## The Annals of Statistics

### Efficient and adaptive nonparametric test for the two-sample problem

#### Abstract

The notion of efficient test for a Euclidean parameter in a semiparametric model was introduced by Stein [Proc. Third Berkeley Symp. Math. Statist. Probab. 1 (1956) 187-195]. Such tests are locally most powerful for a wide class of infinite-dimensional nuisance parameters. The first formal application of this notion to a suitably parametrized two-sample problem was provided by Hájek [Ann. Math. Statist. 33 (1962) 1124-1147]. However, this and subsequent solutions appear to be not well-suited for practical applications. This article aims to show that an adaptive two-sample test introduced recently by Janic-Wróblewska and Ledwina [Scand. J. Statist. 27 (2000) 281-297] is locally most powerful under a more realistic setting.

#### Article information

Source
Ann. Statist., Volume 31, Number 6 (2003), 2036-2058.

Dates
First available in Project Euclid: 16 January 2004

https://projecteuclid.org/euclid.aos/1074290336

Digital Object Identifier
doi:10.1214/aos/1074290336

Mathematical Reviews number (MathSciNet)
MR2036399

Zentralblatt MATH identifier
1065.62079

Subjects
Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties
Secondary: 62G99: None of the above, but in this section

#### Citation

Ducharme, Gilles R.; Ledwina, Teresa. Efficient and adaptive nonparametric test for the two-sample problem. Ann. Statist. 31 (2003), no. 6, 2036--2058. doi:10.1214/aos/1074290336. https://projecteuclid.org/euclid.aos/1074290336

#### References

• Albers, W., Kallenberg, W. C. M. and Martini, F. (2001). Data driven rank tests for classes of tail alternatives. J. Amer. Statist. Assoc. 96 685--696.
• Bajorski, P. (1992). Max-type rank tests in the two-sample problem. Zastos. Mat. 21 371--385.
• Barron, A. R. and Sheu, C. (1991). Approximation of density functions by sequences of exponential families. Ann. Statist. 19 1347--1369.
• Behnen, K. (1975). The Randles--Hogg test and an alternative proposal. Comm. Statist. 4 203--238.
• Behnen, K. (1981). Nichtparametrische Statistik: Zweistichproben Rangtests. Z. Angew. Math. Mech. 61 T203--T212.
• Behnen, K. and Neuhaus, G. (1983). Galton's test as a linear rank test with estimated scores and its local asymptotic efficiency. Ann. Statist. 11 588--599.
• Behnen, K. and Neuhaus, G. (1989). Rank Tests with Estimated Scores and Their Application. Teubner, Stuttgart.
• Bickel, P. J. (1974). Edgeworth expansions in nonparametric statistics. Ann. Statist. 2 1--20.
• 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.
• Book, S. A. (1976). The Cramér--Feller--Petrov large deviation theorem for triangular arrays. Technical report, Dept. Mathematics, California State College, Dominguez Hills.
• Eubank, R. L., LaRiccia, V. N. and Rosenstein, R. B. (1987). Test statistics derived as components of Pearson's phi-squared distance measure. J. Amer. Statist. Assoc. 82 816--825.
• Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman's truncation. J. Amer. Statist. Assoc. 91 674--688.
• Hájek, J. (1962). Asymptotically most powerful rank-order tests. Ann. Math. Statist. 33 1124--1147.
• Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39 325--346.
• Hogg, R. V. and Lenth, R. V. (1984). A review of some adaptive statistical techniques. Comm. Statist. A---Theory Methods 13 1551--1579.
• Hušková, M. (1977). The rate of convergence of simple linear rank statistics under hypothesis and alternatives. Ann. Statist. 5 658--670.
• Hušková, M. (1984). Adaptive methods. In Handbook of Statistics 4. Nonparametric Methods (P. R. Krishnaiah and P. K. Sen, eds.) 347--358. North-Holland, Amsterdam.
• Hušková, M. and Sen, P. K. (1985). On sequentially adaptive asymptotically efficient rank statistics. Sequential Anal. 4 125--151.
• Inglot, T. (1999). Generalized intermediate efficiency of goodness-of-fit tests. Math. Methods Statist. 8 487--509.
• Inglot, T., Kallenberg, W. C. M. and Ledwina, T. (1998). Vanishing shortcoming of data driven Neyman's tests. In Asymptotic Methods in Probability and Statistics (B. Szyszkowicz, ed.) 811--829. North-Holland, Amsterdam.
• Inglot, T., Kallenberg, W. C. M. and Ledwina, T. (2000). Vanishing shortcoming and asymptotic relative efficiency. Ann. Statist. 28 215--238. [Correction (2000) 28 1795.]
• Inglot, T. and Ledwina, T. (1996). Asymptotic optimality of data-driven Neyman's tests for uniformity. Ann. Statist. 24 1982--2019.
• Inglot, T. and Ledwina, T. (2001a). Intermediate approach to comparison of some goodness-of-fit tests. Ann. Inst. Statist. Math. 53 810--834.
• Inglot, T. and Ledwina, T. (2001b). Asymptotic optimality of data driven smooth tests for location--scale family. Sankhyā Ser. A 63 41--71.
• Janic-Wróblewska, A. and Ledwina, T. (2000). Data driven rank test for two-sample problem. Scand. J. Statist. 27 281--297.
• Kallenberg, W. C. M. (1978). Asymptotic Optimality of Likelihood Ratio Tests in Exponential Families. Mathematical Centre Tracts 77. Math. Centrum, Amsterdam.
• Kallenberg, W. C. M. (1982). Cramér type large deviations for simple linear rank statistics. Z. Wahrsch. Verw. Gebiete 60 403--409.
• Kallenberg, W. C. M. (1983). Intermediate efficiency, theory and examples. Ann. Statist. 11 170--182.
• Kallenberg, W. C. M. and Ledwina, T. (1997). Data-driven smooth tests when the hypothesis is composite. J. Amer. Statist. Assoc. 92 1094--1104.
• Neuhaus, G. (1982). $H_0$-contiguity in nonparametric testing problems and sample Pitman efficiency. Ann. Statist. 10 575--582.
• Neuhaus, G. (1987). Local asymptotics for linear rank statistics with estimated score functions. Ann. Statist. 15 491--512.
• Oosterhoff, J. (1969). Combination of One-sided Statistical Tests. Mathematical Centre Tracts 28. Math. Centrum, Amsterdam.
• Oosterhoff, J. and van Zwet, W. R. (1972). The likelihood ratio test for the multinomial distribution. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 1 31--50. Univ. California Press, Berkeley.
• Sansone, G. (1959). Orthogonal Functions. Interscience, New York.
• Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
• Stein, C. (1956). Efficient nonparametric testing and estimation. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 187--195. Univ. California Press, Berkeley.
• Yurinskii, V. V. (1976). Exponential inequalities for sums of random vectors. J. Multivariate Anal. 6 473--499.