Expectile Regression Research Articles

Weighted Expectile Regression Neural Networks for Right Censored Data.

As a favorable alternative to the censored quantile regression, censored expectile regression has been popular in survival analysis due to its flexibility in modeling the heterogeneous effect of covariates. The existing weighted expectile regression (WER) method assumes that the censoring variable and covariates are independent, and that the covariates effects has a global linear structure. However, these two assumptions are too restrictive to capture the complex and nonlinear pattern of the underlying covariates effects. In this article, we developed a novel weighted expectile regression neural networks (WERNN) method by incorporating the deep neural network structure into the censored expectile regression framework. To handle the random censoring, we employ the inverse probability of censoring weighting (IPCW) technique in the expectile loss function. The proposed WERNN method is flexible enough to fit nonlinear patterns and therefore achieves more accurate prediction performance than the existing WER method for right censored data. Our findings are supported by extensive Monte Carlo simulation studies and a real data application.

Renewable estimation in expectile regression model with streaming data sets

Streaming data continually expands over time, making it challenging to apply traditional expectile regression methods due to memory limitations. Expectile regression is advantageous for studying the entire conditional distribution of the response to a predictor, but handling streaming data requires innovative approaches. This paper proposes an online renewable expectile regression strategy, which updates the estimator using both the current data and summary statistics from historical data. An ADMM algorithm is employed to efficiently compute the proposed estimator. We establish the consistency and asymptotic normality of this renewable estimator under regular conditions. Extensive simulations and real-data experiments demonstrate that our method performs comparably to the oracle estimator while maintaining computational efficiency and low storage requirements.

Almost complete convergence of the functional local linear estimator of the expectile regression for data missing at random

In this article, the authors investigate the properties of a local linear estimator for the expectile regression function in scenarios where the scalar response variable is missing at random, and the covariate is a functional variable. The study addresses the challenge of estimating the expectile regression function when the response variable is not fully observed, a common issue in statistical analysis where missing data can significantly impact the accuracy of predictions and inferences. The key contribution of the paper lies in the establishment of the almost complete convergence of the proposed estimator. This convergence resultis derived under a set of appropriate conditions, ensuring that the estimator approaches the true expectile regression function with a probability tending to wards one as the sample size increases. The authors provide a rigorous theoretical foundation for this convergence, including assumptions about the smoothness of the functional covariate and the randomness of the missing data mechanism. To demonstrate the practical applicability of their method, the authors conduct a comprehensive simulation study. This study compares the performance of their local linear estimator with other existing methods in terms of bias, variance, and convergence speed. The simulation resultshighlight the effectiveness of the proposed approach, especially in settings where the response variable is partially missing and the covariates are functional in nature.

Shadow prices of agrochemicals in the Chinese farming sector: A convex expectile regression approach

The use of agrochemical inputs has significantly enhanced agricultural yields in China; however, their excessive utilization has also caused a range of environmental issues. This paper examines the costs associated with reducing agrochemicals by employing shadow prices, which represent the value of the marginal product of agrochemicals, to further develop cost-effective environmental policy measures for reducing their usage. To this end, the shadow prices of agrochemicals have been assessed by adopting a newly developed convex expectile regression approach and using statistical data from 31 provinces in China spanning from 2005 to 2020. Furthermore, the present study investigates the disparities between shadow prices and market prices for different agrochemicals across various regions in China. The findings suggest that the costs of reducing chemical fertilizers are higher than those of reducing pesticides and plastic films. Moreover, the results indicate that central China exhibits relatively high potential for decreasing agrochemical usage. Finally, these findings can inform the Chinese government's restructuring of producer support and environmental policy in a cost-effective way to mitigate agrochemicals use in the future. Additionally, the research method employed in this study holds potential for extension to other agrochemicals-dependent countries.

