Abstract
The sample mean and sample variance are commonly used statistics in our study. In this paper, distributions of the sample mean and sample variance from a skew normal population are derived under closed skew normal (CSN) settings. The noncentral closed skew chi-square distribution is defined, and the distribution of quadratic forms is discussed. Our results generalize the corresponding results given under skew normal settings. Several examples are given for illustration of our results.
References
[1] A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Stat. 12 (1985), 171–178. Search in Google Scholar
[2] A. Azzalini, The Skew-normal and Related Families. Volume 3, Cambridge University, New York, 2013. 10.1017/CBO9781139248891Search in Google Scholar
[3] A. Azzalini and A. Capitanio, Statistical applications of the multivariate skew normal distribution, J. R. Stat. Soc. 61 (1999), 579–602. 10.1111/1467-9868.00194Search in Google Scholar
[4] A. Azzalini and A. Dalla Valle, The multivariate skew-normal distribution, Biometrika 83 (1996), 715–726. 10.1093/biomet/83.4.715Search in Google Scholar
[5] J. T. Chen, A. K. Gupta and T. T. Nguyen, The density of the skew normal sample mean and its applications, J. Stat. Comput. Simul. 74 (2004), 487–494. 10.1080/0094965031000147687Search in Google Scholar
[6] M. G. Genton, Skew-elliptical Distributions and Their Applications: A Journey Beyond Normality, CRC Press, Boca Raton, 2004. 10.1201/9780203492000Search in Google Scholar
[7] M. G. Genton, L. He and X. Liu, Moments of skew-normal random vectors and their quadratic forms, Statist. Probab. Lett. 51 (2001), 319–325. 10.1016/S0167-7152(00)00164-4Search in Google Scholar
[8] M. G. Genton and N. M. Loperfido, Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), 389–401. 10.1007/BF02507031Search in Google Scholar
[9] G. González-Farías, A. Domínguez-Molina and A. K. Gupta, Additive properties of skew normal random vectors, J. Statist. Plann. Inference 126 (2004), 521–534. 10.1016/j.jspi.2003.09.008Search in Google Scholar
[10] A. K. Gupta, G. González-Farías and A. Domínguez-Molina, A multivariate skew normal distribution, J. Multivariate Anal. 89 (2004), 181–190. 10.1016/S0047-259X(03)00131-3Search in Google Scholar
[11] A. K. Gupta and W. J. Huang, Quadratic forms in skew normal variates, J. Math. Anal. Appl. 273 (2002), 558–564. 10.1016/S0022-247X(02)00270-6Search in Google Scholar
[12] W. J. Huang and Y. H. Chen, Quadratic forms of multivariate skew normal-symmetric distributions, Statist. Probab. Lett. 76 (2006), 871–879. 10.1016/j.spl.2005.10.018Search in Google Scholar
[13] N. Loperfido, Quadratic forms of skew-normal random vectors, Statist. Probab. Lett. 54 (2001), 381–387. 10.1016/S0167-7152(01)00103-1Search in Google Scholar
[14] E. Lukacs, A characterization of the normal distribution, Ann. Math. Stat. 13 (1942), 91–93. 10.1214/aoms/1177731647Search in Google Scholar
[15] S. Nadarajah and R. Li, The exact density of the sum of independent skew normal random variables, J. Comput. Appl. Math. 311 (2017), 1–10. 10.1016/j.cam.2016.06.032Search in Google Scholar
[16] W. Tian, C. Wang, M. Wu and T. Wang, The multivariate extended skew normal distribution and its quadratic forms, Causal Inference in Econometrics, Springer, Berlin (2016), 153–169. 10.1007/978-3-319-27284-9_9Search in Google Scholar
[17] W. Tian and T. Wang, Quadratic forms of refined skew normal models based on stochastic representation, Random Oper. Stoch. Equ. 24 (2016), 225–234. 10.1515/rose-2016-0016Search in Google Scholar
[18] T. Wang, B. Li and A. K. Gupta, Distribution of quadratic forms under skew normal settings, J. Multivariate Anal. 100 (2009), 533–545. 10.1016/j.jmva.2008.06.003Search in Google Scholar
[19] C. Wong, J. Masaro and T. Wang, Multivariate versions of Cochran’s theorems, J. Multivariate Anal. 39 (1991), 154–174. 10.1016/0047-259X(91)90011-PSearch in Google Scholar
[20] R. Ye and T. Wang, Inferences in linear mixed models with skew-normal random effects, Acta Math. Sin. (Engl. Ser.) 31 (2015), 576–594. 10.1007/s10114-015-3326-5Search in Google Scholar
[21] R. Ye, T. Wang and A. K. Gupta, Distribution of matrix quadratic forms under skew-normal settings, J. Multivariate Anal. 131 (2014), 229–239. 10.1016/j.jmva.2014.07.001Search in Google Scholar
[22] X. Zhu, B. Li, M. Wu and T. Wang, Plausibility regions on parameters of the skew normal distribution based on inferential models, International Conference of the Thailand Econometrics Society—TES 2018, Springer, Berlin (2018), 287–302. 10.1007/978-3-319-70942-0_21Search in Google Scholar
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