Skip to main content
SpringerLink
Log in

Analysis of correlated Birnbaum–Saunders data based on estimating equations

  • Original Paper
  • Published:
TEST Aims and scope Submit manuscript

Abstract

Estimating equations for analyzing correlated Birnbaum–Saunders (BS) data are derived in this paper. A regression model is proposed for modeling the median of the life time until the failure, and a reweighted iterative process is developed for the joint estimation of the regression coefficients and the shape and correlation parameters. Diagnostic procedures, such as residual analysis and sensitivity studies based on case deletion and local influence, are given. Simulation studies are performed to assess the empirical distributions of the derived estimators and of a Pearson-type residual for correlated data. Finally, a longitudinal data set is analyzed by the procedures developed in the paper and extensions for the double case in which the median and the shape parameter are jointly modeled are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Alencar AP, Singer JM, Rocha FMM (2012) Competing regression models for longitudinal data. Biom J 54:214–229

    Article  MathSciNet  Google Scholar 

  • Artes R (1997) Extensions of the Generalized Estimating Equation Theory to Circular Data and Dispersion Models [unpublished Ph.D. thesis in Portuguese]. Department of Statistics, University of São Paulo, Brazil

  • Artes R, Jorgensen B (2000) Longitudinal data estimating equations for dispersion model. Scand J Stat 27:321–334

    Article  MathSciNet  Google Scholar 

  • Atkinson AC (1981) Two graphical displays for outlying and influential observations in regression. Biometrika 68:13–20

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Kundu D (2019) Birnbaum-Saunders distribution: a review of models, analysis, and applications. Appl Stoch Model Bus 35:4–49

    Article  MathSciNet  Google Scholar 

  • Birnbaum ZW, Saunders SC (1969) A new family of life distributions. J Appl Probab 6:319–327

    Article  MathSciNet  Google Scholar 

  • Cadigan NG, Farrell PJ (2002) Generalized local influence with applications to fish stock cohort analysis. Appl Stat 51:469–483

    MATH  Google Scholar 

  • Cook RD (1977) Detection of influential observations in linear regressions. Technometrics 19:15–18

    MathSciNet  MATH  Google Scholar 

  • Cook RD (1986) Assessment of local influence. J R Stat Soc B 48:133–169

    MathSciNet  MATH  Google Scholar 

  • Crowder M (1987) On linear and quadratic estimating functions. Biometrika 74:591–597

    Article  MathSciNet  Google Scholar 

  • Fitzmaurice GM, Laird NM, Rotnitzky AG (1993) Regression models for discrete longitudinal responses. Stat Sci 8:284–299

    Article  MathSciNet  Google Scholar 

  • Godambe VP (1997) Estimating functions: a synthesis of least squares and maximum likelihood methods. In: Selected proceedings of the symposium on estimating functions. Institute of Mathematical Statistics, Hayward, CA, pp 5–16

  • Greene WH (2012) Econometric analysis. Prentice Hall, New York

    Google Scholar 

  • Hardin JW, Hilbe JM (2012) Generalized estimating equations. Chapman & Hall, Cambridge

    Book  Google Scholar 

  • Jorgensen B, Lundbye-Christensen S, Song PX-K, Sun L (1996) State-space models for multivariate longitudinal data of mixed types. Can J Stat 24:385–402

    Article  MathSciNet  Google Scholar 

  • Kim SK, Huggins R (1998) Diagnostic for autocorrelation regression models. Aust N Z J Stat 40:65–71

    Article  MathSciNet  Google Scholar 

  • Kundu D, Balakrishnan N, Jamalizadeh A (2013) Generalized multivariate Birnbaum–Saunders distributions and related inferential results. J Multivar Anal 116:230–244

    Article  Google Scholar 

  • Leiva V (2016) The Birnbaum–Saunders distribution. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Leiva V, Barros M, Paula GA, Galea M (2007) Influence diagnostics in log-Birnbaum–Saunders regression models with censored data. Comput Stat Data Anal 51:5694–5707