Composite expectile estimation in partial functional linear regression model

Recent research and substantive studies have shown growing interest in expectile regression (ER) procedures. Similar to quantile regression, ER with respect to different expectile levels can provide a comprehensive picture of the conditional distribution of a response variable given predictors. This study proposes three composite-type ER estimators to improve estimation accuracy. The proposed ER estimators include the composite estimator, which minimizes the composite expectile objective function across expectiles; the weighted expectile average estimator, which takes the weighted average of expectile-specific estimators; and the weighted composite estimator, which minimizes the weighted composite expectile objective function across expectiles. Under certain regularity conditions, we derive the convergence rate of the slope function, obtain the mean squared prediction error, and establish the asymptotic normality of the slope vector. Simulations are conducted to assess the empirical performances of various estimators. An application to the analysis of capital bike share data is presented. The numerical evidence endorses our theoretical results and confirm the superiority of the composite-type ER estimators to the conventional least squares and single ER estimators.

Poisson subsampling-based estimation for growing-dimensional expectile regression in massive data

Poisson subsampling-based estimation for growing-dimensional expectile regression in massive data

Inference for high-dimensional linear expectile regression with de-biasing method

The methodology for the inference problem in high-dimensional linear expectile regression is developed. By transforming the expectile loss into a weighted-least-squares form and applying a de-biasing strategy, Wald-type tests for multiple constraints within a regularized framework are established. An estimator for the pseudo-inverse of the generalized Hessian matrix in high dimension is constructed using general amenable regularizers, including Lasso and SCAD, with its consistency demonstrated through a novel proof technique. Simulation studies and real data applications demonstrate the efficacy of the proposed test statistic in both homoscedastic and heteroscedastic scenarios.

Expectile regression averaging method for probabilistic forecasting of electricity prices

AbstractIn this paper we propose a new method for probabilistic forecasting of electricity prices. It is based on averaging point forecasts from different models combined with expectile regression. We show that deriving the predicted distribution in terms of expectiles, might be in some cases advantageous to the commonly used quantiles. We apply the proposed method to the day-ahead electricity prices from the German market and compare its accuracy with the Quantile Regression Averaging method and quantile- as well as expectile-based historical simulation. The obtained results indicate that using the expectile regression improves the accuracy of the probabilistic forecasts of electricity prices, but a variance stabilizing transformation should be applied prior to modelling.

Downside Risk in Australian and Japanese Stock Markets: Evidence Based on the Expectile Regression

The expectile-based Value at Risk (EVaR) has gained popularity as it is more sensitive to the magnitude of extreme losses than the conventional quantile-based VaR (QVaR). This paper applies the expectile regression approach to evaluate the EVaR of stock market indices of Australia and Japan. We use an expectile regression model that considers lagged returns and common risk factors to calculate the EVaR for each stock market and to evaluate the interdependence of downside risk between the two markets. Our findings suggest that both Australian and Japanese stock markets are affected by their past development and the international stock markets. Additionally, ASX 200 index has significant impact on Nikkei 225 in terms of downside tail risk, while the impact of Nikkei 225 on ASX is not significant.

Penalized empirical likelihood for longitudinal expectile regression with growing dimensional data

Penalized empirical likelihood for longitudinal expectile regression with growing dimensional data

Estimation and inference for multi-kink expectile regression with nonignorable dropout

In this paper, we consider parameter estimation, kink points testing and statistical inference for a longitudinal multi-kink expectile regression model with nonignorable dropout. In order to accommodate both within-subject correlations and nonignorable dropout, the bias-corrected generalized estimating equations are constructed by combining the inverse probability weighting and quadratic inference function approaches. The estimators for the kink locations and regression coefficients are obtained by using the generalized method of moments. A selection procedure based on a modified BIC is applied to estimate the number of kink points. We theoretically demonstrate the number selection consistency of kink points and the asymptotic normality of all estimators. A weighted cumulative sum type statistic is proposed to test the existence of kink effects at a given expectile, and its limiting distributions are derived under both the null and the local alternative hypotheses. Simulation studies show that the proposed estimators and test have desirable finite sample performance in both homoscedastic and heteroscedastic errors. An application to the Nation Growth, Lung and Health Study dataset is also presented.

