MathSciNet Google Scholar. Princeton: Princeton University Press. Furthermore, by the Markov inequality, for \(b > 0\), where \(\eta (C_{\varvec{\epsilon }})\) is defined in Assumption 6. We first introduce some preleminary results. In this work, several solutions, based on dimension reduction techniques PCA and PLS, are studied for the calculation of these weights in high dimensional frameworks. Francq, C., Zakoan, J. M. (2010). Zou, H. (2006). To do so, we prove the finite dimensional convergence in distribution of the empirical criterion \({{\mathbb {F}}}_T(\varvec{u})\) to \({{\mathbb {F}}}_{\infty }(\varvec{u})\) with \(\varvec{u}\in {{\mathbb {R}}}^d\), where these quantities are, respectively, defined as, with \(\varvec{Z}_{{{\mathcal {A}}}} \sim {{\mathcal {N}}}(0,{{\mathbb {M}}}_{{{\mathcal {A}}}{{\mathcal {A}}}})\). Coxs regression model for counting processes: A large sample study. \end{aligned}$$, $$\begin{aligned} ({\dot{{{\mathbb {G}}}}}_T l({\hat{\theta }}))_{(k),i} + \frac{\lambda _T}{T} \alpha ^{(k)}_{T,i} \text {sgn}({\hat{\theta }}^{(k)}_{T,i}) + \frac{\gamma _T}{T} \xi _{T,k} \frac{{\hat{\theta }}^{(k)}_i}{\Vert {\hat{\varvec{\theta }}}^{(k)}\Vert _2} = 0. It suffers from three challenges in practical applications: noise, gene grouping, and adaptive gene selection. Econometrica 46(1):3350, Laria JC, Aguilera-Morillo MC, Lillo RE (2019) An iterative sparse-group Lasso. Hjort, N. L., Pollard, D. (1993). Variable selection via nonconcave penalized likelihood and its oracle properties. Note that the square martingale difference condition can be relaxed by \(\alpha \)-mixing and moment conditions. \\&\left. and transmitted securely. Hence \(c = 0\). First, using the same reasoning on the third-order term, we obtain \(\frac{1}{6 T^{1/3}} \nabla '\{\varvec{u}' \ddot{{{\mathbb {G}}}}_T l({\bar{\varvec{\theta }}})\}\varvec{u}\overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} 0\). RAMRSGL: A Robust Adaptive Multinomial Regression Model for Multicancer Classification. \partial ^3_{\theta _{k_2} \theta _{l_2} \theta _{m_2}} l(\varvec{\epsilon }_{t'};{\bar{\varvec{\theta }}}) | \\&\quad \le \frac{1}{T^2} \overset{T}{\underset{t,t'=1}{\sum }} \overset{d}{\underset{k_1,l_1,m_1}{\sum }}\overset{d}{\underset{k_2,l_2,m_2}{\sum }} \varvec{u}_{k_1} \varvec{u}_{l_1} \varvec{u}_{m_1} \varvec{u}_{k_2} \varvec{u}_{l_2} \varvec{u}_{m_2} \upsilon _t(C) \upsilon _{t'}(C), \end{aligned}$$, \(\upsilon _t(C) = \underset{k,l,m=1,\ldots ,d}{\sup } \{ \underset{\varvec{\theta }:\Vert \varvec{\theta }-\varvec{\theta }_0\Vert _2 \le \nu _T C}{\sup } |\partial ^3_{\theta _k \theta _l \theta _m} l(\varvec{\epsilon }_t;\varvec{\theta })|\}\), \(\nu _T = T^{-1/2} + \lambda _T T^{-1} a_T + \gamma _T T^{-1} b_T\), \(\nabla '\{\varvec{u}' \ddot{{{\mathbb {G}}}}_T l({\bar{\varvec{\theta }}}) \varvec{u}\} \varvec{u}= O_p(\Vert \varvec{u}\Vert ^3_2 \eta (C))\), $$\begin{aligned} \frac{1}{6 T^{1/3}}\nabla '\{\varvec{u}' \ddot{{{\mathbb {G}}}}_T l({\bar{\varvec{\theta }}}) \varvec{u}\} \varvec{u}\overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} 0. A Sparse Group Lasso. Acute leukemia; Cancer diagnosis; Gene selection; Sparse group lasso. This paper studies the introduction of sparse group LASSO (SGL) to the quantile regression