### Abstract

Minimum bias (all bias) designs for the linear model were proposed by Box and Draper. In this article we extend their results to generalized linear models. We show that, in the canonical case, the minimum bias design density is: (a) proportional to the weight function reflecting the importance of each design point, and (b) inversely proportional to the observation variance at each point. We also derive minimum bias design densities in non-canonical cases. Implications for binary, Poisson and exponential data are considered in the examples. From these examples we observe that when the experimenter is mainly interested in the mean of the Binomial, Poisson or exponential distribution, rather than the canonical parameter, and if the weight function is chosen to be inversely proportional to the variance of the maximum likelihood estimator of the mean, minimum bias designs are uniform. These uniform designs automatically minimize the bias/standard error ratio and the mean square error/variance ratio.

Original language | English |
---|---|

Pages (from-to) | 587-599 |

Number of pages | 13 |

Journal | Sankhya: The Indian Journal of Statistics |

Volume | 68 |

Issue number | 4 |

Publication status | Published - 2006 |

### Fingerprint

### Keywords

- Binary data
- Exponential data
- Generalized linear model
- Minimum bias design
- Poisson data

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Sankhya: The Indian Journal of Statistics*,

*68*(4), 587-599.

**Minimum bias designs for generalized linear models.** / Abdelbasit, Khidir M.; Butler, Neil A.

Research output: Contribution to journal › Article

*Sankhya: The Indian Journal of Statistics*, vol. 68, no. 4, pp. 587-599.

}

TY - JOUR

T1 - Minimum bias designs for generalized linear models

AU - Abdelbasit, Khidir M.

AU - Butler, Neil A.

PY - 2006

Y1 - 2006

N2 - Minimum bias (all bias) designs for the linear model were proposed by Box and Draper. In this article we extend their results to generalized linear models. We show that, in the canonical case, the minimum bias design density is: (a) proportional to the weight function reflecting the importance of each design point, and (b) inversely proportional to the observation variance at each point. We also derive minimum bias design densities in non-canonical cases. Implications for binary, Poisson and exponential data are considered in the examples. From these examples we observe that when the experimenter is mainly interested in the mean of the Binomial, Poisson or exponential distribution, rather than the canonical parameter, and if the weight function is chosen to be inversely proportional to the variance of the maximum likelihood estimator of the mean, minimum bias designs are uniform. These uniform designs automatically minimize the bias/standard error ratio and the mean square error/variance ratio.

AB - Minimum bias (all bias) designs for the linear model were proposed by Box and Draper. In this article we extend their results to generalized linear models. We show that, in the canonical case, the minimum bias design density is: (a) proportional to the weight function reflecting the importance of each design point, and (b) inversely proportional to the observation variance at each point. We also derive minimum bias design densities in non-canonical cases. Implications for binary, Poisson and exponential data are considered in the examples. From these examples we observe that when the experimenter is mainly interested in the mean of the Binomial, Poisson or exponential distribution, rather than the canonical parameter, and if the weight function is chosen to be inversely proportional to the variance of the maximum likelihood estimator of the mean, minimum bias designs are uniform. These uniform designs automatically minimize the bias/standard error ratio and the mean square error/variance ratio.

KW - Binary data

KW - Exponential data

KW - Generalized linear model

KW - Minimum bias design

KW - Poisson data

UR - http://www.scopus.com/inward/record.url?scp=66249108944&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=66249108944&partnerID=8YFLogxK

M3 - Article

VL - 68

SP - 587

EP - 599

JO - Sankhya: The Indian Journal of Statistics

JF - Sankhya: The Indian Journal of Statistics

SN - 0972-7671

IS - 4

ER -