Abstract
To solve problems of economical representation of large data arrays, multiple hypothesis testing is widely used to identify major features and remove noise. The problem of multiple hypothesis testing is solved by means of FDR, controlled by the Benjamini–Hochberg multiple hypothesis testing algorithm. The observed data are often a modification of the original signal. A case where the original signal is subjected to the action of a linear homogeneous converter is considered. To construct estimates, it is necessary to solve the inversion problem for a linear transformation. A theorem is givenfor estimating the rate of convergence of the risk estimator to the normal distribution.
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REFERENCES
I. M. Johnstone, ‘‘Wavelet shrinkage for correlated data and inverse problems: adaptivity results,’’ Stat. Sin. 9 (1), 51–83 (1999).
S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic Press, New York, 1999). https://doi.org/10.1016/B978-0-12-466606-1.X5000-4
F. Abramovich and B. W. Silverman, ‘‘Wavelet decomposition approaches to statistical inverse problems,’’ Biometrika 85 (1), 115–129 (1998). https://doi.org/10.1093/biomet/85.1.115
O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems, 2nd ed. (Nauka, Moscow, 1996) [in Russian].
F. Abramovich, Y. Benjamini, D. L. Donoho, and I. M. Johnstone, ‘‘Adapting to unknown sparsity by controlling the false discovery rate,’’ Ann. Stat. 34 (2), 584–653 (2006). https://doi.org/10.1214/009053606000000074
M. Jansen, Noise Reduction by Wavelet Thresholding, Lecture Notes in Statistics, Vol. 161 (Springer, New York, 2001). https://doi.org/10.1007/978-1-4613-0145-5
D. L. Donoho and I. M. Johnstone, ‘‘Adapting to unknown smoothness via wavelet shrinkage,’’ J. Am. Stat. Assoc. 90 (432), 1200–1224 (1995). https://doi.org/10.1080/01621459.1995.10476626
J. S. Marron, S. Adak, I. M. Johnstone, M. H. Neumann, and P. Patil, ‘‘Exact risk analysis of wavelet regression,’’ J. Comp. Graph. Stat. 7 (3), 278–309 (1998). https://doi.org/10.1080/10618600.1998.10474777
S. I. Palionnaya and O. V. Shestakov, ‘‘The use of the FDR method of multiple hypothesis testing when inverting linear homogeneous operators,’’ Inform. Primen. 16 (2), 44–51 (2022).
J. Sunklodas, “Approximation of distributions of sums of weakly dependent random variables by the normal distribution,” in Probability Theory – \(6\). Limit Theorems of Probability Theory, Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Naprav. 81, 140–199 (1991);
English transl.: in Limit Theorems of Probability Theory, Ed. by Yu. V. Prokhorov and V. A. Statulevičius (Springer, Berlin, 2000), pp. 113–165. https://doi.org/10.1007/978-3-662-04172-7_3
O. V. Shestakov, ‘‘Approximation to the distribution of a risk estimate in wavelet coefficients thresholding by the normal distribution with using sample variance,’’ Inform. Primen. 4 (4), 72–79 (2010).
Funding
This work was supported by the RF Ministry of Science and Higher Education as part of the Program of the Moscow Center of Fundamental and Applied Mathematics, agreement no. 075-15-2019-1621.
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Translated by I. Tselishcheva
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Palionnaya, S.I. Rate of Convergence of the Risk Estimate Distribution to the Normal Law Using FDR Multiple Hypothesis Testing with Inverting Linear Homogeneous Operators. MoscowUniv.Comput.Math.Cybern. 46, 156–162 (2022). https://doi.org/10.3103/S0278641922030098
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DOI: https://doi.org/10.3103/S0278641922030098