framework. \\&\left. By the ergodic theorem, we deduce \(\ddot{{{\mathbb {G}}}}_T l(\varvec{\theta }_0) \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} {{\mathbb {H}}}\) and by Assumption4, \(\sqrt{T}{\dot{{{\mathbb {G}}}}}_T l(\varvec{\theta }_0) \overset{d}{\longrightarrow } {{\mathcal {N}}}(0,{{\mathbb {M}}})\). \end{aligned}$$, $$\begin{aligned} \lambda _T T^{-1/2}\overset{\varvec{c}_k}{\underset{i=1}{\sum }} \alpha ^{(k)}_{T,i} \sqrt{T}(|\theta ^{(k)}_{0,i} + \varvec{u}^{(k)}_i/\sqrt{T}| - |\theta ^{(k)}_{0,i}|) \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} 0, \end{aligned}$$, \(T^{\eta /2} (|{\tilde{\theta }}^{(k)}_i|)^{\eta } = O_p(1)\), \(\lambda _T T^{(\eta -1)/2} \rightarrow \infty \), $$\begin{aligned}&\lambda _T T^{-1/2} \alpha ^{(k)}_{T,i} \sqrt{T}\left( |\theta ^{(k)}_{0,i} + \varvec{u}^{(k)}_i/\sqrt{T} | - |\theta ^{(k)}_{0,i}|\right) \nonumber \\&\quad = \lambda _T T^{-1/2} |\varvec{u}^{(k)}_i| \frac{T^{\eta /2}}{(T^{1/2}|{\tilde{\theta }}^{(k)}_i|)^{\eta }} \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} \infty . \end{aligned}$$, $$\begin{aligned} \gamma _T T^{-1/2} \sqrt{T} \xi _{T,l} \left( \Vert \varvec{\theta }^{(l)}_0 + \varvec{u}^{(l)}/\sqrt{T}\Vert _2 - \Vert \varvec{\theta }^{(l)}_0\Vert _2 \right) \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} 0. Let \(i \in {{\mathcal {A}}}_k\), then by the asymptotic normality result, \({\hat{\theta }}^{(k)}_i \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} \varvec{\theta }^{(k)}_0\), which implies \({{\mathbb {P}}}(i \in {\hat{{{\mathcal {A}}}}}_k) \rightarrow 1\). developed an adaptive enhanced sparse group LASSO method for the fault diagnosis of rolling bearing [ 32 ]. This is a preview of subscription content, access via your institution. )\) is convex, which implies, Consequently, \(\arg \, \min \{{{\mathbb {G}}}_T \varphi (\varvec{x})\} = O(1)\), such that \({\hat{\varvec{\theta }}} \in {{\mathcal {B}}}_o(\varvec{\theta }_0,C)\) with probability approaching one for C large enough, with \({{\mathcal {B}}}_o(\varvec{\theta }_0,C)\) an open ball centered at \(\varvec{\theta }_0\) and of radius C. Furthermore, as \({{\mathbb {G}}}_{\infty } \varphi (. IEEE/ACM Trans Comput Biol Bioinform. )\) on any compact set \(\varvec{B}\subset \varTheta \), idest, We define \({{\mathcal {C}}}\subset \varTheta \) an open convex set and pick \(\varvec{x}\in {{\mathcal {C}}}\). \\&\left. 2010 pp. Journal of Computational and Graphical Statistics, 22(2), 231245. Our proposed method is memory efficient. By the ergodic theorem of Billingsley (1995), we have, This implies \({{\mathcal {R}}}_T(\varvec{\theta }_0) = o_p(1)\). Let \({\hat{X}}_n\) maximize \(F_n\). Lusso showroom, located at 2219 Oakland Rd, Unit 10, San Jose, CA 95131, has ~12,000 SQ FT displaying Lusso products as well as designs from Toto, Duravit and Hansgrohe, and more. For the \(l^1\) penalty, for any group k, we have for T sufficiently large, under the condition that \(\lambda _T / \sqrt{T} \rightarrow \lambda _0\). Asymptotics for least absolute deviation regression estimators. The proof of this theorem is based on a diagonal argument and Theorem 10.8 of Rockafeller (1970), that is, the pointwise convergence of concave random functions on a dense and countable