Strong consistency rate in functional single index expectile model for spatial data

<abstract><p>Analyzing the real impact of spatial dependency in financial time series data is crucial to financial risk management. It has been a challenging issue in the last decade. This is because most financial transactions are performed via the internet and the spatial dependency between different international stock markets is not standard. The present paper investigates functional expectile regression as a spatial financial risk model. Specifically, we construct a nonparametric estimator of this functional model for the functional single index regression (FSIR) structure. The asymptotic properties of this estimator are elaborated over general spatial settings. More precisely, we establish Borel-Cantelli consistency (BCC) of the constructed estimator. The latter is obtained with the precision of the convergence rate. A simulation investigation is performed to show the easy applicability of the constructed estimator in practice. Finally, real data analysis about the financial data (Euro Stoxx-50 index data) is used to illustrate the effectiveness of our methodology.</p></abstract>

Communication‐efficient low‐dimensional parameter estimation and inference for high‐dimensional Lp$$ {L}^p $$‐quantile regression

AbstractThe ‐quantile regression generalizes both quantile regression and expectile regression, and has become popular for its robustness and effectiveness especially when . In this paper, we consider the data that are inherently distributed and propose two distributed ‐quantile regression estimators for a preconceived low‐dimensional parameter in the presence of high‐dimensional extraneous covariates. To handle the impact of high‐dimensional nuisance parameters, we first investigate regularized projection score for estimating low‐dimensional parameter of main interest in ‐quantile regression. To deal with the distributed data, we further propose two communication‐efficient surrogate projection score estimators and establish their theoretical properties. The finite‐sample performance of the proposed estimators is studied through simulations and an application to Communities and Crime data set is also presented.

Automatic selection by penalized asymmetric Lq -norm in a high-dimensional model with grouped variables

The paper focuses on the automatic selection of the grouped explanatory variables in a high-dimensional model, when the model blue error is asymmetric. After introducing the model and notations, we define the adaptive group LASSO expectile estimator for which we prove the oracle properties: the sparsity and the asymptotic normality. Afterwards, the results are generalized by considering the asymmetric -norm loss function. The theoretical results are obtained in several cases with respect to the number of variable groups. This number can be fixed or dependent on the sample size n, with the possibility that it is of the same order as n. Note that these new estimators allow us to consider weaker assumptions on the data and on the model errors than the usual ones. Simulation study demonstrates the competitive performance of the proposed penalized expectile regression, especially when the samples size is close to the number of explanatory variables and model errors are asymmetrical. An application on air pollution data is considered.

The Financial Risk Measurement EVaR Based on DTARCH Models.

The value at risk based on expectile (EVaR) is a very useful method to measure financial risk, especially in measuring extreme financial risk. The double-threshold autoregressive conditional heteroscedastic (DTARCH) model is a valuable tool in assessing the volatility of a financial asset's return. A significant characteristic of DTARCH models is that their conditional mean and conditional variance functions are both piecewise linear, involving double thresholds. This paper proposes the weighted composite expectile regression (WCER) estimation of the DTARCH model based on expectile regression theory. Therefore, we can use EVaR to predict extreme financial risk, especially when the conditional mean and the conditional variance of asset returns are nonlinear. Unlike the existing papers on DTARCH models, we do not assume that the threshold and delay parameters are known. Using simulation studies, it has been demonstrated that the proposed WCER estimation exhibits adequate and promising performance in finite samples. Finally, the proposed approach is used to analyze the daily Hang Seng Index (HSI) and the Standard & Poor's 500 Index (SPI).

Distributed estimation for large-scale expectile regression

Analysis of large volume of data is very complex due to not only the high level of skewness and heteroscedasticity of variance but also the difficulty of data storage. Expectile regression is a common alternative method to analyze heterogeneous data. Distributed storage can reduce effectively the storage burden of a single machine. In this paper, we consider fitting linear expectile regression model to estimate conditional expectile based on large-scale data. We store the data in a distributed manner and construct a gradient-enhanced loss (GEL) function as a proxy for the global loss function. A distributed algorithm is proposed for the optimization of the GEL function. The asymptotic properties of the proposed estimator are established. Simulation studies are conducted to assess the finite-sample performance of our proposed estimator. Applications to an analysis of the National Health Interview Survey data set demonstrate the practicability of the proposed method.

