Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection
Citations
4 citations
Cites background or methods from "Penalized Composite Quasi-Likelihoo..."
...…on minimising the asymptotic variance of the estimators of only the active set of coefficients, denoted by wMA,2 (Bloznelis et al., 2019) and wC,2 (Bradic et al., 2011) where, with the (k1, k2)th component of A equal to Ak1,k2 = min(τk1 , τk2){1 − max(τk1 , τk2)}, Aε = diag(fε(uτ1), . . . ,…...
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..., 2019) and wC,2 (Bradic et al., 2011) where, with the (k1, k2)th component of A equal to Ak1,k2 = min(τk1 , τk2){1 − max(τk1 , τk2)}, Aε = diag(fε(uτ1), ....
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...This is where our approach differs from Bloznelis et al. (2019) or Bradic et al. (2011), where an irrepresentable condition (needed for consistent model or asymptotically perfect selection) has been used to specify weights and analyze robustness....
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...3 end 4 return wC,1, β̂(αopt, wC,1), and ÂMSE(β̂C(αopt;wcand);β) A possible initial weight vector wC,init is the vector of equal weights or the weight proposed in Bradic et al. (2011); β̂C is estimated by Algorithm 2, and AMSE(β̂C;β) is estimated by (29)....
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...…of the sparse coefficient vector β without assuming that the nonzero entries are selected perfectly; whereas another type of weight choice derived in Bradic et al. (2011); Bloznelis et al. (2019) aims at the lower bound of the variance of the nonzero part of β by imposing the perfect selection…...
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4 citations
Cites background or methods from "Penalized Composite Quasi-Likelihoo..."
...Generalizing the method of CQR, the weighted composite quantile regression (WCQR) was suggested by Bradic et al. (2011) for linear models and Jiang et al. (2012, 2014) for nonlinear models. In terms of nonparametric models, Sun et al. (2013) has shown that the WCQR inherits good properties of CQR to symmetric errors and is applicable to asymmetric errors. However, their works mainly focus on the estimation of nonparametric itself rather than its derivatives. The performance of nonparametric estimation relies on the values chosen for smoothing parameter or bandwidth. For recovering mean response function, much attention has been given to smoothing parameter selection, see Li and Racine (2007) and the references therein. Comparatively, few works have addressed smoothing parameter selection for derivative estimation. In the earlier literature, the studies of choosing smoothing parameter were mainly carried out under the setting of the LS. Rice (1986) introduced a nearly unbiased estimator of the integrated mean square error to select the optimal smoothing parameter for nonparametric derivative estimation. By kernel smoothing, M€ uller et al. (1987) proposed a generalized version of the crossvalidation (CV) technique to estimate the first-order derivative. Fan and Gijbels (1995) developed an integrated residual squares criterion for estimating derivatives employing the local polynomial fitting. Richard et al. (2011) suggested a generalized Cp bandwidth selection criterion to estimate derivative. Through the local LS regression, Henderson et al. (2015) put forward a minimum CV approach for choosing bandwidth of gradient estimation. Whereafter, this work is further promoted by Lin et al. (2015) to discuss the selection of bandwidth for quantile derivative estimation....
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...Generalizing the method of CQR, the weighted composite quantile regression (WCQR) was suggested by Bradic et al. (2011) for linear models and Jiang et al. (2012, 2014) for nonlinear models. In terms of nonparametric models, Sun et al. (2013) has shown that the WCQR inherits good properties of CQR to symmetric errors and is applicable to asymmetric errors. However, their works mainly focus on the estimation of nonparametric itself rather than its derivatives. The performance of nonparametric estimation relies on the values chosen for smoothing parameter or bandwidth. For recovering mean response function, much attention has been given to smoothing parameter selection, see Li and Racine (2007) and the references therein....
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...Generalizing the method of CQR, the weighted composite quantile regression (WCQR) was suggested by Bradic et al. (2011) for linear models and Jiang et al. (2012, 2014) for nonlinear models....
