Abstract
We discuss and study minimax nonparametric goodness-of-fit testing problems under Gaussian models in the sequence space and in the functional space. The unknown signal is assumed to vanish under the null-hypothesis. We consider alternatives under two-side constraints determined by Besov norms. We present the description of the types of sharp asymptotics under the sequence space model and of the rate asymptotics under the functional model. The structures of asymptotically minimax and minimax consistent test procedures are given. These results extend recent results of the paper [12]. The results for an adaptive setting are presented as well.
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References
Burnashev, M.V. (1979). On the minimax detection of an inaccurately known signal in a Gaussian noise background. Theor. Probab. Appl. 24, 107–119
Donoho, D.L. (1993). Asymptotic minimax risk for sup-norm loss: solution via optimal recovery. Probab. Theory Rel. 99, 145–170
Donoho, D.L., Johnstone, I.M. (1992). Minimax estimation via wavelet shrinkage. Technical Report 402, Dep. of Statistics, Stanford University.
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D. (1995). Wavelet shrinkage: asymptopia? (with discussion). J. Roy. Stat. Soc. B 57, 301–369
Härdie, W., Kerkyacharian, G., Picard, D., Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. LNS,129 Springer, New York
Ibragimov, I.A., Khasminskii, R.Z. (1977). One problem of statistical estimation in a white Gaussian noise. Sov. Math. Dokl. 236, 333–337
Ibragimov, I.A., Khasminskii, R.Z. (1980). Asymptotic properties of some non-parametric estimates in a Gaussian white noise. In: Proc. 3rd Summer School on Probab. Theory and Math. Stat. Varna 1978, Sofia. 31–64
Ibragimov, I.A., Khasminskii, R.Z. (1980). On estimation of a probability density. Zapiski Nauchn. Seminar. LOMI. 98, 66–85
Ibragimov, I.A., Khasminskii, R.Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, Berlin-New York
Ingster, Yu.I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III. Math. Method. Stat. 2, 85–114, 171–189, 249–268
Ingster, Yu.I. (1998). Adaptation in minimax non-parametric hypothesis testing. WIAS, Preprint No. 419. Berlin.
Ingster, Yu.I., Suslina, I.A. (2000). Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies. ESAIM: Probab. Stat. 4, 53–135
Juditsky, A. (1997). Wavelet estimators: adapting to unknown smoothness. Math. Method. Stat. 6, 1–25
Korostelev, A.P. (1993). Asymptotically minimax regression estimator in uniform norm up to exact constant. Theor. Probab. Appl. 38, 737–743
Lepski, O.V., Mammen, E., Spokoiny, V.G. (1997). Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. Ann. Stat. 25, 929–947
Lepski, O.V., Spokoiny, V.G. (1997). Optimal pointwize adaptive methods in nonparametric estimation. Ann. Stat. 25, 2512–2546
Lepski, O.V., Spokoiny, V.G. (1999). Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative. Bernoulli 5, 333–358
Lepski, O.V., Tsybakov, A.B. (2000). Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Rel. 117, 17–48
Pinsker, M.S. (1980). Optimal filtration of square-integrable signals in Gaussian noise. Probi. Inf. Transm. 16, 120–133
Spokoiny, V.G. (1996). Adaptive hypothesis testing using wavelets. Ann. Stat. 24,2477–2498
Spokoiny, V.G. (1998). Adaptive and spatially adaptive testing of nonparametric hypothesis. Math. Method. Stat. 7, 245–273
Triebel H. (1992). Theory of Functional Space 2. Birkhäuser, Basel.
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Ingster, Y.I., Suslina, I.A. (2003). Minimax Nonparametric Goodness-of-Fit Testing. In: Haitovsky, Y., Ritov, Y., Lerche, H.R. (eds) Foundations of Statistical Inference. Contributions to Statistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57410-8_13
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DOI: https://doi.org/10.1007/978-3-642-57410-8_13
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-7908-0047-0
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