Abstract
Finding the appropriate threshold is one of the most important issues in the wavelet shrinkage method. Especially when the goal is to estimate the mean matrix parameter for matrix-variate elliptically contoured distributions. In this paper, we introduce a new soft thresholding wavelet shrinkage estimator based on Stein’s unbiased risk estimate (SURE) for the class of matrix-variate elliptically contoured distributions. We want to find a class of soft wavelet shrinkage SURE estimators under the balanced loss function by the wavelet shrinkage method. For this purpose, we find a class of the soft thresholding wavelet shrinkage estimators under the balanced loss function by the wavelet shrinkage method. Also, we obtain the restricted soft thresholding wavelet shrinkage estimator based on non-negative sub matrix of the mean matrix parameter. Finally, the performance of the proposed estimator was investigated using a simulation study.
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References
Abramovich F, Sapatinas T, Silverman BW (1998) Wavelet thresholding via a Bayesian approach. J R Stat Soc Ser B (Stat Methodol) 60(4):725–749
Anderson TW (2003) An Introduction to Multivariate Statistical Analysis, 3rd edn. Wiley, New York
Anderson TW, Fang KT (1982) On the theory of multivariate elliptically contoured distributions and their applications. STANFORD UNIV CA DEPT OF STATISTICS 1982
Benjamini Y, Hochberg Y (1995) Controlling the false discovery rate: a practical and powerful approach to multiple testing. J R Stat Soc Ser B (Methodol) 57(1):289–300
Cai TT, Silverman BW (2001) Incorporating information on neighbouring coefficients into wavelet estimation. Sankhya Ser B 63(2):127–148
Dey DK, Ghosh M, Strawderman WE (1999) On estimation with balanced loss functions. Stat Probab Lett 45(2):97–101
Donoho DL, Johnstone IM (1995) Adapting to unknown smoothness via wavelet shrinkage. J Am Stat Assoc 90(432):1200–1224
Fang KT, Chen H (1984) Relationships among classes of spherical matrix distributions. Acta Math Appl Sin 1(2):138–148
Federer H (2014) Geometric measure theory. Springer, New York
Fourdrinier D, Strawderman WE, Wells MT (2003) Robust shrinkage estimation for elliptically symmetric distributions with unknown covariance matrix. J Multivar Anal 85(1):24–39
Gupta AK, Varga T (1994) A new class of matrix variate elliptically contoured distributions. J Ital Stat Soc 3(2):255–270
Gupta AK, Nagar DK (2018) Matrix variate distributions. Chapman and Hall/CRC, London
Karamikabir H, Afshari M (2019) Wavelet shrinkage generalized Bayes estimation for elliptical distribution parameter’s under LINEX loss. Int J Wavelets Multiresolut Inf Process 17(03):1950009
Karamikabir H, Afshari M (2020) Generalized Bayesian shrinkage and wavelet estimation of location parameter for spherical distribution under balance-type loss: Minimaxity and admissibility. J Multivar Anal 177:104583
Karamikabir H, Afshari M, Lak F (2021) Wavelet threshold based on Stein’s unbiased risk estimators of restricted location parameter in multivariate normal. J Appl Stat 48(10):1712–1729
Karamikabir H, Afshari M (2021) New wavelet SURE thresholds of elliptical distributions under the balance loss. Stat Sin 31(4):1829–1852
Karamikabir H, Asghari AN, Salimi A (2022) Soft thresholding wavelet shrinkage estimation for mean matrix of matrix-variate normal distribution: low and high dimensional. Soft Comput 27(18):1–16
Leung BPK, Spiring FA (2004) Some properties of the family of inverted probability loss functions. Qual Technol Quant Manag 1(1):125–147
Lorenz MO (1905) Methods of measuring the concentration of wealth. Publ Am Stat Assoc 9(70):209–219
Mallat SG (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11(7):674–693
Nadarajah S, Zinodiny S, Rezaei S (2016) Minimax estimation of the mean matrix of the matrix-variate normal distribution. Probab Math Stat 36(2):187–200
Nason GP (1996) Wavelet shrinkage using cross-validation. J R Stat Soc Ser B (Methodol) 58(2):463–479
Nason GP (2008) Wavelet methods in statistics with R
Ouimet F (2022) A symmetric matrix-variate normal local approximation for the Wishart distribution and some applications. J Multivar Anal 189:104923
Stein CM (1956) Inadmissibility of the usual estimator for the mean of a multivariate distribution, vol 1. University California Press, California
Stein CM (1981) Estimation of the mean of a multivariate normal distribution. Ann Stat 9(6):1135–1151
Sutradhar BC, Ali MM (1989) A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate t model. J Multivar Anal 29(1):155–162
Yuasa R, Kubokawa T (2020) Ridge-type linear shrinkage estimation of the mean matrix of a high-dimensional normal distribution. J Multivar Anal 178:104608
Zellner A (1994) Bayesian and non-Bayesian estimation using balanced loss functions, Statistical decision theory and related topics V. Springer, New York, pp 377–390
Zinodiny S, Rezaei S, Nadarajah S (2017) Bayes minimax estimation of the mean matrix of matrix-variate normal distribution under balanced loss function. Stat Probab Lett 125:110–120
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The authors would like to thank the research committee of Persian Gulf university.
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Karamikabir, H. Wavelet Shrinkage Estimation for Mean Matrix of Matrix-Variate Elliptically Contoured Distributions. J Stat Theory Pract 18, 23 (2024). https://doi.org/10.1007/s42519-024-00375-6
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DOI: https://doi.org/10.1007/s42519-024-00375-6