Araştırma Makalesi
A Note On Confidence Regions Based On The Bivariate Chebyshev Inequality. Applications To Order Statistics And Data
Öz
Chebyshev’s inequality was recently extended to the multivariate case. In this paper this new inequality is used to obtain distribution-free confidence regions for an arbitrary bivariate random vector (X;Y ). The regions depend on the means, the variances and the (Pearson) correlation coefficient. The
theoretical method is illustrated by computing the confidence regions for two order statistics obtained from a sample of iid random variables or obtained from a sequence of dependent components. They are also computed for an arbitrary bivariate data set (with or without groups) by obtaining plots similar to univariate box plots.
Anahtar Kelimeler
Chebyshev (Tchebychev) inequality Mahalanobis distance Principal components Ellipsoid Order statistics Bivariate box plots
Kaynakça
- Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N.(2008). A First Course in Order Statistics. Classic ed., SIAM, Philadelphia, Pennsylvania.
- Budny, K. (2014). A generalization of Chebyshev’s inequality for Hilbert-space-valued random elements. Statistics & Probability Letters, 88, 62-65.
- Chen, X. (2011). A new generalization of Chebyshev inequality for random vectors. ArXiv:0707.0805v2.
- Fisher, R.A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7, Part II, 179-188.
- Marshall, A.W. and Olkin, I. (1960). Multivariate Chebyshev inequalities. The Annals of Mathematical Statistics, 31, 1001-1014.
- Navarro, J. (2014). A very simple proof of the multivariate Chebyshev’s inequality. DOI:10.1080/03610926.2013.873135.
- Navarro, J. (2014). Can the bounds in the multivariate Chebyshev inequality be attained?. Statistics & Probability Letters, 91, 1-5.
- Navarro, J. and Balakrishnan, N. (2010). Study of some measures of dependence between order statistics and systems. Journal of Multivariate Analysis, 101, 52-67.
- Prakasa Rao, B.L.S. (2010). Chebyshev’s inequality for Hilbert-space-valued random elements. Statistics & Probability Letters, 80, 1039-1042.
- Zhou, L. and Hu, Z.C. (2012). Chebyshev’s inequality for Banach-space-valued random elements. Statistics & Probability Letters, 82, 925-931.
Yıl 2014,
Cilt: 7 Sayı: 1, 1 - 14, 31.01.2014
Öz
Kaynakça
- Arnold, B.C., Balakrishnan, N. and Nagaraja, H.N.(2008). A First Course in Order Statistics. Classic ed., SIAM, Philadelphia, Pennsylvania.
- Budny, K. (2014). A generalization of Chebyshev’s inequality for Hilbert-space-valued random elements. Statistics & Probability Letters, 88, 62-65.
- Chen, X. (2011). A new generalization of Chebyshev inequality for random vectors. ArXiv:0707.0805v2.
- Fisher, R.A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7, Part II, 179-188.
- Marshall, A.W. and Olkin, I. (1960). Multivariate Chebyshev inequalities. The Annals of Mathematical Statistics, 31, 1001-1014.
- Navarro, J. (2014). A very simple proof of the multivariate Chebyshev’s inequality. DOI:10.1080/03610926.2013.873135.
- Navarro, J. (2014). Can the bounds in the multivariate Chebyshev inequality be attained?. Statistics & Probability Letters, 91, 1-5.
- Navarro, J. and Balakrishnan, N. (2010). Study of some measures of dependence between order statistics and systems. Journal of Multivariate Analysis, 101, 52-67.
- Prakasa Rao, B.L.S. (2010). Chebyshev’s inequality for Hilbert-space-valued random elements. Statistics & Probability Letters, 80, 1039-1042.
- Zhou, L. and Hu, Z.C. (2012). Chebyshev’s inequality for Banach-space-valued random elements. Statistics & Probability Letters, 82, 925-931.
Toplam 10 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 31 Ocak 2014 |
| Yayımlandığı Sayı | Yıl 2014 Cilt: 7 Sayı: 1 |
