TEST

, Volume 27, Issue 1, pp 147–172 | Cite as

Parametric bootstrap edf-based goodness-of-fit testing for sinh–arcsinh distributions

Original Paper
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Abstract

Four-parameter sinh–arcsinh classes provide flexible distributions with which to model skew, as well as light- or heavy-tailed, departures from a symmetric base distribution. A quantile-based method of estimating their parameters is proposed and the resulting estimates advocated as starting values from which to initiate maximum likelihood estimation. Parametric bootstrap edf-based goodness-of-fit tests for sinh–arcsinh distributions are proposed, and their operating characteristics for small- to medium-sized samples explored in Monte Carlo experiments. The developed methodology is illustrated in the analysis of data on the body mass index of athletes and the depth of snow on an Antarctic ice floe.

Keywords

Anderson–Darling statistic Logistic distribution Normal distribution Quantile-based estimation Sinh–arcsinh transformation t-distribution 

Mathematics Subject Classification

62F40 62F03 62F10 

Notes

Acknowledgements

I am most grateful to Dr Chris Banks and Professor Paul Garthwaite for access to the snow depth data, and to two referees and an Associate Editor for their careful reading of the original manuscript and helpful suggestions as to how it might be improved. Financial support for the research which led to the production of this paper was received from the Junta de Extremadura and the European Union in the form of grant GR15013.

