Robust Likelihood Methods Based on the Skew‐t and Related Distributions

The robustness problem is tackled by adopting a parametric class of distributions flexible enough to match the behaviour of the observed data. In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew‐t distribution is explored in more detail and reasons are given to adopt this option as a sensible general‐purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome. Some theoretical arguments, outcomes of a few simulation experiments and various wide‐ranging examples with real data are provided in support of the claim.

[1]  Francis Galton F.R.S..  IV. Statistics by intercomparison, with remarks on the law of frequency of error , 1875 .

[2]  W. R. Buckland,et al.  Statistical Theory and Methodology in Science and Engineering. , 1960 .

[3]  K. Miller.  Some Multivariate $t$-Distributions , 1968 .

[4]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[5]  S. Stigler.  Do Robust Estimators Work with Real Data , 1977 .

[6]  L. Denby,et al.  Robust Estimation of the First-Order Autoregressive Parameter , 1979 .

[7]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[8]  W. Dixon,et al.  Robustness in real life: a study of clinical laboratory data. , 1982, Biometrics.

[9]  H. Künsch.  Infinitesimal Robustness for Autoregressive Processes , 1984 .

[10]  A. Azzalini.  A class of distributions which includes the normal ones , 1985 .

[11]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[12]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[13]  Thomas A. Herring,et al.  Blowing in the wind , 1988, Nature.

[14]  Jeremy MG Taylor,et al.  Robust Statistical Modeling Using the t Distribution , 1989 .

[15]  R. Tibshirani,et al.  Flexible Discriminant Analysis by Optimal Scoring , 1994 .

[16]  A. Goldman.  An Introduction to Regression Graphics , 1995 .

[17]  Yadolah Dodge,et al.  The Guinea Pig of Multiple Regression , 1996 .

[18]  A. Azzalini,et al.  The multivariate skew-normal distribution , 1996 .

[19]  M. Steel,et al.  Multivariate Student -t Regression Models : Pitfalls and Inference , 1999 .

[20]  A. Azzalini,et al.  Statistical applications of the multivariate skew normal distribution , 2009, 0911.2093.

[21]  M. Genton,et al.  Highly Robust Estimation of the Autocovariance Function , 2000 .

[22]  A. Pewsey.  Problems of inference for Azzalini's skewnormal distribution , 2000 .

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

[24]  Trevor Hastie,et al.  The Elements of Statistical Learning , 2001 .

[25]  M. C. Jones.  A dependent bivariate t distribution with marginals on different degrees of freedom , 2002 .

[26]  B. Mallick,et al.  Moments of random vectors with skew t distribution and their quadratic forms , 2003 .

[27]  Arjun K. Gupta.  Multivariate skew t-distribution , 2003 .

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

[29]  M. C. Jones,et al.  A skew extension of the t‐distribution, with applications , 2003 .

[30]  Marc G. Genton,et al.  Comprehensive definitions of breakdown points for independent and dependent observations , 2003 .

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

[32]  Arjun K. Gupta,et al.  A multivariate skew normal distribution , 2004 .

[33]  M. Genton,et al.  Flexible Class of Skew‐Symmetric Distributions , 2004 .

[34]  Samuel Kotz,et al.  Multivariate T-Distributions and Their Applications , 2004 .

[35]  Anna Clara Monti,et al.  Inferential Aspects of the Skew Exponential Power Distribution , 2004 .

[36]  Ananda Sen,et al.  Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality , 2005, Technometrics.

[37]  Monica Chiogna.  A note on the asymptotic distribution of the maximum likelihood estimator for the scalar skew-normal distribution , 2005, Stat. Methods Appl..

[38]  A. Azzalini.  The Skew‐normal Distribution and Related Multivariate Families * , 2005 .

[39]  A. Tsiatis,et al.  Locally Efficient Semiparametric Estimators for Generalized Skew-Elliptical Distributions , 2005 .

[40]  B. Liseo,et al.  A note on reference priors for the scalar skew-normal distribution , 2006 .

[41]  Eric M. Aldrich,et al.  Calibrated Probabilistic Forecasting at the Stateline Wind Energy Center , 2006 .

[42]  N. Sartori.  Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions , 2006 .

[43]  Arthur Pewsey.  Some Observations on a Simple Means of Generating Skew Distributions , 2006 .

[44]  Marc G. Genton,et al.  On Gauss’s characterization of the normal distribution , 2007 .

[45]  Yanyuan Ma,et al.  Constrained local likelihood estimators for semiparametric skew-normal distributions , 2007 .

[46]  C. Bayes.  Bayesian inference for the skewness parameter of the scalar skew-normal distribution , 2007 .

[47]  R. Arellano-Valle,et al.  The centred parametrization for the multivariate skew-normal distribution , 2008 .

[48]  Reinaldo Boris Arellano-Valle,et al.  Shape mixtures of multivariate skew-normal distributions , 2009, J. Multivar. Anal..

[49]  Hsiu J. Ho,et al.  Analysis of multivariate skew normal models with incomplete data , 2009, J. Multivar. Anal..

[50]  Olcay Arslan,et al.  An alternative multivariate skew Laplace distribution: properties and estimation , 2010 .

[51]  Marc G. Genton,et al.  Visualizing Influential Observations in Dependent Data , 2010 .

[52]  Christophe Ley,et al.  On the singularity of multivariate skew-symmetric models , 2010, J. Multivar. Anal..

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

[54]  Christophe Ley,et al.  On Fisher information matrices and profile log-likelihood functions in generalized skew-elliptical models , 2010 .

[55]  William N. Venables,et al.  Modern Applied Statistics with S , 2010 .