F1000Research F1000Research 2046-1402 F1000 Research Limited London, UK 10.12688/f1000research.174712.1 Research Article Articles Bayesian expectile regression with single-index models

[version 1; peer review: awaiting peer review]

 Abdulhasan Zina Conceptualization Formal Analysis Methodology Validation Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0009-0001-0738-3046 a 1 Alhamzawi Rahim Investigation Methodology Project Administration Resources Supervision Writing – Original Draft Preparation 2 Department of Statistics, College of Administration and Economics, University of Al-Qadisiyah, Al Diwaniyah, Iraq Department of Statistics, College of Administration and Economics, University of Al-Qadisiyah, Al Diwaniyah, Iraq statistics.stp.25.8@qu.edu.iq

No competing interests were disclosed.

 15 4 2026 2026 15 516 20 2 2026 Copyright: © 2026 Abdulhasan Z and Alhamzawi R 2026 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Single-index expectile regression models provide a flexible semiparametric regression framework for high-dimensional covariates, and capture parameter heterogeneity and nonlinearity especially when focusing on different parts of the conditional distribution of the outcome of interest. Bayesian approaches have never been studied for such regression models. In this paper, we propose a Bayesian single-index expectile regression model using the asymmetric normal distribution (AND) for the error distribution. We design an MCMC method for posterior estimate. Simulations and real data analysis results show that the proposed approach performs very well compared with some existing approaches.

 Bayesian inference Expectile regression Single-index model Asymetric Normal Distribution The author(s) declared that no grants were involved in supporting this work. 1. Introduction

Expectiles of a probability distribution F, like the quantiles of a probability distribution F, represent different points of a distribution, but they are determined by tail expectations rather than tail probabilities. Expectiles depend on both the tail realizations and their probability, while quantiles only depend on the frequency of tail observations. There exists an one-to-one mapping from expectiles to quantiles, 27 i.e. for each τth expectile, there is a corresponding θ th quantile, where τ and θ ∈ ( 0 , 1 ) . Hence, expectiles can be utilized to estimate quantiles.

The τ th expectile of F is the quantity e τ that satisfies τ = ∫ − ∞ e τ | x − e τ | dF ( x ) ∫ − ∞ ∞ | x − e τ | dF ( x )

Expectiles are a generalization of the mean and the expectile loss is a generalization of the mean squared error in the same way as quantiles are a generalization of the median and the quantile loss is a generalization of the mean absolute error. Standard regression model aims to estimate the conditional expectation of the outcome variable y given the vector of covariates x , i . e . , E ( y | x ) . In many applications, however, it is required to study conditional distributions beyond the mean (conditional expectation). A nice tool for this purpose was offered by 17 in the form of expectile regression. Expectile regression 17 models the relationship between the covariates and the conditional expectiles of the outcome variable. The methodology is a generalization of the mean regression and closely related to quantile regression. It uses expectiles-points that minimize an asymmetric quadratic loss function l τ ( t ) = t 2 . | τ − I ( t < 0 ) | rather than the absolute loss function l θ ( t ) = t . ( θ − I ( t < 0 ) ) used in quantile regression. Both the expectile level τ and the quantile level θ determine the degree of asymmetry of the loss function. Examining the asymmetric quadratic loss function reveals many of the properties that make expectile regression an attractive measure of risk.

Compared to the mean value, expectiles are more sensitive to extreme values, and to the shape of the distribution in general. Furthermore, standard regression implicitly assumes normally distributed residuals, while such an assumption is not necessary in expectile regression. Expectile regression often leads to more efficient estimators compared to quantile regression, especially when the underlying error distribution is close to normal or when you’re interested in extreme values of the conditional distribution. Estimation of expectile regression models can be done using iterative algorithms, such as weighted least squares or stochastic gradient descent, and reliable estimation approaches have been developed in both the classical and Bayesian literatures.

Since its inception, expectile regression has attracted considerable interest in the literature. It has been applied in many different areas: finance and risk management, 3, 6, 14 actuarial science, 4, 7, 16 ecology and environmental studies, 10, 21, 24 social sciences, 19, 28 and so on. In the recent decades, there exists considerable interest in the study of nonparametric and semiparametric models. Single-index model provides an efficient way of coping with high-dimensional nonparametric estimation problems and gives more flexibility and capture parameter heterogeneity and nonlinearity. Expectile regression with single-index models is an efficient method to model asymmetric relationships in data while achieving dimension reduction. There exists a large literature on classical methods for expectile regression with single-index models, and we refer to 12, 13 for an overview. In contrast, a Bayesian method for estimating expectile regression with single index model has not been proposed, yet.

In this paper, we consider a single-index expectile regression model. For a given expectile level 0 < τ < 1 and training data { ( x i , y i ) } i = 1 n , it is given by E y i | x i ( τ ) = ϕ ( x i ′ β ) , i = 1 , … , n .

