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An invariance property of quadratic forms in random vectors with a selection distribution, with application to sample variogram and covariogram estimators. (English) Zbl 1440.62178
Summary: We study conditions under which an invariance property holds for the class of selection distributions. First, we consider selection distributions arising from two uncorrelated random vectors. In that setting, the invariance holds for the so-called \(\mathcal{C}\)-class and for elliptical distributions. Second, we describe the invariance property for selection distributions arising from two correlated random vectors. The particular case of the distribution of quadratic forms and its invariance, under various selection distributions, is investigated in more details. We describe the application of our invariance results to sample variogram and covariogram estimators used in spatial statistics and provide a small simulation study for illustration. We end with a discussion about other applications, for example such as linear models and indices of temporal/spatial dependence.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E05 Probability distributions: general theory
62H10 Multivariate distribution of statistics
62H12 Estimation in multivariate analysis
62M30 Inference from spatial processes
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[1] Allard D., Naveau P. (2007) A new spatial skew-normal random field model. Communications in Statistics Theory and Methods 36: 1821–1834 · Zbl 1124.62064
[2] Arellano-Valle R.B., Azzalini A. (2006) On the unification of families of skew-normal distributions. Scandinavian Journal of Statistics 33: 561–574 · Zbl 1117.62051
[3] Arellano-Valle R.B., Branco M.D., Genton M.G. (2006) A unified view on skewed distributions arising from selections. The Canadian Journal of Statistics 34: 581–601 · Zbl 1121.60009
[4] Arellano-Valle R.B., del Pino G. (2004) From symmetric to asymmetric distributions: a unified approach. In: Genton M.G. (eds) Skew-elliptical distributions and their applications: a journey beyond normality. Chapman & Hall/CRC, Boca Raton, pp 113–130
[5] Arellano-Valle R.B., del Pino G., SanMartĂ­n E. (2002) Definition and probabilistic properties of skew-distributions. Statistics and Probability Letters 58: 111–121 · Zbl 1045.62046
[6] Arellano-Valle R.B., Genton M.G. (2005) On fundamental skew distributions. Journal of Multivariate Analysis 96: 93–116 · Zbl 1073.62049
[7] Azzalini A. (1985) A class of distributions which includes the normal ones. Scandinavian Journal of Statistics. 12: 171–178 · Zbl 0581.62014
[8] Azzalini A., Capitanio A. (1999) Statistical applications of the multivariate skew-normal distribution. Journal of the Royal Statistical Society Series B: 61: 579–602 · Zbl 0924.62050
[9] Azzalini A., Capitanio A. (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. Journal of the Royal Statistical Society Series B 65: 367–389 · Zbl 1065.62094
[10] Azzalini A., Dalla Valle A. (1996) The multivariate skew-normal distribution. Biometrika 83: 715–726 · Zbl 0885.62062
[11] Breusch T.S., Robertson J.C., Welsh A.H. (1997) The emperor’s new clothes: a critique of the multivariate t regression model. Statistica Neerlandica 51: 269–286 · Zbl 0929.62062
[12] Fang, K.-T., Kotz, S., Ng, K.-W. (1990). Symmetric multivariate and related distributions. Monographs on statistics and applied probability, vol 36. London: Chapman and Hall. · Zbl 0699.62048
[13] Genton M.G. (Ed.) (2004) Skew-elliptical distributions and their applications: a journey beyond normality (Edited vol., 416 pp). Chapman & Hall/CRC, Boca Raton, FL · Zbl 1069.62045
[14] Genton M.G., He L., Liu X. (2001) Moments of skew-normal random vectors and their quadratic forms. Statistics and Probability Letters 51: 319–325 · Zbl 0972.62031
[15] Genton M.G., Loperfido N. (2005) Generalized skew-elliptical distributions and their quadratic forms. Annals of the Institute of Statistical Mathematics 57: 389–401 · Zbl 1083.62043
[16] Gorsich D.J., Genton M.G., Strang G. (2002) Eigenstructures of spatial design matrices. Journal of Multivariate Analysis 80: 138–165 · Zbl 1003.62084
[17] Hillier G., Martellosio F. (2006) Spatial design matrices and associated quadratic forms: structure and properties. Journal of Multivariate Analysis 97: 1–18 · Zbl 1078.62103
[18] Khatri C.G. (1980) Quadratic forms in normal variables. In: Krishnaiah P.R. (eds) Handbook of statistics, Vol. 1. North-Holland, Amsterdam, pp 443–469 · Zbl 0465.62045
[19] Kim H.-M., Ha E., Mallick B.K. (2004) Spatial prediction of rainfall using skew-normal processes. In: Genton M.G. (eds) Skew-elliptical distributions and their applications: a journey beyond normality. Chapman & Hall/CRC, Boca Raton, pp 279–289
[20] Wang J., Boyer J., Genton M.G. (2004a) A skew-symmetric representation of multivariate distributions. Statistica Sinica 14: 1259–1270 · Zbl 1060.62059
[21] Wang J., Boyer J., Genton M.G. (2004b) A note on an equivalence between chi-square and generalized skew-normal distributions. Statistics and Probability Letters 66: 395–398 · Zbl 1075.62010
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