Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution

In this paper, we compute doubly truncated moments for the selection elliptical (SE) class of distributions, which includes some multivariate asymmetric versions of well-known elliptical distributions, such as, the normal, Student's t, slash, among others. We address the moments for doubly truncated members of this family, establishing neat formulation for high order moments as well as for its first two moments. We establish sufficient and necessary conditions for the existence of these truncated moments. Further, we propose optimized methods able to deal with extreme setting of the parameters, partitions with almost zero volume or no truncation which are validated with a brief numerical study. Finally, we present some results useful in interval censoring models. All results has been particularized to the unified skew-t (SUT) distribution, a complex multivariate asymmetric heavy-tailed distribution which includes the extended skew-t (EST), extended skew-normal (ESN), skew-t (ST) and skew-normal (SN) distributions as particular and limiting cases.

[1]  G. M. Tallis.  The Moment Generating Function of the Truncated Multi‐Normal Distribution , 1961 .

[2]  Stoyan V. Stoyanov,et al.  Distortion Risk Measures in Portfolio Optimization , 2010 .

[3]  Moments of truncated scale mixtures of skew-normal distributions , 2020 .

[4]  Nadarajah Saralees,et al.  A Truncated Bivariate t Distribution , 2007 .

[5]  D. Dey,et al.  A General Class of Multivariate Skew-Elliptical Distributions , 2001 .

[6]  Simon Broda,et al.  Multivariate Elliptical Truncated Moments , 2016, J. Multivar. Anal..

[7]  Michel Denuit,et al.  Actuarial Theory for Dependent Risks: Measures, Orders and Models , 2005 .

[8]  Arjun K. Gupta,et al.  The Closed Skew-Normal Distribution , 2004 .

[9]  Xiao-Li Meng,et al.  Maximum likelihood estimation via the ECM algorithm: A general framework , 1993 .

[10]  Manjunath B. G.,et al.  Moments Calculation for the Double Truncated Multivariate Normal Density , 2009, Journal of Behavioral Data Science.

[11]  Marc G. Genton,et al.  Multivariate unified skew-elliptical distributions , 2010 .

[12]  C. E. Galarza,et al.  On moments of folded and truncated multivariate Student-t distributions based on recurrence relations , 2021 .

[13]  M. Genton,et al.  Multivariate extended skew-t distributions and related families , 2010 .

[14]  C. E. Galarza,et al.  Finite mixture modeling of censored and missing data using the multivariate skew-normal distribution , 2020, Advances in Data Analysis and Classification.

[15]  Emiliano A. Valdez,et al.  Tail Conditional Expectations for Elliptical Distributions , 2003 .

[16]  Njål Foldnes,et al.  Non-normal Data Simulation using Piecewise Linear Transforms , 2021, Structural Equation Modeling: A Multidisciplinary Journal.

[17]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[18]  J. Heckman.  The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for Such Models , 1976 .

[19]  Derrick S. Tracy,et al.  Recurrence relations for the moments of truncated multinormal distribution , 1976 .

[20]  D. Allard,et al.  Truncated skew-normal distributions: moments, estimation by weighted moments and application to climatic data , 2010 .

[21]  Ming-Hui Chen,et al.  Likelihood-based Inference For Mixed-effects Models With Censored Response Using The Multivariate-t Distribution , 2013 .

[22]  U. Makov,et al.  Extended Generalized Skew-Elliptical Distributions and their Moments , 2016, Sankhya A.

[23]  Larissa A. Matos,et al.  Likelihood‐based inference for spatiotemporal data with censored and missing responses , 2020, Environmetrics.

[24]  Luis M. Castro,et al.  A note on the parameterization of multivariate skewed-normal distributions , 2013 .

[25]  M. Genton,et al.  A unified view on skewed distributions arising from selections , 2006 .

[26]  O. Barndorff-Nielsen,et al.  Normal Variance-Mean Mixtures and z Distributions , 1982 .

[27]  Kun Chen,et al.  Finite mixture modeling of censored data using the multivariate Student-t distribution , 2017, J. Multivar. Anal..

[28]  Raymond Kan,et al.  On Moments of Folded and Truncated Multivariate Normal Distributions , 2016 .

[29]  S. Nadarajah.  A Truncated Bivariate t Distribution , 2007 .

[30]  S. Kullback.  Probability Densities with Given Marginals , 1968 .

[31]  J. Hintze,et al.  Violin plots : A box plot-density trace synergism , 1998 .

[32]  S. Ferrari,et al.  New Results on Truncated Elliptical Distributions , 2019, Communications in Mathematics and Statistics.

[33]  G. M. Tallis.  Plane Truncation in Normal Populations , 1965 .

[34]  Tsung-I Lin,et al.  Some results on the truncated multivariate t distribution , 2012 .

[35]  J. Tobin.  Estimation of Relationships for Limited Dependent Variables , 1958 .

[36]  B. Turnbull.  The Empirical Distribution Function with Arbitrarily Grouped, Censored, and Truncated Data , 1976 .

[37]  G. Pflug.  Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk , 2000 .

[38]  C. J. Adcock,et al.  Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution , 2010, Ann. Oper. Res..

[39]  Ahad Jamalizadeh,et al.  On moments of doubly truncated multivariate normal mean-variance mixture distributions with application to multivariate tail conditional expectation , 2020, J. Multivar. Anal..

[40]  G. M. Tallis.  Elliptical and Radial Truncation in Normal Populations , 1963 .

[41]  D. Tasche,et al.  Expected Shortfall: a natural coherent alternative to Value at Risk , 2001, cond-mat/0105191.

[42]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[43]  Bengt Muthén,et al.  Moments of the censored and truncated bivariate normal distribution , 1990 .

[44]  Haruhiko Ogasawara,et al.  Unified and non-recursive formulas for moments of the normal distribution with stripe truncation , 2021, Communications in Statistics - Theory and Methods.

[45]  S. Sahu,et al.  A new class of multivariate skew distributions with applications to bayesian regression models , 2003 .

[46]  Haruhiko Ogasawara,et al.  A non-recursive formula for various moments of the multivariate normal distribution with sectional truncation , 2021, J. Multivar. Anal..

[47]  W. Breymann,et al.  ghyp: A package on generalized hyperbolic distributions , 2009 .

[48]  B. G. Manjunath,et al.  Moments Calculation for the Doubly Truncated Multivariate Normal Density , 2021 .

[49]  R. Arellano-Valle,et al.  On the Unification of Families of Skew‐normal Distributions , 2006 .

[50]  A. Azzalini,et al.  Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution , 2003, 0911.2342.

[51]  Arjun K. Gupta,et al.  Additive properties of skew normal random vectors , 2004 .

[52]  Dipak K. Dey,et al.  On moments of folded and doubly truncated multivariate extended skew-normal distributions , 2021, Journal of Computational and Graphical Statistics.

[53]  Juan Carlos Arismendi,et al.  Multivariate truncated moments , 2013, J. Multivar. Anal..