Here, Е y i | x i ( τ ) is the τ th expectile function of y i given x i , x i ∈ ℝ p is the covariate vector for the i -th observation, y i ∈ ℝ is the response corresponding to the covariate vector x i , ϕ ( . ) is the unknown univariate link function, and β = ( β 1 , β 2 , ⋯ , β k ) is the parametric index vector which implicitly depends on the desired expectile level τ . Following 15 and 12 for the sake of identifiability, we assume that ‖ β ‖ = 1 and that the first component of β is positive, ∥ · ∥ refers to the Euclidean norm.

The single index regression model is a form of dimension reduction in regression where the covariate vector x i is reduced to a one-dimensional index, so that ϕ ( x i ′ β ) is a univariate function instead of k-variate one, allowing for more interpretable and efficient modeling of the outcome of interest. In this paper, we establish a hierarchical Bayesian model by using asymmetric normal distribution (AND). As shown in Figure 1, the asymmetric normal distribution exhibits different shapes depending on the expectile level τ. For detailed information on Bayesian expectile regression methods, see, 25, 23, 26 and. 20 Following 11 and, 29 we assign a Gaussian process prior distribution on ϕ , to get a flexible nonparametric expectile regression model.

Probability density functions of the AND with <inline-formula> <mml:math display="inline"> <mml:msup> <mml:mi>σ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi>τ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.10</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0.25</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0.50</mml:mn> </mml:math> </inline-formula>. <sup> <xref ref-type="bibr" rid="ref30">30</xref> </sup>

This paper proceeds as follows. In Section 2, we introduce single-index expectile regression, the proposed Bayesian hierarchical model, and derive the corresponding MCMC samplers. Simulation studies are then presented in Section 3 followed by a real data example in Section 4. Conclusions is put in Section 5.

 2. Methods 2.1 Single-Index expectile regression

In the single-index expectile regression, the regression coefficients β can be estimated through optimizing the following empirical loss function min β ∑ i = 1 n Ɩ τ ( y i − ϕ ( x i ′ β ) ) , where the loss function Ɩ τ ( y i − ϕ ( x i ′ β ) ) = ( y i - ϕ ( x i ′ β ) ) 2 | τ − I ( y i − ϕ ( x i ′ β ) < 0 ) | .

Equivalently, we may write (3) as Ɩ τ ( y i − ϕ ( x i ′ β ) ) = { τ ( y i − ϕ ( x i ′ β ) ) 2 , if y i − ϕ ( x i ′ β ) ≥ 0 , ( 1 − τ ) ( y i − ϕ ( x i ′ β ) ) 2 , if y i − ϕ ( x i ′ β ) < 0 .                  

Rather than minimizing the usual expectile loss function (2), we solve the minimization problem by constructing a Markov chain having the joint posterior for the expectile regression coefficients β as its stationary distribution with the minimizer of (2) as its global mode. The quadratic asymmetric loss function (2) is exactly equivalent to the AND; see 23, 25 and. 20 The density function of an AND ( μ , σ 2 , τ ) is p ( y i ) = 2 σ 2 π ( τ ( 1 − τ ) τ + 1 − τ ) exp ( − v i ( y i − ϕ ( x i ′ β ) ) 2 2 σ 2 ) , where, τ is the skew parameter, σ 2 is the scale parameter, μ i = ϕ ( x i ′ β ) is the location parameter and v i = | τ − I ( y i − ϕ ( x i ′ β ) < 0 ) | . Minimizing (2) is equivalent to maximizing the likelihood function of y i by assuming y i from an AND with μ i = ϕ ( x i ′ β ) .

 2.2 Bayesian hierarchical model

Following 11, 29, 5 and, 8 we model the nonparametric link function ϕ by a Gaussian process (GP) prior with mean zero and covariance function C ( · , · ) , i.e. ϕ ~ GP ( 0 , C ( · , · ) ) , where C ( β ′ x i , β ′ x j ) = r exp ( − ( x i − x j ) ′ β β ′ ( x i − x j ) b ) ,

Here, r and b are two hyperparameters. Following 11 and, 29 we replace β / b with a new index vector, still denoted by β , to simplify the estimation procedure. Thus, the Gaussian process (GP) prior in (5) can be written as C ( β ′ x i , β ′ x j ) = r exp ( − ( x i − x j ) ′ β β ′ ( x i − x j ) ) .

To proceed a Bayesian analysis, we assign a Laplace prior for β j , j = 1 , ⋯ , p , of the form 2, 22 f ( β j | σ , λ j ) = ∏ j = 1 k λ j 2 σ exp { − λ i | β j | σ } , which extends Bayesian Lasso 18 by allowing different penalization parameters ( λ j > 0 ) for different regression coefficients. We further put a Gamma prior on the parameter λ j of the form p ( λ j ) ∝ λ j a − 1 exp { − b λ j } , p ( σ 2 ) = 1 / σ 2 on σ 2 and inverse Gamma prior on r , i . e . r ∼ IG ( c , d ) , where c and d are two hyperparameters. Thus, a fully bayesian approach for expectile adaptive lasso regression with single index model can be described as follows: y i | β , σ 2 ~ AND ( ϕ ( x i ′ β ) , σ 2 , τ ) , β | σ 2 , λ ∼ ∏ j = 1 k λ j 2 σ exp { − λ j | β j | σ } , ϕ n | β , r ~ GP ( 0 , C n ( . , . ) ) , σ 2 ~ 1 / σ 2 , λ j ~ ( λ j ) a − 1 exp { − b λ j } , r ~ ( 1 r ) c + 1 exp ( d r ) where λ = ( λ 1 , … , λ k ) ′ , ϕ n = ( ϕ 1 , … , ϕ n ) ′ = ( ϕ ( x i ′ β ) , … , ϕ ( x i ′ β ) ) ′ .

