Abstract
The Student’s t distribution is the most popular model for economic and financial data. In recent years, many generalizations of the Student’s t distribution have been proposed. This paper provides a review of generalizations, including software available for them. A real data application is presented to compare some of the reviewed distributions.
Similar content being viewed by others
References
Aas K, Haff IH (2006) The generalized hyperbolic skew Student’s \(t\) distribution. J Financ Econom 4:275–309
Acitas S, Senoglu B, Arslan O (2015) Alpha-skew generalized \(t\) distribution. Rev Colomb Estad 38:353–370
Ahsanullah M, Kibria BMG, Shakil M (2014) Normal and Student’s \(t\) distributions and their applications. Atlantis Press, Paris
Akahira M (1995) A higher order approximation to a percentage point of the non-central t-distribution. Commun Stat Simul Comput 24:595–605
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723
Arnold BC, Beaver RJ (2002) Skewed multivariate models related to hidden truncation and/or selective reporting. Test 11:7–54
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Statist 12:171–178
Azzalini A, Capitanio A (2003) Statistical applications of the multivariate skew normal distribution. J R Stat Soc B 61:579–602
Baker RD (2016) A new asymmetric generalisation of the \(t\) distribution. arXiv:1606.05203 [stat.ME]
Bergmann DR, de Oliveira MA (2013) Modeling the distribution of Brazilian Stock Returns via scaled Student \(t\). Int Res J Finance Econ 108:27–38
Brazauskas V, Kleefeld A (2011) Folded- and log-folded-\(t\) distributions as models for insurance loss data. Scand Actuar J 2011:59–74
Burnham KP, Anderson DR (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociol Methods Res 33:261–304
Cademartori D, Romo C, Campos R, Galea M (2003) Robust estimation of systematic risk using the \(t\) distribution in the Chilean Stock Markets. Appl Econ Lett 10:447–453
Cassidy DT, Hamp MJ, Ouyed R (2010) Pricing European options with a log Student’s \(t\) distribution: a Gosset formula. Phys A Stat Mech Appl 389:5736–5748
Fang Y (2011) Asymptotic equivalence between cross-validations and Akaike Information Criteria in mixed-effects models. J Data Sci 9:15–21
Fernandez C, Steel MFJ (1998) On Bayesian modelling of fat tails and skewness. J Am Stat Assoc 93:359–371
Gómez HW, Torres FJ, Bolfarine H (2007a) Large-sample inference for the epsilon-skew-\(t\) distribution. Commun Stat Theory Methods 36:73–81
Gómez HW, Varela H, Vidal L (2013) A new class of skew-symmetric distributions and related families. Statistics 47:411–421
Gómez HW, Venegas O, Bolfarine H (2007b) Skew-symmetric distributions generated by the distribution function of the normal distribution. Environmetrics 18:395–407
Gosset WS (1908) The probable error of a mean. Biometrika 6:1–25
Harvey A, Lange R (2015) Volatility modelling with a generalized \(t\) distribution. Cambridge Working Papers in Economics, Faculty of Economics, Cambridge University
Hasan AM (2013) A study of non-central skew \(t\) distributions and their applications in data analysis and change point detection. Ph.D. Thesis, Bowling Green State University, OH, USA
Ho HJ, Pyne S, Lin TI (2012) Maximum likelihood inference for mixtures of skew Student-\(t\)-normal distributions through practical EM-type algorithms. Stat Comput 22:287–299
Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2, 2nd edn. Wiley, New York
Jones MC, Faddy MJ (2003) A skew extension of the \(t\) distribution, with applications. J R Stat Soc B 65:159–174
Jones MC, Pewsey A (2009) Sinh-arcsinh distributions. Biometrika 96:761–780
Kim H (2008) Moments of truncated Student-\(t\) distribution. J Korean Stat Soc 37:81–87
Ku YHH (2008) Student-\(t\) distribution based VAR-MGARCH: an application of the DCC model on international portfolio risk management. Appl Econ 40:1685–1697
Levy KJ, Narula SC (1974) Probability density plots of the non-central t-distribution. Int Stat Rev 42:305–306
Li R, Nadarajah S (2016) The Kumaraswamy skew \(G\) distributions. Bull Braz Math Soc (submitted)
McDonald JB, Newey WK (1988) Partially adaptive estimation of regression models via the generalized \(t\) distribution. Econ Theory 4:428–457
Mittnik S, Paolella MS (2000) Conditional density and Value-at-Risk prediction of Asian currency exchange rates. J Forecast 19:313–333
Nadarajah S (2008) On the generalized \(t\) (GT) distribution. Statistics 42:467–473
Ord JK (1968) The discrete Student’s \(t\) distribution. Ann Math Stat 39:1513–1516
Paolella M (1997) Tail estimation and conditional modeling of heteroscedastic time series. Ph.D. thesis, Institute of Statistics and Econometrics, Christian Albrechts University at Kiel
Papastathopoulos I, Tawn JA (2013) A generalised Student’s \(t\)-distribution. Stat Probab Lett 83:70–77
Psarakis S, Panaretos J (2007) The folded \(t\) distribution. Commun Stat Theory Methods 19:2717–2734
R Development Core Team (2016) A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna
Roozegar R, Nematollahi A, Jamalizadeh A (2016) Properties and inference for a new class of skew-\(t\) distributions. Commun Stat Simul Comput 45:3217–3237
Rosco JF, Jones MC, Pewsey A (2011) Skew \(t\) distributions via the sinh-arcsinh transformation. Test 20:630–652
Schlüter S, Fischer M (2012) A tail quantile approximation for the Student t distribution. Commun Stat Theory Methods 41:2617–2625
Schwarz GE (1978) Estimating the dimension of a model. Ann Stat 6:461–464
Sepanski JH, Kong L (2007) A family of generalized beta distributions for income. arXiv:0710.4614v1
Shafiei S, Doostparast M (2014) Balakrishnan skew-\(t\) distribution and associated statistical characteristics. Commun Stat Theory Methods 43:4109–4122
Shittu OI, Adepoju KA, Adeniji OE (2014) On beta skew-\(t\) distribution in modelling stock returns in Nigeria. Int J Mod Math Sci 11:94–102
Theodossiou P (1998) Financial data and the skewed generalized \(t\) distribution. Manag Sci 44:1650–1661
Zhu D, Galbraith JW (2010) A generalized asymmetric Student-\(t\) distribution with application to financial econometrics. J Econ 157:297–305
Acknowledgements
The authors would like to thank the Editor and the referee for careful reading and comments which greatly improved the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, R., Nadarajah, S. A review of Student’s t distribution and its generalizations. Empir Econ 58, 1461–1490 (2020). https://doi.org/10.1007/s00181-018-1570-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00181-018-1570-0