The Annals of Statistics

Adaptive hypothesis testing using wavelets

V. G. Spokoiny

Full-text: Open access


Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set are separated away from zero in an integral (e.g., $L_2$) norm and also possess some smoothness properties. The minimax rate of testing for this problem was evaluated in earlier papers by Ingster and by Lepski and Spokoiny under different kinds of smoothness assumptions. It was shown that both the optimal rate of testing and the structure of optimal (in rate) tests depend on smoothness parameters which are usually unknown in practical applications. In this paper the problem of adaptive (assumption free) testing is considered. It is shown that adaptive testing without loss of efficiency is impossible. An extra log log-factor is inessential but unavoidable payment for the adaptation. A simple adaptive test based on wavelet technique is constructed which is nearly minimax for a wide range of Besov classes.

Article information

Ann. Statist., Volume 24, Number 6 (1996), 2477-2498.

First available in Project Euclid: 16 September 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Adaptive testing signal detection minimax hypothesis testing nonparametric alternative thresholding wavelet decomposition


Spokoiny, V. G. Adaptive hypothesis testing using wavelets. Ann. Statist. 24 (1996), no. 6, 2477--2498. doi:10.1214/aos/1032181163.

Export citation


  • AMOSOVA, N. N. 1972. On limit theorem for probabilities of moderate deviations. Vestnik Z. Leningrad. Univ. 13 5 14. In Russian. Z.
  • BROWN, L. D. and LOW, M. G. 1996. Asy mptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384 2398. Z.
  • BROWN, L. D. and LOW, M. G. 1992. Superefficiency and lack of adaptability in functional estimation. Technical report, Cornell Univ.Z.
  • COHEN, A., DAUBECHIES, I., JAWERTH, B. and VIAL, P. 1993. Multiresolution analysis, wavelets, and fast algorithms on an interval. C. R. Acad. Sci. Paris Ser. I Math. 316 417 421. ´ Z.
  • COHEN, A., DAUBECHIES, B. and VIAL, P. 1993. Wavelets on the interval and fast wavelet transforms. Applied and Computational Harmonic Analy sis Ser. I Math. 1 54 81. ´ Z.
  • DAUBECHIES, I. 1992. Ten Lectures on Wavelets. SIAM, Philadelphia. Z.
  • DELy ON, B. and JUDITSKY, A. 1995. Wavelet estimators, global error measures: revisited. J. Applied Comp. and Harmonic Analy sis. To appear. Z.
  • DONOHO, D. L. and JOHNSTONE, I. M. 1994. Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425 455. Z.
  • DONOHO, D. L. and JOHNSTONE, I. M. 1995. Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200 1204. Z.
  • DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. 1994. Wavelet shrinkage: Z. asy mptopia? with discussion. J. Roy. Statist. Soc. Ser. B 57 301 369. Z.
  • EFROMOVICH, S. and PINSKER, M. S. 1984. Learning algorithm for nonparametric filtering. Automat. Remote Control 11 1434 1440. Z.
  • ERMAKOV, M. S. 1990. Minimax detection of a signal in a white Gaussian noise. Theory Probab. Appl. 35 667 679. Z.
  • GOLUBEV, G. K. 1990. Quasilinear estimates of signal in L. Problems Inform. Transmission 26 2 15 20. Z.
  • HARDLE, W. and MAMMEN, E. 1993. Comparing nonparametric versus parametric regression ¨ fits. Ann. Statist. 21 1926 1947. Z.
  • INGSTER, YU. I. 1982. Minimax nonparametric detection of signals in white Gaussian noise. Problems Inform. Transmission 18 130 140. Z.
  • INGSTER, YU. I. 1984a. Asy mptotic minimax testing of nonparametric hy pothesis on the distribution density of an independent sample. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Z. Inst. Steklov 136 74 96. In Russian. Z.
  • INGSTER, YU. I. 1984b. An asy mptotic minimax test of nonparametric hy pothesis about spectral density. Theory Probab. Appl. 29 846 847. Z.
  • INGSTER, YU. I. 1993. Asy mptotically minimax hy pothesis testing for nonparametric alternatives I III. Math. Methods Statist. 2 85 114; 3 171 189; 4 249 268. Z.
  • KERKy ACHARIAN, G. and PICARD D. 1993. Density estimation by kernel and wavelet method, optimality in Besov space. Statist. Probab. Lett. 18 327 336. Z.
  • LEHMANN, E. L. 1959. Testing Statistical Hy pothesis. Wiley, New York.
  • LEPSKI, O. V. 1990. One problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 459 470. Z.
  • LEPSKI, O. V. 1991. Asy mptotic minimax adaptive estimation. 1. Upper bounds. Theory Probab. Appl. 36 645 659. Z.
  • LEPSKI, O. V., MAMMEN, E. and SPOKOINY, V. G. 1997. Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. To appear. Z.
  • LEPSKI, O. V. and SPOKOINY, V. G. 1995a. Optimal pointwise adaptive methods in nonparametric estimation. Unpublished manuscript. Z.
  • LEPSKI, O. V. and SPOKOINY, V. G. 1995b. Minimax nonparametric hy pothesis testing: the case of an inhomogeneous alternative. Bernoulli. To appear. Z.
  • MARRON, J. S. 1988. Automatic smoothing parameter selection: a survey. Empir. Econ. 13 187 208. Z.
  • MEy ER, Y. 1990. Ondelettes. Herrmann, Paris. Z.
  • NEUMANN, M. and SPOKOINY, V. 1995. On the efficiency of wavelet estimators under arbitrary error distributions. Math. Methods Statist. 4 137 166. Z.
  • NUSSBAUM, M. 1996. Asy mptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399 2430. Z.
  • PETROV, V. V. 1975. Sums of Independent Random Variables. Springer, New York. Z.
  • POLJAK, B. T. and TSy BAKOV, A. B. 1990. Asy mptotic optimality of C -test for the orthogonal p series estimation of regression. Theory Probab. Appl. 35 293 306. Z.
  • TRIEBEL, H. 1992. Theory of Function Spaces II. Birkhauser, Basel. ¨