subset of an open set implies uniform convergence on any compact subset of the open set. \end{aligned}$$, \(\lambda _T / T \rightarrow \lambda _0 \ge 0\), \(\gamma _T / T \rightarrow \gamma _0 \ge 0\), $$\begin{aligned} |{{\mathbb {G}}}_T \varphi (\varvec{x}) - {{\mathbb {G}}}_{\infty }\varphi (\varvec{x})| \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} 0. On the asymptotic properties of the Group Lasso estimator for linear models. Billingsley, P. (1961). M-estimation for autoregressions with infinite variance. Abstract: In this paper, we study sparse group Lasso for high-dimensional double sparse linear regression, where the parameter of interest is simultaneously element-wise and group-wise sparse. (2015). Yuan, M., Lin, Y. We have, where \({{\mathbb {F}}}_T(. Fan, J., Li, R. (2001). Bookshelf Suppose f has a unique maximum at \(x_0 \in E\). )\), which can be expanded as. Then by convexity of \({{\mathbb {G}}}_T \varphi (. \, \varvec{z}^{(k)} {\left\{ \begin{array}{ll} = \frac{\varvec{u}^{(k)}}{\Vert \varvec{u}^{(k)}\Vert _2} \, \text {if} \, \varvec{u}^{(k)} \ne 0,\\ \in \{\varvec{z}^{(k)} : \Vert \varvec{z}^{(k)}\Vert _2 \le 1\} \, \text {if} \, \varvec{u}^{(k)} = 0, \end{array}\right. } Inequalities and limit theorems for weakly dependent sequences. https://doi.org/10.1007/s10463-018-0692-7, DOI: https://doi.org/10.1007/s10463-018-0692-7. Because an additive component corresponds to a vector of coe cients, which can be treated as a group of variables, we employ the group LASSO method to select nonzero vectors of coe . This problem is an important instance of the simultaneously structured model -- an actively studied topic in statistics and machine learning. Adaptive estimators are usually focused on the study of the oracle property under asymptotic and double asymptotic frameworks. For \(k \in {{\mathcal {S}}}\), that is, the vector \(\varvec{\theta }^{(k)}_0\) is at least nonzero, then, Consequently, if \(\varvec{u}^{(k) *}_i = 0, \forall i \in {{\mathcal {A}}}^c_k\), with \(k \in {{\mathcal {S}}}\), then the conditions (13) become, Combining relationships in (12), we obtain, The same reasoning applies for active groups with inactive components, so that combining relationships in (13), we obtain, Under the assumption that \(\lambda _0 < \infty \) and \(\gamma _0 < \infty \), we obtain, Thus \(c < 1\), which proves (10), that is proposition1. I gratefully acknowledge the Ecodec Laboratory for its support and the Japan Society for the Promotion of Science. Neumann (2013) proposed such a central limit theorem for weakly dependent sequences of arrays. We then deduce \(\Vert {\hat{\varvec{\theta }}} - \varvec{\theta }_0\Vert = O_p(\nu _T)\). Then by Corollary2 of Andersen and Gill, we obtain, We denote \(\nu _T = T^{-1/2} + \lambda _T T^{-1} a + \gamma _T T^{-1} b\), with \(a = \text {card}({{\mathcal {A}}})(\underset{k}{\max } \; \alpha _k)\) and \(b = \text {card}({{\mathcal {A}}})(\underset{l}{\max } \; \xi _l)\). This is a trusted computer. \(T^{-1/2} \sum ^{T}_{t=1} x_t \overset{d}{\rightarrow } {{\mathcal {N}}}(0,\sigma ^2_x)\), \(F_n(x) \overset{{{\mathbb {P}}}}{\underset{n \rightarrow \infty }{\longrightarrow }} f(x)\), $$\begin{aligned} \underset{x \in A}{\sup } |F_n(x) - f(x)| \overset{{{\mathbb {P}}}}{\underset{n \rightarrow \infty }{\longrightarrow }} 0. We first consider the unpenalized empirical criterion of \({{\mathbb {F}}}_T(. Wellner, J. Department of Statistics, University Carlos III of Madrid, Madrid, Spain, uc3m-Santander Big Data Institute, Madrid, Spain, Alvaro Mendez-Civieta,M. Carmen Aguilera-Morillo&Rosa E. Lillo, Department of Applied Statistics and Operational Research, and Quality, Universitat Politecnica de Valencia, Valencia, Spain, You can also search for this author in The Annals of Statistics, 10(4), 11001120. Did you know that with a free Taylor & Francis Online account you can gain access to the following benefits? Additionally, a more flexible version, an adaptive SGL is proposed based on the adaptive idea, this is, the usage of adaptive weights in the penalization. Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japan, You can also search for this author in We consider variable selection using the adap-tive Lasso, where the L1 norms in the penalty are re-weighted by data-dependent weights. It only requires the lower-semicontinuity and convexity of the empirical criterion. Fan, J., Peng, H. (2004). Journal of the American Statistical Association, 101(476), 14181429. An adaptive sparse group LASSO (ASGL) for quantile regression estimator is defined, working especially on enabling the usage of the ASGL estimator in high dimensional scenarios (with p\gg n ). Annals of the Institute of Statistical Mathematics And I thank warmly Jean-David Fermanian for his significant help and helpful comments. probability_of_significance_calculator Function. )\} \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} \arg \, \min \, \{{{\mathbb {G}}}_{\infty } \varphi (.)\}\). J R Stat Soc Ser B (Methodol) 58(1):267288, Wang L, Wu Y, Li R (2012) Quantile regression for analyzing heterogeneity in ultra-high dimension. (2014). Springer series in statistics Berlin: Springer. structures, where those in the same group are correlated. Code definitions. No description, website, or topics provided. UNC313-4E-2361, No. Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine. J Comput Graph Stat 28:722731, Li Y, Zhu J (2008) L\(_1\)-Norm quantile regression. \end{aligned}$$, \(\gamma _T T^{-1/2} \rightarrow \gamma _0 \ge 0\), $$\begin{aligned} \gamma _T \overset{m}{\underset{l = 1}{\sum }} \xi _l \left[ \Vert \varvec{\theta }^{(l)}_0 + u^{(l)}/\sqrt{T}\Vert _2 - \Vert \varvec{\theta }^{(l)}_0\Vert _2 \right]= & {} \gamma _0 \overset{m}{\underset{l = 1}{\sum }} \xi _l \left\{ \Vert u^{(l)}\Vert _2 {\mathbf {1}}_{\theta ^{(l)}_{0,k} = {\mathbf {0}}}\right. {\hat{{{\mathcal {A}}}}}^c_k, \theta ^{(k)}_{0,i} {=} 0\right\} . Hence under the assumption \(\lambda _T T^{(\eta -1)/2} \rightarrow \infty \), we obtain, As for the \(l^1/l^2\) quantity, we remind that \(\xi _{T,l} = \Vert {\tilde{\varvec{\theta }}}^{(l)}\Vert ^{-\mu }_2\), so that for \(l \in {{\mathcal {S}}}\), \({\tilde{\varvec{\theta }}}^{(l)} \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} \varvec{\theta }^{(l)}_0\), and in this case, Consequently, using \(\gamma _T T^{-1/2} \rightarrow 0\), and for \(l \in {{\mathcal {S}}}\), we obtain, Combining the fact \(k \in {{\mathcal {S}}}\) and \(\varvec{\theta }^{(k)}_0\) is partially zero, that is \(i \in {{\mathcal {A}}}^c_k\), we obtain the divergence given in (15). The Annals of Statistics, 1(5), 799821. \end{aligned}$$, $$\begin{aligned} a_1= & {} {{\mathbb {P}}}(l \in {{\mathcal {S}}}^c, \Vert {{\mathbb {H}}}_{(l) {{\mathcal {S}}}} {{\mathbb {H}}}^{-1}_{{{\mathcal {S}}}{{\mathcal {S}}}} (\varvec{Z}_{{{\mathcal {S}}}} + \lambda _0 \tau _{{{\mathcal {S}}}} + \gamma _0 \zeta _{{{\mathcal {S}}}}) - \varvec{Z}_{(l)} -\lambda _0 \alpha _l \varvec{w}^{(l)}\Vert _2 \le \gamma _0 \xi _l)< 1, \\ a_2= & {} {{\mathbb {P}}}(k \in {{\mathcal {S}}}, i \in {{\mathcal {A}}}^c_k, |({{\mathbb {H}}}_{{{\mathcal {A}}}^c_k {{\mathcal {A}}}_k} {{\mathbb {H}}}^{-1}_{{{\mathcal {A}}}_k {{\mathcal {A}}}_k} (\varvec{Z}_{{{\mathcal {A}}}_k} + \lambda _0 \alpha _k \text {sgn}(\varvec{\theta }_{0,{{\mathcal {A}}}_k}) \\&+ \gamma _0 \xi _k \frac{\varvec{\theta }_{0,{{\mathcal {A}}}_k}}{\Vert \varvec{\theta }_{0,{{\mathcal {A}}}_k}\Vert _2}) - \varvec{Z}_{{{\mathcal {A}}}^c_k})_i| \le \lambda _0 \alpha _k) < 1. Use Git or checkout with SVN using the web URL. Wainwright, M. J. WGRLR: A Weighted Group Regularized Logistic Regression for Cancer Diagnosis and Gene Selection. Google Scholar. In a first step, we prove the uniform convergence of \({{\mathbb {G}}}_T \varphi (. This paper aims to solve the above problems by developing the logistic regression with adaptive sparse group lasso penalty (LR-ASGL). {{\mathcal {A}}}^c_k, {\hat{\theta }}^{(k)}_i {=} 0\right\} \cap \left\{ \forall k {=}1,\ldots ,m, \, i \!\in \! Abstract: We study the asymptotic properties of the adaptive Lasso estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase with the sample size. 2022 Springer Nature Switzerland AG. Proceedings of the American Mathematical Society, 12, 788792. If nothing happens, download GitHub Desktop and try again. Journal of the Royal Statistical Society. We study the asymptotic properties of a new version of the Sparse Group Lasso estimator (SGL), called adaptive SGL. Toronto, Canada, Restore content access for purchases made as guest, Medicine, Dentistry, Nursing & Allied Health, 48 hours access to article PDF & online version, Choose from packages of 10, 20, and 30 tokens, Can use on articles across multiple libraries & subject collections. Using \(\varvec{p}_1(\lambda _T,\alpha ,0) = 0\) and \(\varvec{p}_2(\gamma _T,\xi ,0) = 0\), by a Taylor expansion to \({{\mathbb {G}}}_T l(\varvec{\theta }_0 + \nu _T \varvec{u})\), we obtain, where \({\bar{\varvec{\theta }}}\) is defined as \(\Vert {\bar{\varvec{\theta }}} - \varvec{\theta }_0\Vert \le \Vert \varvec{\theta }_T - \varvec{\theta }_0\Vert \). Geyer, C. J. \end{aligned}$$, \(\xi _{T,l} = \Vert {\tilde{\varvec{\theta }}}^{(l)}\Vert ^{-\mu }_2\), \({\tilde{\varvec{\theta }}}^{(l)} \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} \varvec{\theta }^{(l)}_0\), $$\begin{aligned} \sqrt{T} \left\{ \Vert \varvec{\theta }^{(l)}_0 + \varvec{u}^{(l)}/\sqrt{T}\Vert _2 - \Vert \varvec{\theta }^{(l)}_0\Vert _2\right\} = \frac{\varvec{u}^{(l) '} \varvec{\theta }^{(l)}_0}{\Vert \varvec{\theta }^{(l)}_0\Vert _2} + o\left( T^{-1/2}\right) . (2009). 