### Citations

465 |
Bayes and Empirical Bayes Methods for Data Analysis,
- Carlin, Louis
- 1996
(Show Context)
Citation Context ...θÞ were known, then the Bayes estimator of θnþ1 which delivers the minimal mean squared risk would be given by the following equation: tðyÞ R∞ −∞ θqðyjθÞgðθÞ dθR∞ −∞ qðyjθÞgðθÞ dθ ð1:1Þ (see, e.g., =-=Carlin and Louis, 2000-=-, or Maritz and Lwin, 1989). Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jspi Journal of Statistical Planning and Inference 0378-3758/$ - see front mat... |

226 |
Introduction to Nonparametric Estimation.
- Tsybakov
- 2009
(Show Context)
Citation Context ... lower bound for the minimax risk when g∈Gr . In order to construct lower bound ΔnðyÞ for (3.1), we use Theorem 2.7 of Tsybakov (2008) which we reformulate here for the case of squared risk. Lemma 7 (=-=Tsybakov, 2008-=-, Theorem 2.7). Assume that Ξ contains elements ξ0; ξ1;…; ξn0 , n0≥1, such that (i) dðξi; ξi0 Þ≥2χ40, for 0≤io i0≤n0; (ii) Pi5P0, for i 1;…;n0, and KðPi; P0Þ≤Cn0 ; where Pi Pξi , i 0;1;…;n0; and C... |

204 |
Parametric empirical Bayes inference, theory and applications.
- Morris
- 1983
(Show Context)
Citation Context ... prior distribution belongs, is specified a priori. The past data is then used to estimate the values of the unknown parameters, usually using the maximum likelihood approach (see, e.g., Louis, 1984; =-=Morris, 1983-=-; Casella, 1985; Efron and Morris, 1977 among others). In nonparametric EB estimation, prior distribution is completely unspecified. One of the approaches to nonparametric EB estimation is based on es... |

136 | An empirical Bayes approach to statistics - Robbins - 1956 |

115 | Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. The Annals of Statistics, - Lepski, Mammen, et al. - 1997 |

74 | The empirical Bayes approach to statistical decision problems - Robbins - 1964 |

59 |
Stein’s paradox in statistics
- Efron, Morris
- 1977
(Show Context)
Citation Context ...is specified a priori. The past data is then used to estimate the values of the unknown parameters, usually using the maximum likelihood approach (see, e.g., Louis, 1984; Morris, 1983; Casella, 1985; =-=Efron and Morris, 1977-=- among others). In nonparametric EB estimation, prior distribution is completely unspecified. One of the approaches to nonparametric EB estimation is based on estimation of the numerator and the denom... |

58 |
An introduction to empirical Bayes data analysis.
- Casella
- 1985
(Show Context)
Citation Context ...ution belongs, is specified a priori. The past data is then used to estimate the values of the unknown parameters, usually using the maximum likelihood approach (see, e.g., Louis, 1984; Morris, 1983; =-=Casella, 1985-=-; Efron and Morris, 1977 among others). In nonparametric EB estimation, prior distribution is completely unspecified. One of the approaches to nonparametric EB estimation is based on estimation of the... |

45 |
Estimating a Population of Parameter Values Using Bayes and Empirical Bayes Methods
- Louis
- 1984
(Show Context)
Citation Context ... to which the prior distribution belongs, is specified a priori. The past data is then used to estimate the values of the unknown parameters, usually using the maximum likelihood approach (see, e.g., =-=Louis, 1984-=-; Morris, 1983; Casella, 1985; Efron and Morris, 1977 among others). In nonparametric EB estimation, prior distribution is completely unspecified. One of the approaches to nonparametric EB estimation ... |

27 | Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means
- Brown, Greenshtein
- 2009
(Show Context)
Citation Context ... is based on estimation of the numerator and the denominator in the ratio in (1.4). This approach was introduced by Robbins (1955, 1964) himself and later developed by a number of authors (see, e.g., =-=Brown and Greenshtein, 2009-=-; Datta, 1991, 2000; Ma and Balakrishnan, 2000; Nogami, 1988; Pensky, 1997a,b; Raykar and Zhao, 2011; Singh, 1976, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with go... |

19 |
Robust empirical Bayes estimation of means from stratified samples
- Ghosh, Lahiri
- 1987
(Show Context)
Citation Context ... X1;…;Xn. The technique is extremely efficient computationally and was immediately put to practical use, for instance, for prediction of the finite population mean (see, e.g., Ghosh and Meeden, 1986; =-=Ghosh and Lahiri, 1987-=-; Karunamuni and Zhang, 2003). However, a linear EB estimator has a large bias since, due to its very simple form, it has a limited ability to approximate the Bayes estimator t(y). For this reason, li... |

