Tailor-made tests for goodness of fit to semiparametric hypotheses. (English) Zbl 1092.62050

Summary: We introduce a new framework for constructing tests of general semiparametric hypotheses which have nontrivial power on the \(n^{-1/2}\) scale in every direction, and can be tailored to put substantial power on alternatives of importance. The approach is based on combining test statistics based on stochastic processes of score statistics with bootstrap critical values.


62G10 Nonparametric hypothesis testing
62G09 Nonparametric statistical resampling methods
62G20 Asymptotic properties of nonparametric inference
62M99 Inference from stochastic processes
Full Text: DOI arXiv


[1] Bickel, P. J. and Chernoff, H. (1993). Asymptotic distribution of the likelihood ratio statistic in a prototypical non-regular problem. In Statistics and Probability : A Raghu Raj Bahadur Festschrift (J. K. Ghosh, S. K. Mitra, K. R. Parthasarathy and B. L. S. Prakasa Rao, eds.) 83–96. Wiley, New York.
[2] Bickel, P. J., Götze, F. and van Zwet, W. R. (1997). Resampling fewer than \(n\) observations: Gains, losses and remedies for losses. Statist. Sinica 7 1–31. · Zbl 0927.62043
[3] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models . Johns Hopkins Univ. Press, Baltimore, MD. · Zbl 0786.62001
[4] Bickel, P. J. and Ren, J.-J. (1996). The \(m\) out of \(n\) bootstrap and goodness of fit tests with doubly censored data. Robust Statistics , Data Analysis , and Computer Intensive Methods. Lecture Notes in Statist. 109 35–47. Springer, New York. · Zbl 0839.62054
[5] Bickel, P. J. and Ren, J.-J. (2001). The bootstrap in hypothesis testing. In State of the Art in Probability and Statistics. Festschrift for Willem R. van Zwet (M. de Gunst, C. A. J. Klaassen and A. van der Vaart, eds.) 91–112. IMS, Beachwood, OH. · Zbl 1380.62183
[6] Bickel, P. J. and Ritov, Y. (1992). Testing for goodness of fit: A new approach. In Nonparametric Statistics and Related Topics (A. K. Md. E. Saleh, ed.) 51–57. North-Holland, Amsterdam.
[7] Bickel, P. J., Ritov, Y. and Stoker, T. (2005). Nonparametric testing of an index model. In Identification and Inference for Econometric Models : A Festschrift in Honor of Thomas Rothenberg (D. W. K. Andrews and J. H. Stock, eds.). Cambridge Univ. Press. · Zbl 1119.62039
[8] Choi, S., Hall, W. J. and Schick, A. (1996). Asymptotically uniformly most powerful tests in parametric and semiparametric models. Ann. Statist. 24 841–861. · Zbl 0860.62020
[9] Durbin, J. (1973). Distribution Theory for Tests Based on the Sample Distribution Function . SIAM, Philadelphia. · Zbl 0267.62002
[10] Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153–193. · Zbl 1029.62042
[11] Feuerverger, A. and Mureika, R. A. (1977). The empirical characteristic function and its applications. Ann. Statist. 5 88–97. · Zbl 0364.62051
[12] Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests . Academic Press, New York. · Zbl 0161.38102
[13] Hart, J. (1997). Nonparametric Smoothing and Lack-of-Fit Tests . Springer, New York. · Zbl 0886.62043
[14] Ibragimov, I. A. and Hasminskii, R. Z. (1981). Statistical Estimation : Asymptotic Theory . Springer, New York. · Zbl 0467.62026
[15] Ingster, Yu. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III. Math. Methods Statist. 2 85–114, 171–189, 249–268., Mathematical Reviews (MathSciNet): Mathematical Reviews (MathSciNet): MR1259685 · Zbl 0798.62059
[16] Janssen, A. (2000). Global power functions of goodness-of-fit tests. Ann. Statist. 28 239–254. · Zbl 1106.62329
[17] Kac, M., Kiefer, J. and Wolfowitz, J. (1955). On tests of normality and other tests of goodness-of-fit based on distance methods. Ann. Math. Statist. 26 189–211. · Zbl 0066.12301
[18] Kallenberg, W. and Ledwina, T. (1999). Data-driven rank tests for independence. J. Amer. Statist. Assoc. 94 285–301. JSTOR: · Zbl 1072.62574
[19] Khmaladze, E. V. (1979). The use of \(\omega^2\) tests for testing parametric hypotheses. Theory Probab. Appl. 24 283–301. · Zbl 0447.62049
[20] Klaassen, C. A. J. and Wellner, J. A. (1997). Efficient estimation in the bivariate normal copula model: Normal margins are least favourable. Bernoulli 3 55–77. · Zbl 0877.62055
[21] Neyman, J. (1959). Optimal asymptotic tests of composite statistical hypotheses. In Probability and Statistics : The Harald Cramér Volume (U. Grenander, ed.) 213–234. Almqvist and Wiksell, Stockholm. · Zbl 0104.12602
[22] Pfanzagl, J. (with the assistance of W. Wefelmayer) (1982). Contributions to a General Asymptotical Theory . Lecture Notes in Statist. 13 . Springer, New York. · Zbl 0512.62001
[23] Rao, C. R. (1973). Linear Statistical Inference and Its Applications , 2nd ed. Wiley, New York. · Zbl 0256.62002
[24] Rayner, J. C. W. and Best, D. J. (1989). Smooth Tests of Goodness-of-Fit . Oxford Univ. Press, New York. · Zbl 0731.62064
[25] Roy, S. N. (1957). Some Aspects of Multivariate Analysis . Wiley, New York. · Zbl 0083.19305
[26] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
[27] Wald, A. (1941). Asymptotically most powerful tests of statistical hypotheses. Ann. Math. Statist. 12 1–19. · Zbl 0024.42904
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.