### Abstract

Confidence bands for the simple linear model are examined to assess their degrees of robustness to departures from the model. All calculations are made under an interval constraint on the range of interest for the predictor variable. The true model is taken to be a quadratic polynomial, and departure from the linear case is considered in terms of increasing magnitude of a curvature parameter. Proposed measures of robustness include the actual coverage probability under the true model and a measure of percentage coverage over the predictor variable axis when the band fails to cover the true quadratic model. The different band functions considered include hyperbolic bands constructed from Scheffd’s 5 method, linear-segmented bands, and fixed-width (minimax) and minimax-regret bands. In terms of preserving coverage probability under quadratic mis-specification, the fixed-width and linear-segment bands perform best, the former being preferred when the constraint interval on the predictor variable is small. When coverage is lost over some portion of the constraint interval, the fixed-width bands are also shown to preserve the greatest percentage of covered (predictor) axis values. The favorable performance of the fixed-width bands may be due to their generally wider, more rigid shape, relative to more curvilinear competitors. This may allow the bands to retain more extreme quadratic misspecifications than other bands.

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

Pages (from-to) | 879-885 |

Number of pages | 7 |

Journal | Journal. American Statistical Association |

Volume | 82 |

Issue number | 399 |

DOIs | |

State | Published - 1987 |

Externally published | Yes |

### Fingerprint

### Keywords

- Coverage probability
- Mean axis coverage
- Quadratic regression
- Simple linear regression

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Model robustness for simultaneous confidence bands.** / Piegorsch, Walter W.

Research output: Contribution to journal › Article

*Journal. American Statistical Association*, vol. 82, no. 399, pp. 879-885. https://doi.org/10.1080/01621459.1987.10478512

}

TY - JOUR

T1 - Model robustness for simultaneous confidence bands

AU - Piegorsch, Walter W

PY - 1987

Y1 - 1987

N2 - Confidence bands for the simple linear model are examined to assess their degrees of robustness to departures from the model. All calculations are made under an interval constraint on the range of interest for the predictor variable. The true model is taken to be a quadratic polynomial, and departure from the linear case is considered in terms of increasing magnitude of a curvature parameter. Proposed measures of robustness include the actual coverage probability under the true model and a measure of percentage coverage over the predictor variable axis when the band fails to cover the true quadratic model. The different band functions considered include hyperbolic bands constructed from Scheffd’s 5 method, linear-segmented bands, and fixed-width (minimax) and minimax-regret bands. In terms of preserving coverage probability under quadratic mis-specification, the fixed-width and linear-segment bands perform best, the former being preferred when the constraint interval on the predictor variable is small. When coverage is lost over some portion of the constraint interval, the fixed-width bands are also shown to preserve the greatest percentage of covered (predictor) axis values. The favorable performance of the fixed-width bands may be due to their generally wider, more rigid shape, relative to more curvilinear competitors. This may allow the bands to retain more extreme quadratic misspecifications than other bands.

AB - Confidence bands for the simple linear model are examined to assess their degrees of robustness to departures from the model. All calculations are made under an interval constraint on the range of interest for the predictor variable. The true model is taken to be a quadratic polynomial, and departure from the linear case is considered in terms of increasing magnitude of a curvature parameter. Proposed measures of robustness include the actual coverage probability under the true model and a measure of percentage coverage over the predictor variable axis when the band fails to cover the true quadratic model. The different band functions considered include hyperbolic bands constructed from Scheffd’s 5 method, linear-segmented bands, and fixed-width (minimax) and minimax-regret bands. In terms of preserving coverage probability under quadratic mis-specification, the fixed-width and linear-segment bands perform best, the former being preferred when the constraint interval on the predictor variable is small. When coverage is lost over some portion of the constraint interval, the fixed-width bands are also shown to preserve the greatest percentage of covered (predictor) axis values. The favorable performance of the fixed-width bands may be due to their generally wider, more rigid shape, relative to more curvilinear competitors. This may allow the bands to retain more extreme quadratic misspecifications than other bands.

KW - Coverage probability

KW - Mean axis coverage

KW - Quadratic regression

KW - Simple linear regression

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U2 - 10.1080/01621459.1987.10478512

DO - 10.1080/01621459.1987.10478512

M3 - Article

VL - 82

SP - 879

EP - 885

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 399

ER -