17 | Some thoughts on empirical Bayes estimation - Robbins - 1983 |

16 |
Empirical bayes estimation in finite population sampling.
- Ghosh, Meeden
- 1986
(Show Context)
Citation Context ...he basis of observations X1;…;Xn. The technique is extremely efficient computationally and was immediately put to practical use, for instance, for prediction of the finite population mean (see, e.g., =-=Ghosh and Meeden, 1986-=-; Ghosh and Lahiri, 1987; Karunamuni and Zhang, 2003). However, a linear EB estimator has a large bias since, due to its very simple form, it has a limited ability to approximate the Bayes estimator t... |

8 | Empirical Bayes estimation in Lebesgue-exponential family with rates near the best possible rate - Singh - 1979 |

8 |
Wavelets and other orthogonal systems
- Walter, Shen
- 2001
(Show Context)
Citation Context ... choose a twice continuously differentiable scaling function φ with bounded support and s vanishing moments, so that supp φ∈M1;M2; ð2:1ÞZ ∞ −∞ xℵ∑ k∈Z φðx−kÞφðz−kÞ dx zℵ; 0≤ℵ≤s−1 ð2:2Þ (see, e.g., =-=Walter and Shen, 2001-=-). Approximate t(y) by a wavelet series, for some fixed m≥0, tmðyÞ ∑ k∈Z am;kφm;kðyÞ ð2:3Þ where φm;kðyÞ 2m=2φð2my−kÞ, and estimate coefficients of tmðyÞ by minimizing the integrated mean squared ... |

6 |
Bayes empirical Bayes estimation for natural exponential families with quadratic variance function
- Walter, Hamedani
- 1991
(Show Context)
Citation Context ... later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; Datta, 1991, 2000; Ma and Balakrishnan, 2000; Nogami, 1988; Pensky, 1997a,b; Raykar and Zhao, 2011; Singh, 1976, 1979; =-=Walter and Hamedani, 1991-=- among others). The method provides estimators with good convergence rates, however, it requires relatively tedious three-step procedure: estimation of the top and the bottom of the fraction and then ... |

5 |
Empirical Bayes estimation with convergence rates in non-continuous Lebesgue-exponential families
- Singh
- 1976
(Show Context)
Citation Context ..., 1964) himself and later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; Datta, 1991, 2000; Ma and Balakrishnan, 2000; Nogami, 1988; Pensky, 1997a,b; Raykar and Zhao, 2011; =-=Singh, 1976-=-, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with good convergence rates, however, it requires relatively tedious three-step procedure: estimation of the top and the... |

4 |
Convergence rates for empirical Bayes estimation in the uniform 17(0,0) distribution
- Nogami
- 1988
(Show Context)
Citation Context ...n (1.4). This approach was introduced by Robbins (1955, 1964) himself and later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; Datta, 1991, 2000; Ma and Balakrishnan, 2000; =-=Nogami, 1988-=-; Pensky, 1997a,b; Raykar and Zhao, 2011; Singh, 1976, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with good convergence rates, however, it requires relatively tediou... |

4 |
A general approach to nonparametric empirical Bayes estimation
- Pensky
- 1997
(Show Context)
Citation Context ...approach was introduced by Robbins (1955, 1964) himself and later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; Datta, 1991, 2000; Ma and Balakrishnan, 2000; Nogami, 1988; =-=Pensky, 1997-=-a,b; Raykar and Zhao, 2011; Singh, 1976, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with good convergence rates, however, it requires relatively tedious three-step p... |

4 |
Empirical Bayes estimation of a location parameter
- Pensky
- 1997
(Show Context)
Citation Context ...approach was introduced by Robbins (1955, 1964) himself and later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; Datta, 1991, 2000; Ma and Balakrishnan, 2000; Nogami, 1988; =-=Pensky, 1997-=-a,b; Raykar and Zhao, 2011; Singh, 1976, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with good convergence rates, however, it requires relatively tedious three-step p... |

3 | Asymptotic mimimax adaptive estimation. I: Upper bounds. Optimally adaptive estimates. Theory Probab - Lepski - 1991 |

3 | On an empirical Bayes test for a normal mean. - Liang - 2000 |

3 |
Empirical Bayes estimation for truncation parameters
- Ma, Balakrishnan
- 2000
(Show Context)
Citation Context ... denominator in the ratio in (1.4). This approach was introduced by Robbins (1955, 1964) himself and later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; Datta, 1991, 2000; =-=Ma and Balakrishnan, 2000-=-; Nogami, 1988; Pensky, 1997a,b; Raykar and Zhao, 2011; Singh, 1976, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with good convergence rates, however, it requires rel... |

