# zbMATH — the first resource for mathematics

Gaussianization machines for non-Gaussian function estimation models. (English) Zbl 1440.62115
Summary: A wide range of nonparametric function estimation models have been studied individually in the literature. Among them the homoscedastic nonparametric Gaussian regression is arguably the best known and understood. Inspired by the asymptotic equivalence theory, L. D. Brown et al. [Ann. Stat. 36, No. 5, 2055–2084 (2008; Zbl 1148.62019); ibid. 38, No. 4, 2005–2046 (2010; Zbl 1202.62050); Probab. Theory Relat. Fields 146, No. 3–4, 401–433 (2010; Zbl 1180.62055)] developed a unified approach to turn a collection of non-Gaussian function estimation models into a standard Gaussian regression and any good Gaussian nonparametric regression method can then be used. These Gaussianization Machines have two key components, binning and transformation. When combined with BlockJS, a wavelet thresholding procedure for Gaussian regression, the procedures are computationally efficient with strong theoretical guarantees. Technical analysis given in [loc. cit.] shows that the estimators attain the optimal rate of convergence adaptively over a large set of Besov spaces and across a collection of non-Gaussian function estimation models, including robust nonparametric regression, density estimation, and nonparametric regression in exponential families. The estimators are also spatially adaptive. The Gaussianization Machines significantly extend the flexibility and scope of the theories and methodologies originally developed for the conventional nonparametric Gaussian regression. This article aims to provide a concise account of the Gaussianization Machines developed in [loc. cit.].
##### MSC:
 62G07 Density estimation 62G08 Nonparametric regression and quantile regression 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text:
##### References:
 [1] Anscombe, F. J. (1948). The transformation of Poisson, binomial and negative-binomial data. Biometrika 35 246-254. · Zbl 0032.03702 [2] Bartlett, M. S. (1936). The square root transformation in analysis of variance. Suppl. J. R. Stat. Soc. 3 68-78. · JFM 63.1085.01 [3] Berk, R. and MacDonald, J. M. (2008). Overdispersion and Poisson regression. J. Quant. Criminol. 24 269-284. [4] Besbeas, P., De Feis, I. and Sapatinas, T. (2004). A comparative simulation study of wavelet shrinkage estimators for Poisson counts. Int. Stat. Rev. 72 209-237. · Zbl 1211.62055 [5] Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Institute of Mathematical Statistics Lecture Notes—Monograph Series 9. IMS, Hayward, CA. · Zbl 0685.62002 [6] Brown, L. D., Cai, T. T. and Zhou, H. H. (2008). Robust nonparametric estimation via wavelet median regression. Ann. Statist. 36 2055-2084. · Zbl 1148.62019 [7] Brown, L. D., Cai, T. T. and Zhou, H. H. (2010). Nonparametric regression in exponential families. Ann. Statist. 38 2005-2046. · Zbl 1202.62050 [8] Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398. · Zbl 0867.62022 [9] Brown, L. D., Wang, Y. and Zhao, L. H. (2003). On the statistical equivalence at suitable frequencies of GARCH and stochastic volatility models with the corresponding diffusion model. Statist. Sinica 13 993-1013. · Zbl 1034.62105 [10] Brown, L. D. and Zhang, C.-H. (1998). Asymptotic nonequivalence of nonparametric experiments when the smoothness index is $$1/2$$. Ann. Statist. 26 279-287. · Zbl 0932.62061 [11] Brown, L. D., Cai, T. T., Low, M. G. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688-707. · Zbl 1029.62044 [12] Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074-2097. · Zbl 1062.62083 [13] Brown, L., Cai, T. T., Zhang, R., Zhao, L. and Zhou, H. (2010). The root-unroot algorithm for density estimation as implemented via wavelet block thresholding. Probab. Theory Related Fields 146 401-433. · Zbl 1180.62055 [14] Cai, T. T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach. Ann. Statist. 27 898-924. · Zbl 0954.62047 [15] Cai, T. T. and Zhou, H. H. (2009). Asymptotic equivalence and adaptive estimation for robust nonparametric regression. Ann. Statist. 37 3204-3235. · Zbl 1191.62070 [16] Cai, T. T. and Zhou, H. H. (2010). Nonparametric regression in natural exponential families. In Borrowing Strength: Theory Powering Applications—a Festschrift for Lawrence D. Brown. Inst. Math. Stat. (IMS) Collect. 