Advances and challenges in inferences for elliptically contoured \(t\) distributions. (English) Zbl 1407.62237
Sutradhar, Brajendra C. (ed.), Advances and challenges in parametric and semi-parametric analysis for correlated data. Proceedings of the 2015 international symposium in statistics, ISS 2015, St. John’s, Canada, July 6–8, 2015. Cham: Springer. Lect. Notes Stat. 218, 3-39 (2016).
Summary: When a multivariate elliptical such as \(t\) response is taken from each of \(n\) individuals, the inference for the parameters of the \(t\) distribution including the location (or regression effects), scale and degrees of freedom (or shape) depends on the assumption whether \(n\) multi-dimensional responses are independent or uncorrelated but dependent. In the former case, that is, when responses are independent, the exact sampling theory based inference is extremely complicated, whereas in the later case the derivation of the exact sampling distributions for the standard statistics is manageable but the estimators based on certain standard statistics such as sample covariance matrix may be inconsistent for the respective parameters. In this paper we provide a detailed discussion on the advances and challenges in inferences using uncorrelated but dependent \(t\) samples. We then propose a clustered regression model where the multivariate \(t\) responses in the cluster are uncorrelated but such clustered responses are taken from a large number of independent individuals. The inference including the consistent estimation of the parameters of this proposed model is also presented.
For the entire collection see [Zbl 1347.62015].
For the entire collection see [Zbl 1347.62015].
MSC:
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
| 62F03 | Parametric hypothesis testing |
Keywords:
clustered regression model with uncorrelated \(t\) errors; consistent estimation; elliptically contoured distribution; multivariate \(t\) and normal as special cases; normality based testing yielding degrees of freedom based power property; regression effects; scale matrix; shape or degrees of freedom parameterReferences:
| [1] | Amemiya, T.: Advanced Econometrics. Harvard University Press, Cambridge, MA (1985) |
| [2] | Anderson, T.W., Fang, K.T., Hsu, H.: Maximum likelihood estimates and likelihood ratio criteria for multivariate elliptically contoured distributions. Can. J. Stat. 14, 55-59 (1986) · Zbl 0587.62104 · doi:10.2307/3315036 |
| [3] | Chib, S., Tiwari, R.C., Jammalamadaka, S.R.: Bayes prediction in regressions with elliptical errors. J. Econ. 38, 349-360 (1988) · Zbl 0677.62030 · doi:10.1016/0304-4076(88)90050-4 |
| [4] | Chib, S., Osiewalski, J., Steel, M.F.J.: Posterior inference on the degrees of freedom parameter in multivariate-\(t\) regression model. Econ. Lett. 37, 391-397 (1991) · Zbl 0748.62033 · doi:10.1016/0165-1765(91)90076-W |
| [5] | Cornish, E.A.: The multivariate \(t\) distribution associated with a set of normal sample deviates. Aust. J. Phys. 7, 531-542 (1954) · Zbl 0059.13201 |
| [6] | Dunnett, C.W., Sobel, M.: A bivariate generalization of Student’s \(t\)-distribution with tables for certain special cases. Biometrika 41, 153-169 (1954) · Zbl 0056.36703 · doi:10.1093/biomet/41.1-2.153 |
| [7] | Fisher, R.A.: Applications of “Student’s” distribution. Metron 5, 90-104 (1925) · JFM 51.0387.01 |
| [8] | Fisher, R.A., Healy, M.J.R.: New tables of Behren’s test of significance. J. R. Stat. Soc. B 18, 212-216 (1956) · Zbl 0072.36006 |
| [9] | Ghosh, B.K.: On the distribution of the difference of two \(t\)-variables. J. Am. Stat. Assoc. 70, 463-467 (1975) · Zbl 0324.62010 |
| [10] | Johnson, N.L., Kotz, S.: Continuous Multivariate Distributions. Wiley, New York (1972) · Zbl 0248.62021 |