    Article  MathSciNet  Google Scholar 

  • Liang KY, Zeger SL (1986) Longitudinal analysis using generalized linear models. Biometrika 73:13–22

    Article  MathSciNet  Google Scholar 

  • Liang NY, Zeger SL, Qaqish (1992) Multivariate regression analyses for categorical data. J R Stat Soc B 54:3–40

    MathSciNet  MATH  Google Scholar 

  • Manghi RF, Paula GA, Cysneiros JFA (2016) On elliptical multilevel models. J Appl Stat 43:2150–2171

    Article  MathSciNet  Google Scholar 

  • Marchant C, Leiva V, Cysneiros FJA (2016) A multivariate log-linear model for Birnbaum–Saunders distributions. IEEE Trans Reliab 652:816–827

    Article  Google Scholar 

  • Marchant C, Leiva V, Cysneiros FJA, Vivanco JF (2016) Diagnostics in multivariate Birnbaum–Saunders regression models. J Appl Stat 43:2829–2849

    Article  MathSciNet  Google Scholar 

  • Moral RA, Hinde J, Demétrio CGB (2017) Half-normal plots and overdispersed models in R: the hnp package. J Stat Softw 81:1–23

    Article  Google Scholar 

  • Munnell AH (1990) Why has productivity declined? Productivity and public investment. N Engl Econ Rev Jan/Feb 3–22

  • Paula GA, Medeiros M, Vilca-Labra FE (2009) Influence diagnostics for linear models with first-order autoregressive elliptical errors. Stat Probab Lett 79:339–346

    Article  MathSciNet  Google Scholar 

  • Poon W, Poon YS (1999) Conformal normal curvature and assessment of local influence. J R Stat Soc B 61:51–61

    Article  MathSciNet  Google Scholar 

  • Pregibon D (1981) Logistic regression diagnostics. Ann Stat 9:705–724

    Article  MathSciNet  Google Scholar 

  • Preisser JS, Qaqish BF (1996) Deletion diagnostics for generalised estimating equations. Biometrika 83:551–562

    Article  Google Scholar 

  • R Core Team (2017) R: a language and environment for statistical computing. https://www.R-project.org/

  • Rieck JR, Nedelman JR (1991) A log-linear model for the Birnbaum–Saunders distribution. Technometrics 33:51–60

    MATH  Google Scholar 

  • Venezuela MK, Botter DA, Sandoval MC (2007) Diagnostic techniques in generalized estimating equations. J Stat Comput Simul 77:879–888

    Article  MathSciNet  Google Scholar 

  • Venezuela MK, Sandoval MC, Botter DA (2011) Local influence in estimating equations. Comput Stat Data Anal 55:1867–1883

    Article  MathSciNet  Google Scholar 

  • Villegas C, Paula GA, Leiva V (2011) Birnbaum–Saunders mixed models for censored reliability data analysis. IEEE Trans Reliab 60:748–758

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by FAPESP and CNPq, Brazil. The authors are grateful to the Associate Editor and reviewers for their helpful comments.

Funding

Funding was provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico (Grant No. 305757/2014-8 and No. 310359/2017-1).

Author information

Authors and Affiliations

  1. Departamento de Estatística, Universidade Federal de Pernambuco, Recife, Brazil

    Aline B. Tsuyuguchi

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil

    Gilberto A. Paula

  3. Departamento de Estatística, Universidade Federal de Campina Grande, Campina Grande, Brazil

    Michelli Barros

Authors
  1. Aline B. Tsuyuguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilberto A. Paula
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michelli Barros
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilberto A. Paula.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 428 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsuyuguchi, A.B., Paula, G.A. & Barros, M. Analysis of correlated Birnbaum–Saunders data based on estimating equations. TEST 29, 661–681 (2020). https://doi.org/10.1007/s11749-019-00675-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11749-019-00675-1

Keywords

Mathematics Subject Classification

Search

Navigation