References

  1. Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12:171–178MathSciNetMATHGoogle Scholar
  2. Azzalini A, Capitanio A (2014) The skew-normal and related families. Cambridge University Press, CambridgeMATHGoogle Scholar
  3. Babu GJ, Rao CR (2004) Goodness-of-fit tests when parameters are estimated. Sankhyā 66:63–74MathSciNetMATHGoogle Scholar
  4. Bagkeris E, Malyuta R, Volokha A, Cortina-Borja M, Bailey H, Townsend CL, Thorne C (2015) Pregnancy outcomes in HIV-positive women in Ukraine, 2000–12 (European Collaborative Study in EuroCoord): an observational cohort study. Lancet HIV 2:385–392CrossRefGoogle Scholar
  5. Balanda KP, MacGillivray HL (1988) Kurtosis: a critical review. Am Stat 42:111–119MATHGoogle Scholar
  6. Beran R (1986) Simulated power functions. Ann Stat 14:151–173MathSciNetCrossRefMATHGoogle Scholar
  7. Cook RD, Weisberg S (1994) An introduction to regression graphics. Wiley, New YorkCrossRefMATHGoogle Scholar
  8. D’Agostino RB, Stephens MA (eds) (1986) Goodness-of-fit techniques. Dekker, New YorkMATHGoogle Scholar
  9. Fischer M, Herrmann K (2013) The HS-SAS and GSH-SAS distribution as model for unconditional and conditional return distributions. Austrian J Stat 42:33–45CrossRefGoogle Scholar
  10. Fischer MJ (2014) Generalized hyperbolic secant distributions: with applications to finance. Springer, HeidelbergCrossRefMATHGoogle Scholar
  11. Georgikopoulos NI, Voudouri V (2014) Demand dynamics and peer effects in consumption: historic evidence from a non-parametric model. Arch Econ Hist 26:27–59Google Scholar
  12. Harkness WL, Harkness ML (1968) Generalized hyperbolic secant distributions. J Am Stat Assoc 63:329–337MathSciNetMATHGoogle Scholar
  13. Hope AC (1968) A simplified Monte Carlo significance test procedure. J R Stat Soc B 30:582–598MATHGoogle Scholar
  14. Jöckel K-H (1986) Finite sample properties and asymptotic efficiency of Monte Carlo tests. Ann Stat 14:336–347MathSciNetCrossRefMATHGoogle Scholar
  15. Jones MC (2015) On families of distributions with shape parameters. Int Stat Rev 83:175–192MathSciNetCrossRefGoogle Scholar
  16. Jones MC, Pewsey A (2009) Sinh–arcsinh distributions. Biometrika 96:761–780MathSciNetCrossRefMATHGoogle Scholar
  17. Jones MC, Rosco JF, Pewsey A (2011) Skewness-invariant measures of kurtosis. Am Stat 65:89–95MathSciNetCrossRefGoogle Scholar
  18. Ke X, Cortina-Borja M, Silva BC, Lowe R, Rakyan V, Dalding D (2013) Integrated analysis of genome-wide genetic and epigenetic association data for identification of disease mechanisms. Epigenetics 8:1236–1244CrossRefGoogle Scholar
  19. Knowles RL, Day T, Wade A, Bull C, Wren C, Dezateux C (2014) Patient-reported quality of life outcomes for children with serious congenital heart defects. Arch Dis Child 99:413–419CrossRefGoogle Scholar
  20. Marriott FH (1979) Barnard’s Monte Carlo tests: how many simulations? J R Stat Soc C 28:75–77Google Scholar
  21. Matsumoto K, Voudouris V, Stasinopoulos D, Rigby R, Di Maio C (2012) Exploring crude oil production and export capacity of the OPEC Middle East countries. Energ Policy 48:820–828CrossRefGoogle Scholar
  22. McKeague IW (2015) Central limit theorems under special relativity. Stat Probab Lett 99:149–155MathSciNetCrossRefMATHGoogle Scholar
  23. Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313MathSciNetCrossRefMATHGoogle Scholar
  24. Perks W (1932) On some experiments in the graduation of mortality statistics. J Inst Act 63:12–57CrossRefGoogle Scholar
  25. Pewsey A (2000) Problems of inference for Azzalini’s skew-normal distribution. J Appl Stat 27:859–870CrossRefMATHGoogle Scholar
  26. Pewsey A, Abe T (2015) The sinh–arcsinhed logistic family of distributions: properties and inference. Ann Inst Stat Math 67:573–594MathSciNetCrossRefMATHGoogle Scholar
  27. Pewsey A, Neuhäuser M, Ruxton GD (2013) Circular statistics in R. Oxford University Press, OxfordMATHGoogle Scholar
  28. Pingel R (2014) Some approximations of the logistic distribution with application to the covariance matrix of logistic regression. Stat Probab Lett 85:63–68MathSciNetCrossRefMATHGoogle Scholar
  29. Romano JP (1988) A bootstrap revival of some nonparametric distance tests. J Am Stat Assoc 83:698–708MathSciNetCrossRefMATHGoogle Scholar
  30. Rosco JF, Jones MC, Pewsey A (2011) Skew \(t\) distributions via the sinh–arcsinh transformation. Test 20:630–652MathSciNetCrossRefMATHGoogle Scholar
  31. Rubio FJ, Ogundimu EO, Hutton JL (2016) On modelling asymmetric data using two-piece sinh–arcsinh distributions. Braz J Probab Stat 30:485–501MathSciNetCrossRefMATHGoogle Scholar
  32. Santos-Fernández E, Govindaraju K, Jones G (2014) A new variables acceptance sampling plan for food safety. Food Control 44:249–257CrossRefGoogle Scholar
  33. Spinelli JJ, Stephens MA (1983) Tests for exponentiality when origin and scale parameters are unknown. Technometrics 29:471–476MathSciNetCrossRefGoogle Scholar
  34. Stephens MA (1974) EDF statistics for goodness-of-fit and some comparisons. J Am Stat Assoc 69:730–737CrossRefGoogle Scholar
  35. Stephens MA (1979) Tests of fit for the logistic distribution based on the empirical distribution function. Biometrika 66:591–595CrossRefMATHGoogle Scholar
  36. Stute W, González-Manteiga W, Presedo-Quindimil M (1993) Bootstrap based goodness-of-fit-tests. Metrika 40:243–256MathSciNetCrossRefMATHGoogle Scholar
  37. Szűcs G (2008) Parametric bootstrap tests for continuous and discrete distributions. Metrika 67:63–81MathSciNetCrossRefMATHGoogle Scholar
  38. Talacko J (1956) Perks’ distributions and their role in the theory of Wiener’s stochastic variables. Trab Estad 7:159–174MathSciNetCrossRefMATHGoogle Scholar
  39. Tarnopolski M (2016) Analysis of gamma-ray burst duration distribution using mixtures of skewed distributions. Mon Not R Astron Soc 458:2024–2031CrossRefGoogle Scholar
  40. Thas O (2010) Comparing distributions. Springer, New YorkCrossRefMATHGoogle Scholar
  41. Vaughan DC (2002) The generalized secant hyperbolic distribution and its properties. Commun Stat Theory Methods 31:219–238MathSciNetCrossRefMATHGoogle Scholar
  42. Voudouris V, Stasinopoulos D, Rigby R, Di Maio C (2011) The ACEGES laboratory for energy policy: exploring the production of crude oil. Energ Policy 39:5480–5489CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  1. 1.Mathematics Department, Escuela PolitécnicaUniversity of ExtremaduraCáceresSpain

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