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![Histograms for 1000 Monte Carlo replications of |A^m|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\widehat{\mathcal{A}}_m|$$\end{document} for adaptive LASSO expectile (a), adaptive LASSO quantile (b) and adaptive LASSO LS (c) estimations, when the design X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {X}}$$\end{document} is D2, m=300\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=300$$\end{document}, p=250\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=250$$\end{document}, |A0|=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathcal{A}^0|=3$$\end{document}, ε∼N(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \sim \mathcal{N}(0,1)$$\end{document}](publication/357886069/figure/fig3/AS:1179939881648146@1658330998872/Histograms-for-1000-Monte-Carlo-replications-of-Amdocumentclass12ptminimal.png)
Histograms for 1000 Monte Carlo replications of |A^m|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\widehat{\mathcal{A}}_m|$$\end{document} for adaptive LASSO expectile (a), adaptive LASSO quantile (b) and adaptive LASSO LS (c) estimations, when the design X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {X}}$$\end{document} is D2, m=300\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=300$$\end{document}, p=250\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=250$$\end{document}, |A0|=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mathcal{A}^0|=3$$\end{document}, ε∼N(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \sim \mathcal{N}(0,1)$$\end{document}
Source publication
In the present paper we address the real-time detection problem of a change-point in the coefficients of a linear model with the possibility that the model errors are asymmetrical and that the explanatory variables number is large. We build test statistics based on the cumulative sum (CUSUM) of the expectile function derivatives calculated on the r...
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Citations
... If the regression function f is linear in β ∈ , then the asymptotic behaviour in the proposition reduces to a special case of Proposition 1 from Ciuperca (2022). Similarly, if the regression function f in (1) is nonlinear in β ∈ , but the random error terms follow some normal distribution N (0, σ 2 ) with σ 2 < ∞, the asymptotic behaviour in Proposition 1 gives the results of Theorem 2.1 in Seber and Wild (2003). ...
... The test statistic in Theorem 2 is based on the expectile estimator β m of the true parameter vector β 0 ∈ calculated from the historical data. However, the limit process is the same as for the expectile estimator in the linear model considered in Ciuperca (2022), or the quantile estimator proposed in Zhou et al (2015). On the other hand, the test statistic is different from that proposed by Ciuperca (2013) or Horváth et al (2004) where the authors rather considered the CUSUM type statistic based on the least squares residuals of the linear model or the nonlinear model respectively. ...
... Next, similarly as in the proof of Lemma 1 in Ciuperca (2022), under Assumptions (A1) and (A2), using the last relation above, we have that for all constants C 1 , C 4 > 0, there exists a constant C > 0 such that ...
An online changepoint detection procedure based on conditional expectiles is introduced. The key contribution is threefold: nonlinearity of the underlying model improves the overall flexibility while a parametric form of the unknown regression function preserves a simple and straightforward interpretation; The conditional expectiles, well-known in econometrics for being the only coherent and elicitable risk measure, introduce additional robustness—especially with respect to asymmetric error distributions common in various types of data; The proposed statistical test is proved to be consistent and the distribution under the null hypothesis does not depend on the functional form of the underlying model nor the unknown parameters. Empirical properties of the proposed real-time changepoint detection test are investigated in a simulation study and a practical applicability is illustrated using the Covid-19 prevalence data from Prague.
