Abstract
We consider a two-sample semiparametric model involving a real parameter θ and a nuisance parameter F which is a distribution function. This model includes the proportional hazard, proportional odds, linear transformation and Harrington-Fleming models (1982, Biometrika, 69, 533–546). We propose two types of estimates based on ranks. The first is a rank approximation to Huber's M-estimates (1981, Robust Statistics, Wiley) and the second is a Hodges-Lehmann type rank inversion estimate (1963, Ann. Math. Statist., 34, 598–611). We obtain asymptotic normality and efficiency results. The estimates are consistent and asymptotically normal generally but fully efficient only for special cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anscombe, F. J. and Tukey, J. W. (1954). The criticism of transformation, unpublished manuscript.
Begun, J. M. (1981). A Class of Rank Estimators of Relative Risk, Institute of Statistics Mimeo Series No. 1381, University of North Carolina at Chapel Hill.
Begun, J. M. and Wellner, J. A. (1983). Asymptotic efficiency of relative risk estimates, Contributions to Statistics: Essays in Honor of Norman L. Johnson, (ed. P. K., Sen), North-Holland, Amsterdam.
Begun, J. M., Hall, W. J., Huang, W. M. and Wellner, J. A. (1983). Information and asymptotic efficiency in parametric—nonparametric models, Ann. Statist., 11, 432–452.
Bennett, S. (1983). Log-logistic regression models for survival data, Appl. Statist., 32, 165–171.
Bickel, P. J. (1986). Efficient testing in a class of transformation models, Tech. Report, University of California, Berkeley.
Bickel, P. J. and Doksum, K. A. (1981). An analysis of transformations revisited, J. Amer. Statist. Assoc., 76, 293–311.
Box, G. E. P. and Cox, D. R. (1964). An analysis of transformation, J. Roy. Statist. Soc. Ser. B, 26, 211–252.
Clayton, D. and Cuzick, J. (1986). The semiparametric Pareto model for regression analysis of survival times, Proc. ISI., Amsterdam.
Cox, D. R. (1972). Regression models and life tables, J. Roy. Statist. Soc. Ser. B, 34, 187–202.
Cox, D. R. (1975). Partial likelihood, Biometrika, 62, 269–276.
Dabrowska, D. M. and Doksum, K. A. (1988). Partial likelihood in transformation models with censored data. Scand. J. Statist., 15, 1–23.
Doksum, K. A. (1987). An extension of partial likelihood methods for proportional hazard model to general transformation models, Ann. Statist., 15, 325–345.
Doksum, K. A. and Naboya, S. (1984). Estimation in proportional hazard and log-linear models, J. Statist. Plann. Inference, 9, 297–303.
Efron, B. (1977). The efficiency of Cox's likelihood function for censored data, J. Amer. Statist. Assoc., 72, 557–565.
Ferguson, T. S. (1967). Mathematical Statistics, Academic Press, New York.
Harrington, D. P. and Fleming, T. R. (1982). A class of rank test procedures for censored survival data, Biometrika, 69, 533–546.
Hodges, J. L.Jr. and Lehmann, E. L. (1963). Estimates of location based on rank tests, Ann. Math. Statist., 34, 598–611.
Huber, P. J. (1981). Robust Statistics, Wiley, New York.
Lehmann, E. L. (1953). The power of rank tests, Ann. Math. Statist, 24, 23–43.
Miura, R. (1985). Hodges-Lehmann type estimates and Lehmann's alternative models: A special lecture presented at the annual meeting of Japanese Mathematical Society, the Division of Statistical Mathematics, April, 1985 (in Japanese).
Pettitt, A. N. (1984). Proportional odds model for survival data and estimates using ranks, J. Roy. Statist. Soc. Ser. C, 33, 169–175.
Ruymgaart, F. H., Shorack, G. R. and Van, Zwet, W. R. (1972). Asymptotic normality of nonparametric tests for independence, Ann. Math. Statist., 43, 1122–1135.
Savage, I. R. (1956). Contributions to the theory of rank order statistics: The two sample case, Ann. Math. Statist., 31, 415–426.
Shorack, G. R. (1972). Functions of order statistics, Ann. Math. Statist., 43, 412–427.
Tukey, J. W. (1957). On the comparative anatomy of transformations, Ann. Math. Statist., 28, 602–632.
Wellner, J. A. (1986). Semiparametric models; progress and problems, Proc. ISI., Amsterdam.
Wong, W. H. (1986). Theory of partial likelihood, Ann. Statist., 14, 88–123.
Author information
Authors and Affiliations
Additional information
Research partially supported by National Science Foundation Grant DMS-86-02083 and National Institute of General Medical Sciences Grant SSS-Y1RO1GM35416-01
About this article
Cite this article
Dabrowska, D.M., Doksum, K.A. & Miura, R. Rank estimates in a class of semiparametric two-sample models. Ann Inst Stat Math 41, 63–79 (1989). https://doi.org/10.1007/BF00049110
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00049110