zbMATH — the first resource for mathematics

Skewed distributions generated by the normal kernel. (English) Zbl 1048.62014
Summary: Following a recent paper by A. K. Gupta et al. [Random Oper. Stoch. Equ. 10, 133–140 (2002)] we generate skew probability density functions (pdfs) of the form \(2f(u)G(\lambda u)\), where \(f\) is taken to be a normal pdf while the cumulative distributive function \(G\) is taken to come from one of normal, Student’s \(t\), Cauchy, Laplace, logistic or uniform distribution. The properties of the resulting distributions are studied. In particular, expressions for the \(n\) th moment and the characteristic function are derived. We also provide graphical illustrations and quantify the range of possible values of skewness and kurtosis.

62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
Full Text: DOI
[1] Arnold, B.C.; Beaver, R.J., Some skewed multivariate distributions, Amer. J. math. management sci., 20, 27-38, (2000) · Zbl 1189.62087
[2] Arnold, B.C.; Beaver, R.J., The skew-Cauchy distribution, Statist. probab. lett., 49, 285-290, (2000) · Zbl 0969.62037
[3] Arnold, B.C.; Beaver, R.J.; Groeneveld, R.A.; Meeker, W.Q., The nontruncated marginal of a truncated bivariate normal distribution, Psychometrika, 58, 471-488, (1983) · Zbl 0794.62075
[4] Azzalini, A., A class of distributions which includes the normal ones, Scand. J. statist., 12, 171-178, (1985) · Zbl 0581.62014
[5] Azzalini, A., Further results on a class of distributions which includes the normal ones, Statistica, 46, 199-208, (1986) · Zbl 0606.62013
[6] Balakrishnan, N., Ambagaspitiya, R.S., 1994. On skew-Laplace distributions. Technical Report, Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada.
[7] Gupta, A.K.; Chang, F.C.; Huang, W.J., Some skew-symmetric models, Random operators stochastic equations, 10, 133-140, (2002) · Zbl 1118.60300
[8] Hill, M.A.; Dixon, W.J., Robustness in real lifea study of clinical laboratory data, Biometrics, 38, 377-396, (1982)
[9] Mukhopadhyay, S.; Vidakovic, B., Efficiency of linear Bayes rules for a normal meanskewed priors class, The Statistician, 44, 389-397, (1995)
[10] O’Hagan, A.; Leonard, T., Bayes estimation subject to uncertainty about parameter constraints, Biometrika, 63, 201-203, (1976) · Zbl 0326.62025
[11] Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I., 1990. Integrals and Series, Vols. 1-3. Gordon and Breach Science Publishers, Amsterdam. · Zbl 0733.00004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.