## The Annals of Statistics

### Confidence bands in generalized linear models

#### Abstract

Generalized linear models (GLM) include many useful models. This paper studies simultaneous confidence regions for the mean response function in these models. The coverage probabilities of these regions are related to tail probabilities of maxima of Gaussian random fields, asymptotically, and hence, the so-called tube formula is applicable without any modification. However, in the generalized linear models, the errors are often nonadditive and non-Gaussian and may be discrete. This poses a challenge to the accuracy of the approximation by the tube formula in the moderate sample situation. Here two alternative approaches are considered. These approaches are based on an Edgeworth expansion for the distribution of a maximum likelihood estimator and a version of Skorohod’s representation theorem, which are used to convert an error term (which is of order $n^{-1 /2}$ in one-sided confidence regions and of \$n^{-1} in two-sided confidence regions) from the Edgeworth expansion to a “bias” term. The bias is then estimated and corrected in two ways to adjust the approximation formula. Examples and simulations show that our methods are viable and complementary to existing methods. An application to insect data is provided. Code for implementing our procedures is available via the software parfit

#### Article information

Source
Ann. Statist., Volume 28, Number 2 (2000), 429-460.

Dates
First available in Project Euclid: 15 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1016218225

Digital Object Identifier
doi:10.1214/aos/1016218225

Mathematical Reviews number (MathSciNet)
MR1790004

Zentralblatt MATH identifier
1106.62343

#### Citation

Sun, Jiayang; Loader, Catherine; McCormick, William P. Confidence bands in generalized linear models. Ann. Statist. 28 (2000), no. 2, 429--460. doi:10.1214/aos/1016218225. https://projecteuclid.org/euclid.aos/1016218225

#### References

• Bliss, C. L. (1935). The calculation ofthe dosage-mortality curve. Ann. Appl. Biology 22 134-167.
• Brown, L. D. (1986). Fundamentals of Statistical Exponential Families: with Applications in Statistical Decision Theory. IMS, Hayward, CA.
• Faraway, J. and Sun, J (1995). Simultaneous confidence bands for linear regression with heteroscedastic errors. J. Amer. Statist. Assoc. 90 1094-1098.
• Hall, P. and Titterington, D. M. (1988). On confidence bands in nonparametric density estimation and regression. J. Multivariate Anal. 27 228-254.
• Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
• H¨ardle, W. and Marron, J. S. (1991). Bootstrap simultaneous error bars for nonparametric regression. Ann. Statist. 19 778-796.
• Johansen, S. and Johnstone, I. (1990). Hotelling's theorem on the volume oftubes: some illustrations in simultaneous inference and data analysis. Ann. Statist. 18 652-684.
• Knafl, G., Sacks, J. and Ylvisaker, D. (1985). Confidence bands for regression functions. J. Amer. Statist. Assoc. 80 683-691.
• Knowles, M. and Siegmund, D. (1989). On Hotelling's geometric approach to testing for a nonlinear parameter in regression. Internat. Statist. Rev. 57 205-220.
• Kreyszig, E. (1968). Introduction to Differential Geometry and Riemannian Geometry. Univ. Toronto Press.
• Loader, C. (1993). Nonparametric regression, confidence bands and bias correction. In Computing Science and Statistics. Proceedings of the 25th Symposium on the Interface 131-136. Interface Foundation of North America, Fairfax Station, VA.
• Loader, C. and Sun, J. (1997). Robustness oftube formula. J. Comput. Graph. Statist. 6 242-250.
• Naiman, D. Q. (1987). Simultaneous confidence bounds in multiple regression using predictor variable constraints. J. Amer. Statist. Assoc. 82 214-219.
• Naiman, D. Q. (1990). On volumes oftubular neighborhoods ofspherical polyhedra and statistical inference. Ann. Statist. 18 685-716.
• Naiman, D. Q. and Wynn, H. P. (1997). Abstract tubes, improved inclusion-exclusion identities and inequalities and importance sampling. Ann. Statist. 25 1954-1983.
• Scheff´e, H. (1959). The Analysis of Variance. Wiley, New York. Skovgaard, I. M. (1981a). Transformation ofan Edgeworth expansion by a sequence ofsmooth functions. Scand. J. Statist. 8 207-217. Skovgaard, I. M. (1981b). Edgeworth expansions ofthe distribution ofmaximum likelihood estimators in the general (non i.i.d.) case. Scand. J. Statist. 8 227-236.
• Sun, J. (1993). Tail probabilities ofthe maxima ofGaussian random fields. Ann. Probab. 21 34-71.
• Sun, J. and Loader, C. (1994). Simultaneous confidence bands for linear regression and smoothing. Ann. Statist. 22 1328-1345.
• Sun, J., Loader, C. and McCormick, W. (1998). Confidence bands for generalized confidence bands. Available at http://sun.cwru.edu/ jiayang/glm.ps.
• Sun, J., Raz., J. and Faraway, J. (1999). Simultaneous confidence bands for growth and response curves. Statist. Sinica 9 679-698.
• Weyl, H. (1939). On the volume oftubes. Amer. J. Math. 61 461-472.