Robust Bayesian inference on scale parameters. (English) Zbl 0981.62024
Summary: We represent random vectors \(Z\) that take values in \({\mathfrak R}^n- \{0\}\) as \(Z=RY\), where \(R\) is a positive random variable and \(Y\) takes values in an \((n-1)\)-dimensional space \({\mathcal Y}\). By fixing the distribution of either \(R\) or \(Y\), while imposing independence between them, different classes of distributions on \({\mathfrak R}^n\) can be generated. As examples the spherical, \(l_q\)-spherical, \(v\)-spherical and anisotropic classes can be interpreted in this unifying framework.
We present a robust Bayesian analysis on a scale parameter in the pure scale model and in the regression model. In particular, we consider robustness of posterior inference on the scale parameter when the sampling distribution ranges over classes related to those mentioned above. Some links between Bayesian and sampling-theory results are also highlighted.
We present a robust Bayesian analysis on a scale parameter in the pure scale model and in the regression model. In particular, we consider robustness of posterior inference on the scale parameter when the sampling distribution ranges over classes related to those mentioned above. Some links between Bayesian and sampling-theory results are also highlighted.
MSC:
| 62F15 | Bayesian inference |
| 62J05 | Linear regression; mixed models |
| 60E05 | Probability distributions: general theory |
References:
| [1] | Barlow, R. E.; Mendel, M. B., The operational Bayesian approach, (Freeman, P. R.; Smith, A. F.M., Aspects of Uncertainty—A Tribute to D. V. Lindley (1994), Wiley: Wiley New York), 19-28 · Zbl 0841.62001 |
| [2] | Berger, J. O., Statistical Decision Theory and Bayesian Analysis (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0572.62008 |
| [3] | Box, G. E.P.; Tiao, G. C., Bayesian Inference in Statistical Analysis (1973), Addison-Wesley: Addison-Wesley Reading · Zbl 0178.22003 |
| [4] | Fang, K. T.; Bentler, P. M., A largest characterization of spherical and related distributions, Statist. Probab. Lett., 11, 107-110 (1991) · Zbl 0712.62045 |
| [5] | Fang, K. T.; Kotz, S.; Ng, K. W., Symmetric Multivariate and Related Distributions (1990), Chapman and Hall: Chapman and Hall London · Zbl 0699.62048 |
| [6] | Fang, K. T.; Li, R., Bayesian statistical inference on elliptical matrix distributions, J. Multivariate Anal., 70, 66-85 (1999) · Zbl 0958.62004 |
| [7] | Fang, K. T.; Zhang, Y. T., Generalized Multivariate Analysis (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0724.62054 |
| [8] | Fernández, C.; Osiewalski, J.; Steel, M. F.J., Modeling and inference with \(υ\)-spherical distributions, J. Amer. Statist. Assoc., 90, 1331-1340 (1995) · Zbl 0868.62045 |
| [9] | Fernández, C.; Osiewalski, J.; Steel, M. F.J., Classical and Bayesian inference robustness in multivariate regression models, J. Amer. Statist. Assoc., 92, 1434-1444 (1997) · Zbl 0912.62042 |
| [10] | Fernández, C.; Steel, M. F.J., Multivariate Student-\(t\) regression models: Pitfalls and inference, Biometrika, 86, 153-167 (1999) · Zbl 0917.62020 |
| [11] | Gupta, A. K.; Song, D., \(L_p\)-norm spherical distribution, J. Statist. Plann. Inference, 60, 241-260 (1997) · Zbl 0900.62270 |
| [12] | Gupta, A. K.; Song, D., Characterization of \(p\)-generalized normality, J. Multivariate Anal., 60, 61-71 (1997) · Zbl 0883.62048 |
| [13] | Kelker, D., Distribution theory of spherical distributions and a location-scale generalization, Sankhyā A, 32, 419-430 (1970) · Zbl 0223.60008 |
| [14] | Mendel, M. B., Development of Bayesian Parametric Theory with Applications to Control (1989), M.I.T: M.I.T Cambridge |
| [15] | Nachtsheim, C. J.; Johnson, M. E., A new family of multivariate distributions with applications to Monte Carlo studies, J. Amer. Statist. Assoc., 83, 984-989 (1988) · Zbl 0669.62030 |
| [16] | Ng, K. W.; Fraser, D. A.S., Inference for linear models with radially decomposable error, Multivariate Analysis and Its Applications. Multivariate Analysis and Its Applications, IMS Lecture Notes, 24 (1994), Inst. Math. Statist: Inst. Math. Statist Hayward, p. 359-367 |
| [17] | Osiewalski, J.; Steel, M. F.J., Robust Bayesian inference in \(l_q\)-spherical models, Biometrika, 80, 456-460 (1993) · Zbl 0778.62026 |
| [18] | Osiewalski, J.; Steel, M. F.J., Posterior moments of scale parameters in elliptical sampling models, (Berry, D.; Chaloner, K.; Geweke, J., Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner (1996), Wiley: Wiley New York), 323-335 |
| [19] | Takemura, A.; Kuriki, S., Theory of Cross Sectionally Contoured Distributions and its Applications. Theory of Cross Sectionally Contoured Distributions and its Applications, Discussion Paper F-Series, 96-F-15 (1996), University of TokyoFaculty of Economics |
| [20] | Yue, X. N.; Ma, C. S., Multivariate \(L_p\)-norm symmetrical distributions, Statist. Probab. Lett., 24, 281-288 (1995) · Zbl 0837.62045 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