6 199-215. IMS, Beachwood, OH. [17] Dalalyan, A. and Reiß, M. (2006). Asymptotic statistical equivalence for scalar ergodic diffusions. Probab. Theory Related Fields 134 248-282. · Zbl 1081.62002 [18] Dalalyan, A. and Reiß, M. (2007). Asymptotic statistical equivalence for ergodic diffusions: The multidimensional case. Probab. Theory Related Fields 137 25-47. · Zbl 1105.62004 [19] Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. SIAM, Philadelphia, PA. · Zbl 0776.42018 [20] Delattre, S. and Hoffmann, M. (2002). Asymptotic equivalence for a null recurrent diffusion. Bernoulli 8 139-174. · Zbl 1040.60067 [21] Donoho, D. L. (1993). Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data. In Different Perspectives on Wavelets (San Antonio, TX, 1993). Proc. Sympos. Appl. Math. 47 173-205. Amer. Math. Soc., Providence, RI. · Zbl 0786.62094 [22] Efromovich, S. and Samarov, A. (1996). Asymptotic equivalence of nonparametric regression and white noise model has its limits. Statist. Probab. Lett. 28 143-145. · Zbl 0849.62023 [23] Fryzlewicz, P. and Nason, G. P. (2004). A Haar-Fisz algorithm for Poisson intensity estimation. J. Comput. Graph. Statist. 13 621-638. [24] Genon-Catalot, V., Laredo, C. and Nussbaum, M. (2002). Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift. Ann. Statist. 30 731-753. · Zbl 1029.62071 [25] Golubev, G. K., Nussbaum, M. and Zhou, H. H. (2010). Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Statist. 38 181-214. · Zbl 1181.62152 [26] Grama, I. G. and Neumann, M. H. (2006). Asymptotic equivalence of nonparametric autoregression and nonparametric regression. Ann. Statist. 34 1701-1732. · Zbl 1246.62105 [27] Grama, I. and Nussbaum, M. (2002). Asymptotic equivalence for nonparametric regression. Math. Methods Statist. 11 1-36. · Zbl 1005.62039 [28] Hoyle, M. H. (1973). Transformations-an introduction and a bibliography. Int. Stat. Rev. 41 203-223. · Zbl 0267.62005 [29] Johnstone, I. M. (2011). Gaussian Estimation: Sequence and Wavelet Models. Unpublished manuscript. [30] Kolaczyk, E. D. (1999a). Bayesian multiscale models for Poisson processes. J. Amer. Statist. Assoc. 94 920-933. · Zbl 1072.62630 [31] Kolaczyk, E. D. (1999b). Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds. Statist. Sinica 9 119-135. · Zbl 0927.62081 [32] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $$\text{RV}$$’s and the sample $$\text{DF}$$. I. Z. Wahrsch. Verw. Gebiete 32 111-131. · Zbl 0308.60029 [33] Kroll, M. (2019). Non-parametric Poisson regression from independent and weakly dependent observations by model selection. J. Statist. Plann. Inference 199 249-270. · Zbl 1418.62119 [34] Le Cam, L. (1974). On the information contained in additional observations. Ann. Statist. 2 630-649. · Zbl 0286.62004 [35] Low, M. G. and Zhou, H. H. (2007). A complement to Le Cam’s theorem. Ann. Statist. 35 1146-1165. · Zbl 1194.62007 [36] Mason, D. M. and Zhou, H. H. (2012). Quantile coupling inequalities and their applications. Probab. Surv. 9 439-479. · Zbl 1307.62036 [37] Meister, A. (2011). Asymptotic equivalence of functional linear regression and a white noise inverse problem. Ann. Statist. 39 1471-1495. · Zbl 1221.62011 [38] Milstein, G. and Nussbaum, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535-543. · Zbl 1053.62556 [39] Morris, C. N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10 65-80. · Zbl 0498.62015 [40] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399-2430. · Zbl 0867.62035 [41] Reiß, M. (2008). Asymptotic equivalence for nonparametric regression with multivariate and random design. Ann. Statist. 36 1957-1982. · Zbl 1142.62023 [42] Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 772-802. · Zbl 1215.62113 [43] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability. CRC Press, London. · Zbl 0617.62042 [44] Triebel, H. (1983). Theory of Function Spaces. Monographs in Mathematics 78. Birkhäuser, Basel. · Zbl 0546.46028 [45] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, New York. Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. [46] Ver Hoef, J. M. and Boveng, P. L. (2007). Quasi-Poisson vs. negative binomial regression: How should we model overdispersed count data? Ecology 88 2766-2772. [47] Wang, Y. (2002). Asymptotic nonequivalence of GARCH models and diffusions. Ann. Statist. 30 754-783. · Zbl 1029.62006 [48] Winkelmann, R. · Zbl 1032.62108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.