3 | Empirical Bayesian thresholding for sparse signals using mixture loss functions Statistica Sinica
- Raykar, Zhao
- 2011
(Show Context)
Citation Context ...oduced by Robbins (1955, 1964) himself and later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; Datta, 1991, 2000; Ma and Balakrishnan, 2000; Nogami, 1988; Pensky, 1997a,b; =-=Raykar and Zhao, 2011-=-; Singh, 1976, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with good convergence rates, however, it requires relatively tedious three-step procedure: estimation of th... |

2 | Empirical Bayes estimation with non-identical components - Datta - 2000 |

2 |
Wavelet based empirical Bayes estimation for the uniform distribution
- Huang
- 1997
(Show Context)
Citation Context ...be expressed as the derivative of the denominator. Wavelets provide an opportunity to construct adaptive wavelet-based EB estimators with better computational properties in this framework (see, e.g., =-=Huang, 1997-=-; Pensky, 1998, 2000, 2002) but the necessity of estimation of the ratio in (1.4) remains. Another nonparametric approach developed in Jiang and Zhang (2009) is based on application of nonparametric M... |

2 | Optimal rates of convergence of empirical Bayes tests for the continuous one-parameter exponential family - Karunamuni - 1996 |

2 |
Optimal linear Bayes and empirical Bayes estimation and prediction of the finite population
- Karunamuni, Zhang
- 2003
(Show Context)
Citation Context ...is extremely efficient computationally and was immediately put to practical use, for instance, for prediction of the finite population mean (see, e.g., Ghosh and Meeden, 1986; Ghosh and Lahiri, 1987; =-=Karunamuni and Zhang, 2003-=-). However, a linear EB estimator has a large bias since, due to its very simple form, it has a limited ability to approximate the Bayes estimator t(y). For this reason, linear EB estimators are optim... |

2 | F.(2005). Convergence rates of empirical Bayes estimation in exponential family - Li, Gupta, et al. |

2 | On the lower bounds for mean square error of empirical Bayes estimators - Penskaya - 1995 |

2 | Adaptive wavelet empirical Bayes estimation of a location or a scale parameter - Pensky - 2000 |

2 | Locally adaptive wavelet Bayes estimation of location parameter - Pensky - 2002 |

2 | Rates of convergence of empirical Bayes tests for a normal mean - Pensky - 2003 |

2 |
Generalization of linear empirical Bayes estimation via wavelet series
- Pensky, Alotaibi
- 2005
(Show Context)
Citation Context ...addou, M. Pensky / Journal of Statistical Planning and Inference 143 (2013) 1672–16881674 Author's personal copy to zero, we need to keep more terms in the system of equations (2.5) (see Lemma A.3 in =-=Pensky and Alotaibi, 2005-=- for more details). The entries (2.6) of the matrix B are unknown and can be estimated by sample means bBj;k n−1 ∑n l 1 φm;kðXlÞφm;jðXlÞ: ð2:9Þ In order to estimate cj, one needs to find functio... |

2 | Extended linear empirical Bayes estimation - Pensky, Ni - 2000 |

1 |
Nonparametric empirical Bayes estimation with Oðn−1=2Þ rate of a truncation parameter
- Datta
- 1991
(Show Context)
Citation Context ...e numerator and the denominator in the ratio in (1.4). This approach was introduced by Robbins (1955, 1964) himself and later developed by a number of authors (see, e.g., Brown and Greenshtein, 2009; =-=Datta, 1991-=-, 2000; Ma and Balakrishnan, 2000; Nogami, 1988; Pensky, 1997a,b; Raykar and Zhao, 2011; Singh, 1976, 1979; Walter and Hamedani, 1991 among others). The method provides estimators with good convergenc... |

1 | Journal of Statistical Planning and Inference 143 (2013) 1672–1688 1687 Author's personal copy Jiang - Benhaddou, Pensky - 2009 |

1 |
Empirical Bayes estimation based on wavelets. Sankhyā A60
- Pensky
- 1998
(Show Context)
Citation Context ...as the derivative of the denominator. Wavelets provide an opportunity to construct adaptive wavelet-based EB estimators with better computational properties in this framework (see, e.g., Huang, 1997; =-=Pensky, 1998-=-, 2000, 2002) but the necessity of estimation of the ratio in (1.4) remains. Another nonparametric approach developed in Jiang and Zhang (2009) is based on application of nonparametric MLE technique w... |