Abstract
This paper studies regression, where the reciprocal of the mean of a dependent variable is considered to be a linear function of the regressor variables, and the observations on the dependent variable are assumed to have an inverse Gaussian distribution. The large sample theory for the pseudo maximum likelihood estimators is available in the literature, only when the number of replications increase at a fixed rate. This is inadequate for many practical applications. This paper establishes consistency and derives the asymptotic distribution for the pseudo maximum likelihood estimators under very general conditions on the design points. This includes the case where the number of replications do not grow large, as well as the one where there are no replications. The bootstrap procedure for inference on the regression parameters is also investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Babu, G. J. and Bai, Z. D. (1992). Edgeworth expansions for errors-in-variables models, J. Multivariate Anal., 42, 226–244.
Bhattacharyya, G. K. and Fries, A. (1982a). Inverse Gaussian regression and accelerated life tests, Survival Analysis (eds. J. Crowly and R. A. Johnson), IMS Lecture Notes-Monograph Series, 2, 101–118, Institute of Mathematical Statistics, Hayward, California.
Bhattacharyya, G. K. and Fries, A. (1982b). Fatigue failure models—the Birnbaum Saunders versus the inverse Gaussian, IEEE Transactions on Reliability, R- 31, 439–441.
Bhattacharyya, G. K. and Fries, A. (1986). On the inverse Gaussian multiple regression and model checking procedures, Reliability and Quality Control, (ed. A. P.Basu), 86–100, Elsevier, North-Holland.
Bunke, H. and Bunke, O. (1986). Statistical Inference in Linear Models; Statistical Methods of Model Building, Vol. 1, Wiley, New York.
Chaubey, Y. P., Nebebe, F. and Chen, P. (1993). Small area estimation under an inverse Gaussian model in finite population sampling (submitted).
Chhikara, R. S. and Folks, J. L. (1977). The inverse Gaussian distribution as a lifetime model, Technometrics, 19, 461–468.
Chhikara, R. S. and Folks, J. L. (1989). The Inverse Gaussian Distribution; Theory, Methodology and Applications, Marcel Dekker, New York.
Davis, A. S. (1977). Linear statistical inference as related to the inverse Gaussian distribution, Ph.D. Thesis, Department of Statistics, Oklahoma State University.
Folks, J. L. and Chhikara, R. S. (1978). The inverse Gaussian distribution and its statistical application—a review. J. Roy. Statist. Soc., B 40, 263–289.
Fries, A. and Bhattacharyya, G. K. (1983). Analysis of two-factor experiments under an inverse Gaussian model, J. Amer. Statist. Assoc., 78, 820–826.
Fuller, W. A. and Rao, J. N. K. (1978). Estimation for a linear regression model with unknown diagonal covariance matrix. Ann. Statist., 6, 1149–1158.
Iyengar, S. and Patwardhan, G. (1988). Recent developments in the inverse Gaussian distribution, Handbook of Statistics 7, (eds. P. R. Krishnaiah and C. R. Rao), 479–490, Elsevier, North-Holland, Amsterdam.
Jacquez, J. A., Mather, F. J. and Crawford, C. R. (1968). Linear regression with nonconstant, unknown error variances: Sampling experiments with least squares, weighted least squares and maximum likelihood estimators, Biometrics, 24, 607–626.
Nelson, W. B. (1971). Analysis of accelerated life test data, IEEE Transactions on Electrical Insulation. EI- 6, 165–181.
Shuster, J. J. (1968). On the inverse Gaussian distribution function. J. Amer. Statist. Assoc., 63, 1514–1516.
Tweedie, M. C. K. (1957a). Statistical properties of inverse Gaussian distributions—I, Ann. Math. Statist., 28, 362–377.
Tweedie, M. C. K. (1957b). Statistical properties of inverse Gaussian distributions—II, Ann. Math. Statist., 28, 696–705.
Whitmore, G. A. (1979). An inverse Gaussian model for labour turnover, J. Roy. Statist. Soc. Ser. A, 142, 468–478.
Whitmore, G. A. (1983). A regression method for censored inverse Gaussian data, Canad. J. Statist., 11, 305–315.
Whitmore, G. A. and Yalovsky, M. (1978). A normalizing logarithmic transformation for inverse Gaussian random variables, Technometrics, 20, 207–208.
Author information
Authors and Affiliations
Additional information
Research supported in part by NSF Grant DMS-9208066.
Research supported in part by NSERC of Canada.
About this article
Cite this article
Babu, G.J., Chaubey, Y.P. Asymptotics and bootstrap for inverse Gaussian regression. Ann Inst Stat Math 48, 75–88 (1996). https://doi.org/10.1007/BF00049290
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00049290