### Abstract

We consider likelihood and Bayesian inferences for seemingly unrelated (linear) regressions for the joint multivariate tor (e.g. Zellner, 1976) and the independent tor (e.g. Maronna, 1976) models. For likelihood inference, the scale matrix and the shape parameter for the joint tor model cannot be consistently estimated because of the lack of adequate information to identify the latter. The joint tor model also yields the same MLEs for the regression coefficients and the scale matrix as for the independent normal error model, which are not robust against, outliers. Further, linear hypotheses with respect to the regression coefficients also give rise to the same null distributions as for the independent normal error model, though the MLE has a non-normal limiting distribution. In contrast to the striking similarities between the joint tor and the independent normal error models, the independent tor model yields MLEs that are robust against outliers. Since the MLE of the shape parameter reflects the tails of the data distributions, this model extends the independent normal error model for modeling data distributions with relatively thicker tails. These differences are also discussed with respect to the posterior and predictive distributions for Bayesian inference.

Original language | English (US) |
---|---|

Pages (from-to) | 2119-2140 |

Number of pages | 22 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 28 |

Issue number | 9 |

State | Published - 1999 |

Externally published | Yes |

### Fingerprint

### Keywords

- Bayesian inference
- GMANOVA
- Growth curves models
- Maximum likelihood
- Multivariate normal distribution
- Robust Inference

### ASJC Scopus subject areas

- Statistics and Probability
- Safety, Risk, Reliability and Quality

### Cite this

*Communications in Statistics - Theory and Methods*,

*28*(9), 2119-2140.

**On the difference in inference and prediction between the joint and independent t-error models for seemingly unrelated regressions.** / Kowalski, Jeanne; Mendoza-Blanco, José R.; Tu, Xin M.; Gleser, Leon J.

Research output: Contribution to journal › Article

*Communications in Statistics - Theory and Methods*, vol. 28, no. 9, pp. 2119-2140.

}

TY - JOUR

T1 - On the difference in inference and prediction between the joint and independent t-error models for seemingly unrelated regressions

AU - Kowalski, Jeanne

AU - Mendoza-Blanco, José R.

AU - Tu, Xin M.

AU - Gleser, Leon J.

PY - 1999

Y1 - 1999

N2 - We consider likelihood and Bayesian inferences for seemingly unrelated (linear) regressions for the joint multivariate tor (e.g. Zellner, 1976) and the independent tor (e.g. Maronna, 1976) models. For likelihood inference, the scale matrix and the shape parameter for the joint tor model cannot be consistently estimated because of the lack of adequate information to identify the latter. The joint tor model also yields the same MLEs for the regression coefficients and the scale matrix as for the independent normal error model, which are not robust against, outliers. Further, linear hypotheses with respect to the regression coefficients also give rise to the same null distributions as for the independent normal error model, though the MLE has a non-normal limiting distribution. In contrast to the striking similarities between the joint tor and the independent normal error models, the independent tor model yields MLEs that are robust against outliers. Since the MLE of the shape parameter reflects the tails of the data distributions, this model extends the independent normal error model for modeling data distributions with relatively thicker tails. These differences are also discussed with respect to the posterior and predictive distributions for Bayesian inference.

AB - We consider likelihood and Bayesian inferences for seemingly unrelated (linear) regressions for the joint multivariate tor (e.g. Zellner, 1976) and the independent tor (e.g. Maronna, 1976) models. For likelihood inference, the scale matrix and the shape parameter for the joint tor model cannot be consistently estimated because of the lack of adequate information to identify the latter. The joint tor model also yields the same MLEs for the regression coefficients and the scale matrix as for the independent normal error model, which are not robust against, outliers. Further, linear hypotheses with respect to the regression coefficients also give rise to the same null distributions as for the independent normal error model, though the MLE has a non-normal limiting distribution. In contrast to the striking similarities between the joint tor and the independent normal error models, the independent tor model yields MLEs that are robust against outliers. Since the MLE of the shape parameter reflects the tails of the data distributions, this model extends the independent normal error model for modeling data distributions with relatively thicker tails. These differences are also discussed with respect to the posterior and predictive distributions for Bayesian inference.

KW - Bayesian inference

KW - GMANOVA

KW - Growth curves models

KW - Maximum likelihood

KW - Multivariate normal distribution

KW - Robust Inference

UR - http://www.scopus.com/inward/record.url?scp=0011873003&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011873003&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0011873003

VL - 28

SP - 2119

EP - 2140

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 9

ER -