Functional additive expectile regression in the reproducing kernel Hilbert space

In the literature, the functional additive regression model has received much attention. Most current studies, however, only estimate the mean function, which may not adequately capture the heteroscedasticity and/or asymmetries of the model errors. In light of this, we extend functional additive regression models to their expectile counterparts and obtain an upper bound on the convergence rate of its regularized estimator under mild conditions. To demonstrate its finite sample performance, a few simulation experiments and a real data example are provided.

Dimension reduction techniques for conditional expectiles

Marginalizing the importance of characterizing tail events can lead to catastrophic repercussions. Look no further than examples from meteorology and climatology (polar reversals, natural disasters), economics (2008 subprime mortgage crisis), or even medical-diagnostics (low/high risk patients in survival analysis). Investigating these events can become even more challenging when working with high-dimensional data, making it necessary to use dimension reduction techniques. Although research has recently turned to dimension reduction techniques that use conditional quantiles, there is a surprisingly limited amount of research dedicated to the underexplored research area of expectile regression (ER). Therefore, we present the first comprehensive work about dimension reduction techniques for conditional expectiles. Specifically, we introduce the central expectile subspace, i.e., the space that spans the fewest linear combinations of the predictors that contain all the information about the response that is available from the conditional expectile. We then introduce a nonlinear extension of the proposed methodology that extracts nonlinear features. The performance of the algorithms are demonstrated through extensive simulation examples and a real data application. The results suggest that ER is an effective tool for describing tail events and is a competitive alternative to quantile regression.

Retire: Robust expectile regression in high dimensions

High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. The former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this paper, we propose and study (penalized) robust expectile regression (retire), with a focus on iteratively reweighted ℓ1-penalization which reduces the estimation bias from ℓ1-penalization and leads to oracle properties. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which d≪n; (ii) high-dimensional regime in which s≪n≪d with s denoting the number of significant predictors. In the high-dimensional setting, we thoroughly analyze the statistical properties of the solution path of iteratively reweighted ℓ1-penalized retire estimation, adapted from the local linear approximation algorithm for folded-concave regularization. Under a mild minimum signal strength condition, we demonstrate that with as few as log(logd) iterations, the final iterate of our proposed approach achieves the oracle convergence rate. At each iteration, we solve the weighted ℓ1-penalized convex program using a semismooth Newton coordinate descent algorithm. Numerical studies demonstrate the promising performance of the proposed procedure in comparison to both non-robust and quantile regression based alternatives.

Parametric expectile regression and its application for premium calculation

Premium calculation has been a popular topic in actuarial sciences over the decades. Generally, a two-stage model is used to develop the premium calculation process. It can be decomposed into estimating the probability of having at least one claim by the logistic regression in the first stage and calibrating the severity in the second stage. Existing methods for the second stage include generalized linear models (GLMs) and quantile regression. However, the GLMs fail to provide information associated with the extreme claim amount and the quantile is not sensitive to the size of the extreme losses. Given that the magnitude of the extreme claim amount may lead to a huge loss for the insurer, we introduce the expectile risk measure into premium calibration and propose an expectile-based risk premium, which outperforms other methods for heavy-tailed distributions. Naively adopting the conventional expectile regression in the second stage is not preferred because it would be over-parameterized and time-consuming if the portfolio contains a large number of risk classes. Thus, we put forward a two-stage parametric expectile regression (TSPER) with parametric expectile regression (PER) method used in the second stage. The consistency and asymptotic normality of the proposed PER estimator are established under mild conditions. We also propose a novel Expectile Premium Principle to allocate the total premium for each policy. In the analysis of the automobile insurance data set, the proposed TSPER method outperforms other two-stage methods in terms of the ordered Lorenz curve and the Gini index.