Additive models for extremal quantile regression with Pareto-type distributions
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Additive models for extremal quantile regression with Pareto‑type distributions Takuma Yoshida1 Received: 26 December 2019 / Accepted: 9 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Estimating conditional quantiles in the tail of a distribution is an important problem for several applications. However, data sparsity indicates that the predictions of tail behavior are more difficult compared with those for the mean or center quantiles, in particular, when a multivariate covariate is used. As additive models are known to be an efficient approach for multiple regression, this study considers an additive model for extremal quantile regression. The conditional quantile function is first estimated using a two-stage estimation method for the intermediate-order (not too extreme) quantile. Subsequently, the extreme-order quantile estimator is constructed by extrapolating from the intermediate-order quantile estimator. By combining the asymptotic and extreme value theories, the theoretical properties of the intermediate- and extreme-order quantile estimators are evaluated. A simulation study is conducted to confirm the performance of the estimators, and an application using real data is provided. Keywords Additive model · Asymptotic normality · Extrapolation · Extremal quantile regression · Extreme value theory
1 Introduction Quantile regression is a useful tool for investigating the relationship between response and covariate variables. Since it was first reported by Koenker and Bassett (1978), several authors have developed related methods and theories, and the fundamental properties of quantile regression have been clarified by Koenker (2005). The advantage of quantile regression is that it is robust to outliers, unlike mean regression. However, this property is limited to the quantile level close to the center. For the upper and lower tail quantile levels, the robustness of the quantile * Takuma Yoshida [email protected]‑u.ac.jp 1
Graduate School of Science and Engineering, Kagoshima University, Kagoshima 890‑8580, Japan
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regression fails, and the prediction becomes more difficult compared with that for the central quantile because of data sparsity. However, predicting the tail quantile of the conditional distribution is important in several applications, such as determining financial risk, identifying extreme rainfall events, and investigating birth weights. Accordingly, we propose a quantile regression method for extreme quantiles. We focus on high quantiles, although the discussion for low quantiles is similar. In extremal quantile regression, there are two scenarios for the rate of the quantile. For a quantile level 𝜏 and sample size n, if n → ∞, 𝜏 → 1, n(1 − 𝜏) → ∞ , we have the so-called intermediate-order quantile. When n → ∞, 𝜏 → 1, n(1 − 𝜏) → c ∈ [0, ∞) , we have an extreme-order quantile. Thus, the extreme-order quantile is at a higher level than the intermediate-order quantile. In addition, we need to impose the tail behavio
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