3099067 This argument relies on the convexity lemma, which is a key result to obtain an asymptotic distribution when the objective function is not differentiable. Part of Springer Nature. Additionally, a more flexible version, an adaptive SGL is proposed based on the. Weak convergence and empirical processes. It can be viewed as an improved version of sparse group Lasso (SGL) and uses data-dependent weights to improve selection performance. This new version includes two distinct regularization parameters, one for the Lasso penalty and one for the Group Lasso penalty, and we consider the adaptive version of this regularization, where both penalties are weighted by preliminary random coefficients. Asymptotics for Lasso-type estimators. Unpublished manuscript. Estimating such time-varying sparse vectors requires the development of suitable adaptive ltering algorithms with sparse regularization. (1973). However, the periodic impulses are rather weak in the time domain when the noise is strong. Journal of the Royal Statistical Society. Convex analysis. Econometric Theory, 7(2), 186199. New York: Wiley. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. a sparse functional additive model, and our method for estimating the additive components in the model. In practice this implies the usage of a non penalized estimator that limits the adaptive solutions to low dimensional scenarios. We now prove the finite dimensional distribution of \({{\mathbb {F}}}_T\) to \({{\mathbb {F}}}_{\infty }\) to apply Lemma1. Unpublished manuscript. The Annals of Statistics, 32(3), 928961. Correspondence to )\), we obtain, We pick \(\alpha \) such that \(\Vert {\bar{\varvec{u}}}\Vert = C_{\varvec{\epsilon }}\) with \({\bar{\varvec{u}}} := \alpha \varvec{\theta }_1 + (1-\alpha ) \varvec{\theta }_0\). )\}\), $$\begin{aligned} |{{\mathbb {G}}}_T \varphi (\varvec{\theta })| \overset{{{\mathbb {P}}}}{\underset{\Vert \varvec{\theta }\Vert \rightarrow \infty }{\longrightarrow }} \infty . J Am Stat Assoc 103(484):16651673, Koenker R (2005) Quantile regression. This is a link to the ASGL package, that implements all the penalized models that can be observed in these simulations. Now based on the convexity of the objective function, we have, a relationship that allows us to work with a fixed \(\Vert \varvec{u}\Vert _2\). In: IEEE 11th international conference on data mining workshops. We obtain, Then we proved that \({{\mathbb {F}}}_T(\varvec{u}) \overset{d}{\longrightarrow } {{\mathbb {F}}}_{\infty }(\varvec{u})\), for a fixed \(\varvec{u}\). )\) is convex, continuous, then \(\underset{\varvec{x}\in B}{\arg \, \min } \, \{{{\mathbb {G}}}_{\infty } \varphi (\varvec{x})\}\) exists and is unique. Shiryaev (1991) proposed a version of the central limit theorem for dependent sequence of arrays, provided this sequence is a square integrable martingale difference satisfying the so-called Lindeberg condition. \end{aligned}$$, \(\varvec{p}^{(k)}_i = \partial _{\varvec{u}_i} \{ |\varvec{u}^{(k)}_i| {\mathbf {1}}_{\theta ^{(k)}_{0,i} = 0} + \varvec{u}^{(k)}_i \text {sgn}(\theta ^{(k)}_{0,i}){\mathbf {1}}_{\theta ^{(k)}_{0,i} \ne 0} \}\), \(\varvec{u}^{(m) *} = 0, \forall m \notin {{\mathcal {S}}}\), $$\begin{aligned} \left\{ \begin{array}{llll} {{\mathbb {H}}}_{{{\mathcal {S}}}{{\mathcal {S}}}} \varvec{u}^*_{{{\mathcal {S}}}} + \varvec{Z}_{{{\mathcal {S}}}} + \lambda _0 \tau _{{{\mathcal {S}}}} + \gamma _0 \zeta _{{{\mathcal {S}}}} = 0, &{} &{}\\ \Vert -{{\mathbb {H}}}_{(l) {{\mathcal {S}}}} \varvec{u}^*_{{{\mathcal {S}}}} - \varvec{Z}_{(l)} -\lambda _0 \alpha _l \varvec{w}^{(l)}\Vert _2 \le \gamma _0 \xi _l, \, \text {as} \, \Vert \varvec{z}^{(l)}\Vert _2 \le 1, \, l \in {{\mathcal {S}}}^c, &{} &{} \end{array}\right. \end{aligned}$$, $$\begin{aligned} {{\mathbb {P}}}(\exists \varvec{u}, \Vert \varvec{u}\Vert _2= & {} C_{\varvec{\epsilon }}: |\varvec{p}_1(\lambda _T,\alpha ,\varvec{\theta }_T)-\varvec{p}_1(\lambda _T,\alpha ,\varvec{\theta }_0)|> \nu _T \delta _T/8) \\\le & {} {{\mathbb {P}}}(\exists \varvec{u}, \Vert \varvec{u}\Vert _2 = C_{\varvec{\epsilon }}: \text {card}({{\mathcal {S}}}) \{ \underset{k \in {{\mathcal {S}}}}{\max } \; \alpha _k \} \lambda _T T^{-1} \nu _T \Vert \varvec{u}\Vert _1> \nu _T \delta _T/8)< \varvec{\epsilon }/5, \\&{{\mathbb {P}}}(\exists \varvec{u}, \Vert \varvec{u}\Vert _2 = C_{\varvec{\epsilon }}: |\varvec{p}_2(\gamma _T,\xi ,\varvec{\theta }_T)-\varvec{p}_2(\gamma _T,\xi ,\varvec{\theta }_0)|> \nu _T \delta _T/8) \\\le & {} {{\mathbb {P}}}(\exists \varvec{u}, \Vert \varvec{u}\Vert _2 = C_{\varvec{\epsilon }}: \text {card}({{\mathcal {S}}}) \{ \underset{l\in {{\mathcal {S}}}}{\max } \; \xi _l \} \gamma _T T^{-1} \nu _T \Vert \varvec{u}\Vert _2 > \nu _T \delta _T/8) < \varvec{\epsilon }/5. Ann Stat 36(2):587613, Huang J, Ma S, Zhang C-H (2008b) Adaptive Lasso for sparse high-dimensional regression. Proc Natl Acad Sci 103(39):1442914434, Simon N, Friedman J, Hastie T, Tibshirani R (2013) A sparse-group lasso. Tibshirani, R. (1996). Under such settings, variable selection should be conducted at both the group-level and within-group-level, that is, a bi-level selection. 2018. Then the KarushKuhnTucker conditions for \({{\mathbb {G}}}_T \psi ({\hat{\varvec{\theta }}})\) are given by, Using the same reasoning as previously, \(T^{1/2}({\dot{{{\mathbb {G}}}}}_T l({\hat{\varvec{\theta }}}))_{(k),i}\) is also asymptotically normal, and \({\tilde{\varvec{\theta }}}^{(k)} \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} \varvec{\theta }^{(k)}_0\) for \(k \in {{\mathcal {S}}}\), and besides, so that we obtain the same when adding \(\gamma _T T^{-1/2}\xi _{T,k} \frac{{\hat{\theta }}^{(k)}_i}{\Vert {\hat{\varvec{\theta }}}^{(k)}\Vert _2}\). Learn more. Bhlmann, P., van de Geer, S. (2011). With applications to statistics. Then by Assumption4, we have the central limit theorem of Billingsley (1961), \(\sqrt{T} {\dot{{{\mathbb {G}}}}}_T l(\varvec{\theta }_0) \overset{d}{\longrightarrow } {{\mathcal {N}}}(0,{{\mathbb {M}}})\), and by the ergodic theorem \(\ddot{{{\mathbb {G}}}}_T l(\varvec{\theta }_0) \overset{{{\mathbb {P}}}}{\underset{T \rightarrow \infty }{\longrightarrow }} {{\mathbb {H}}}\). Then by \(\gamma _T T^{(\mu -1)/2} \rightarrow \infty \) we have, We deduce the pointwise convergence \({{\mathbb {F}}}_T (\varvec{u}) \overset{d}{\longrightarrow } {{\mathbb {F}}}_{\infty }(\varvec{u})\), where \({{\mathbb {F}}}_{\infty }(. (Shiryaev 1991) Let a sequence of square integrable martingale differences \(\xi ^n = (\xi _{nk},{{\mathcal {F}}}^n_k),n \ge 1\), with \({{\mathcal {F}}}^n_k = \sigma (\xi _{ns},s \le k)\), satisfy the Lindeberg condition for any \(0
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