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...Generalizing the method of CQR, the weighted composite quantile regression (WCQR) was suggested by Bradic et al. (2011) for linear models and Jiang et al. (2012, 2014) for nonlinear models. In terms of nonparametric models, Sun et al. (2013) has shown that the WCQR inherits good properties of CQR to symmetric errors and is applicable to asymmetric errors. However, their works mainly focus on the estimation of nonparametric itself rather than its derivatives. The performance of nonparametric estimation relies on the values chosen for smoothing parameter or bandwidth. For recovering mean response function, much attention has been given to smoothing parameter selection, see Li and Racine (2007) and the references therein. Comparatively, few works have addressed smoothing parameter selection for derivative estimation. In the earlier literature, the studies of choosing smoothing parameter were mainly carried out under the setting of the LS. Rice (1986) introduced a nearly unbiased estimator of the integrated mean square error to select the optimal smoothing parameter for nonparametric derivative estimation....
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...Generalizing the method of CQR, the weighted composite quantile regression (WCQR) was suggested by Bradic et al. (2011) for linear models and Jiang et al. (2012, 2014) for nonlinear models. In terms of nonparametric models, Sun et al. (2013) has shown that the WCQR inherits good properties of CQR to symmetric errors and is applicable to asymmetric errors. However, their works mainly focus on the estimation of nonparametric itself rather than its derivatives. The performance of nonparametric estimation relies on the values chosen for smoothing parameter or bandwidth. For recovering mean response function, much attention has been given to smoothing parameter selection, see Li and Racine (2007) and the references therein. Comparatively, few works have addressed smoothing parameter selection for derivative estimation. In the earlier literature, the studies of choosing smoothing parameter were mainly carried out under the setting of the LS. Rice (1986) introduced a nearly unbiased estimator of the integrated mean square error to select the optimal smoothing parameter for nonparametric derivative estimation. By kernel smoothing, M€ uller et al. (1987) proposed a generalized version of the crossvalidation (CV) technique to estimate the first-order derivative. Fan and Gijbels (1995) developed an integrated residual squares criterion for estimating derivatives employing the local polynomial fitting. Richard et al. (2011) suggested a generalized Cp bandwidth selection criterion to estimate derivative. Through the local LS regression, Henderson et al. (2015) put forward a minimum CV approach for choosing bandwidth of gradient estimation....
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4 citations
4 citations
Cites methods from "Penalized Composite Quasi-Likelihoo..."
...Bradic et al. (2011) addressed robustness and efficiency of penalized model selection method and proposed a data-driven weighted linear combination of convex loss functions, along with weighted L1 penalty....
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4 citations
Cites methods from "Penalized Composite Quasi-Likelihoo..."
...Koenker (1984) discussed the optimal weight in composite quantile regression, and Bradic, Fan, and Wang (2011) proposed a data-driven method to estimate the optimal weights....
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References
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"Penalized Composite Quasi-Likelihoo..." refers background or methods in this paper
...…(16) can be recast as a penalized weighted least square regression argmin β n∑ i=1 w1∣∣∣Yi −XTi β̂ (0) ∣∣∣ + w2 ( Yi −XTi β )2 + n p∑ j=1 γλ(|β(0)j |)|βj | which can be efficiently solved by pathwise coordinate optimization (Friedman et al., 2008) or least angle regression (Efron et al., 2004)....
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...) are all nonnegative. This class of problems can be solved with fast and efficient computational algorithms such as pathwise coordinate optimization (Friedman et al., 2008) and least angle regression (Efron et al., 2004). One particular example is the combination of L 1 and L 2 regressions, in which K= 2, ρ 1(t) = |t−b 0|andρ 2(t) = t2. Here b 0 denotes themedian of error distributionε. Iftheerror distribution is sym...
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...i=1 w 1 Yi −XT i βˆ (0) +w 2 Yi −XT i β 2 +n Xp j=1 γλ(|β (0) j |)|βj| which can be efficiently solved by pathwise coordinate optimization (Friedman et al., 2008) or least angle regression (Efron et al., 2004). If b 0 6= 0, the penalized least-squares problem ( 16) is somewhat different from (5) since we have an additional parameter b 0. Using the same arguments, and treating b 0 as an additional parameter ...
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...This class of problems can be solved with fast and efficient computational algorithms such as pathwise coordinate optimization (Friedman et al., 2008) and least angle regression (Efron et al., 2004)....
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