| [11] | Kariya, T.: Robustness of multivariate tests. Ann. Stat. 9, 1267-1275 (1981) · Zbl 0474.62048 · doi:10.1214/aos/1176345643 |
| [12] | Kelker, D.: Distribution theory of spherical distribution and a location scale parameter generalization. Sankhya A 32, 419-430 (1970) · Zbl 0223.60008 |
| [13] | Kotz, S., Nadarajah, S.: Multivariate T-Distributions and Their Applications. Cambridge University Press, Cambridge (2004) · Zbl 1100.62059 · doi:10.1017/CBO9780511550683 |
| [14] | Kubokawa, T., Srivastava, M.S.: Robust improvements in estimation of mean and covariance matrices in elliptically contoured distribution. Technical Report No. 9714. Department of Statistics, University of Toronto (1977) |
| [15] | Mardia, K.V.: Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 47-55 (1970) · Zbl 0214.46302 · doi:10.1093/biomet/57.3.519 |
| [16] | Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982) · Zbl 0556.62028 · doi:10.1002/9780470316559 |
| [17] | Pearson, K.: Tables of the Incomplete \(β\)-Function. University Press, London (1934) · Zbl 0008.30403 |
| [18] | Raiffa, H., Schlaifer, R.: Applied Statistical Decision Theory. Harvard University Press,Cambridge, MA (1961) · Zbl 0181.21801 |
| [19] | Student: Probable error of a correlation coefficient. Biometrika 6, 302-310 (1908) |
| [20] | Sutradhar, B.C.: On the characteristic function of multivariate Student \(t\)-distribution. Can. J. Stat. 14, 329-337 (1986) · Zbl 0621.60018 · doi:10.2307/3315191 |
| [21] | Sutradhar, B.C.: Testing linear hypothesis with \(t\) error variable. Sankhya B 40, 175-180 (1988) · Zbl 0654.62027 |
| [22] | Sutradhar, B.C.: Discrimination of observations into one of two \(t\) populations. Biometrics 46, 827-835 (1990) · doi:10.2307/2532099 |
| [23] | Sutradhar, B.C.: Score test for the covariance matrix of the elliptical \(t\)-distribution. J. Multivar. Anal. 46, 1-12 (1993) · Zbl 0778.62052 · doi:10.1006/jmva.1993.1043 |
| [24] | Sutradhar, B.C.: On cluster regression and factor analysis models with elliptic t errors. In: Anderson, T.W., Fang, K.T., Olkin, I. (eds.) IMS Lecture Notes-Monograph Series, Institute of Mathematical Statistics, Hayward, California, vol. 24, pp. 369-383 (1994) |
| [25] | Sutradhar, B.C.: Multivariate \(t\) distribution. In: update volume 2 of Encyclopedia of Statistical Sciences. Wiley, New York, pp. 440-448 (1998) |
| [26] | Sutradhar, B.C.: On the consistency of the estimators of the parameters of elliptically contoured distributions. In the special issue of Journal of Statistical Sciences published in honour of Professor M. S. Haq, U. W. O., ed. B.K. Sinha, Indian Statistical Institute, 3, 91-103 (2004) |
| [27] | Sutradhar, B.C.: Dynamic Mixed Models for Familial Longitudinal Data. Springer Series in Statistics, 512 pp. Springer, New York (2011) · Zbl 1277.62026 |
| [28] | Sutradhar, B.C., Ali, M.M.: Estimation of the parameters of a regression model with a multivariate \(t\) error variable. Commun. Stat. Theory Methods 15, 429-450 (1986) · Zbl 0608.62061 · doi:10.1080/03610928608829130 |
| [29] | Sutradhar, B.C., Ali, M.M.: A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate \(t\) model. J. Multivar. Anal. 29, 155-162 (1989) · Zbl 0667.62036 · doi:10.1016/0047-259X(89)90082-1 |
| [30] | Walker, G.A., Saw, J.G.: The distribution of linear combinations of \(t\) variables. J. Am. Stat. Assoc. 73, 876-878 (1978) · Zbl 0391.62013 |
| [31] | Yuen, K.K., Murthy, V.K.: Percentage points of the \(t\)-distribution of the \(t\) statistic when the parent is Student’s \(t\). Technometrics 16, 495-497 (1974) · Zbl 0305.62032 |
| [32] | Zellner, A.: Bayesian and non-Bayesian analysis of the regression model with multivariate Student-\(t\) error terms. J. Am. Stat. Assoc. 71, 400-405 (1976) · Zbl 0348.62026 |
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