BS distribution has different shapes for its PDF, which cover high, medium, and low asymmetry; (v) the new parameterization of the BS distribution based on the mean permits us to analyze data in their original scale, avoiding problems of interpretation in models based on logged data; (vi) in frailty models, the BS distribution is highly competitive in terms of fitting…”

Section: Preliminariesmentioning

confidence: 99%

Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data

Leão

Leiva

Saulo

et al. 2018

Statistics in Medicine

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Cure rate models have been widely studied to analyze time-to-event data with a cured fraction of patients. Our proposal consists of incorporating frailty into a cure rate model, as an alternative to the existing models to describe this type of data, based on the Birnbaum-Saunders distribution. Such a distribution has theoretical arguments to model medical data and has shown empirically to be a good option for their analysis. An advantage of the proposed model is the possibility to jointly consider the heterogeneity among patients by their frailties and the presence of a cured fraction of them. In addition, the number of competing causes is described by the negative binomial distribution, which absorbs several particular cases. We consider likelihood-based methods to estimate the model parameters and to derive influence diagnostics for this model. We assess local influence on the parameter estimates under different perturbation schemes. Deriving diagnostic tools is needed in all statistical modeling, which is another novel aspect of our proposal. Numerical evaluation of the considered model is performed by Monte Carlo simulations and by an illustration with melanoma data, both of which show its good performance and its potential applications. Particularly, the illustration confirms the importance of statistical diagnostics in the modeling.

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“…Consider a reparameterized version of the BS (RBS) distribution by setting κ = 2/τ and σ = τ µ/[τ + 1], such that τ = 2/κ 2 and µ = σ[1 + κ 2 /2], where τ > 0 is a shape and precision parameter and µ > 0 is a scale parameter and the mean of the distribution; see Leiva et al (2014a) and SantosNeto et al (2016). In this case, the PDF of X ∼ RBS(µ, τ ) is given by…”

Section: A Reparameterized Bs Distributionmentioning

confidence: 99%

“…Bhatti (2010) suggested that (B2) might possibly: (C1) improve the model fit, since for asymmetric, heavy-tailed distributions, as occurs with TD data, the median is often considered as a better measure of central tendency than the mean; and (C2) increase the forecasting ability due to the fact that the mean is greater than the median for skew distributions. In this context, we consider two models: A first new mean-based model (BSACD1 in short) specified in terms of a time-varying conditional mean duration, as usual in ACD models, using a reparameterized version of the BS distribution (see Leiva et al, 2014a;Santos-Neto et al, 2016); and a second median-based model (BSACD2 in short) specified in terms of a time-varying conditional median duration. Thus, the primary objective of this paper is to compare both BSACD1 and BSACD2 models.…”

Section: Introductionmentioning

confidence: 99%

Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data

et al. 2017

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Modern financial markets now record the precise time of each stock trade, along with price and volume, with the aim of analysing the structure of the times between trading events -leading to a big data problem. In this paper, we propose and compare two Birnbaum-Saunders autoregressive conditional duration models specified in terms of time-varying conditional median and mean durations. These models provide a novel alternative to the existing autoregressive conditional duration models due to their flexibility and ease of estimation. Influence diagnostic tools are developed to allow goodness-of-fit assessment and to detect departures from assumptions, including the presence of outliers and influential cases. Both global and local influence tools are considered based on the parameter estimates under different perturbation schemes. A thorough Monte Carlo study is presented to evaluate the performance of the maximum likelihood estimators, and the forecasting ability of the models is assessed using the traditional and density forecast evaluation techniques. The Monte Carlo study suggests that the parameter estimators are asymptotically unbiased, consistent and normally distributed. Finally, a full analysis of a real-world financial transaction data set, from the German DAX in 2016, is presented to illustrate the proposed approach and to compare the fitting and forecasting performances with existing models in the literature. One case related to the duration time is identified as potentially influential, but its removal does not change resulting inferences demonstrating the robustness of the proposed approach. Fitting and forecasting performances favor the proposed models and, in particular, the median-based approach gives additional protection against outliers, as expected.

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Birnbaum–Saunders statistical modelling: a new approach

Cited by 72 publications

References 39 publications

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

Incorporation of frailties into a cure rate regression model and its diagnostics and application to melanoma data

Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data

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