2.3 MCMC sampling

The posterior distribution of all parameters of interest is found via MCMC sampling algorithm and the details of full conditional distributions are given below.

Sample the regression coefficients β from their posteriors using a random walk Metropolis-Hastings steps,

f ( β | σ 2 , λ , r ) ∝ ∫ f ( y | σ 2 , ϕ n ) f ( ϕ n | β , r ) d ϕ n f ( β | σ 2 , λ )                                                                                             ∝ ( det [ V + C n ] ) − 1 2 exp { − y ′ ( V + C n ) y 2 σ 2 } ∏ j = 1 k exp { − λ j | β j | σ 2 } , where the weight matrix V = diag ( v 1 , … , v n ) adjusts for the expectile loss so that v i = | τ − I ( y i − ϕ ( x i ′ β ) < 0 | .

Sample the hyperparameter r from the posterior using a random walk Metropolis-Hastings steps,

                                       f ( r | β , σ 2 , λ ) ∝ ∫ f ( y | σ 2 , ϕ n ) f ( ϕ n | β , r   ) d ϕ n f ( r )                                                                                 ∝ ( det [ V + C n ] ) − 1 2 exp { − y ′ ( V + C n ) y 2 σ 2 } ( 1 r ) c + 1 exp ( d r )

Sample the nonparametric link function ϕ n from a multivariate normal distribution with mean μ n = C n ( V + C n ) − 1 y and variance Σ n = C n ( V + C n ) − 1 V

Sample σ 2 from inverse Gamma (IG) with shape parameter ( n − 1 + k ) / 2 and scale parameter ( ∑ i = 1 n v i ( y i − ϕ ( x i ′ β ) ) 2 / 2 ) + ∑ j = 1 k λ j | β j |

Sample λ j from Gamma distribution with shape parameter a + 1 and scale parameter b + | β j | / σ 2

3. Simulation studies

In this section, we investigate the prediction accuracy of the proposed approach (BESIM) and compare its performance with a non-Bayesian single-index expectile regression 12 referred as “ESIM”. We simulate data from the model y i = ϕ ( x i ′ β ) + σ ( x i ′ β ) u i ,               i = 1 , … , n .

The covariates are simulatted independently from the uniform distribution on [ 0 , 1 ] and u i is i.i.d. N ( 0 , 3 ) . We experiment with four different scenarios by varying the sample size ( n = 50 , 150 , 250 , 500 ) and simulations are repeated 150 times for each of given n and τ ∈ ( 0.10 , 0.20 , 0.30 , 0.40 , 0.50 , 0.60 , 0.70 , 0.80 , 0.90 ) .

3.1 Simulation 1

The simulation setup is similar to Example 1 in 11, 29 with different parameter values for the regression coefficients and error distribution. We generate data sets from model (16), where ϕ ( s ) = sin ( π ( s − A ) C − A ) , β = ( β 1 , β 2 , β 3 ) = 1 3 ( 1 , 0 , 0 ) , A = 3 2 − 1.645 12 , C = 3 2 + 1.645 12 , σ ( x i ′ β ) = 0.5 and u i ~ N ( 0 , 3 ) .

For a Bayesian point estimator we consider the posterior mean using 15,000 iterations of the MCMC after 1,000 iterations as burn-in. The resulting estimates are summarized in boxplot Figures 2 and 3 based on 100 replications. These boxplots display the estimated coefficients, comparing BESIM and ESIM, with τ ∈ { 0.10 , 0.50 , 0.90 } . In general, the boxplots give the impression that the Bayesian estimates (BESIM) produces more precise and stable estimates than classical estimates (ESIM). Mean squared errors (MSE) of the estimates based on the 100 replications in each case are shown in Tables 1 and 2 for four sample sizes n = 50 , 150 , 250 and 500 and all nine expectiles. MSE results show that the proposed method generally behaves better than the ESIM method in terms of the MMAD.

Comparison of MSE results in Simulation 1 for BESIM and ESIM based on <inline-formula> <mml:math display="inline"> <mml:mn>150</mml:mn> <mml:mspace width="0.25em"/> </mml:math> </inline-formula>replications when <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>50</mml:mn> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>150</mml:mn> </mml:math> </inline-formula>.
n = 50 n = 150
τ Methods β 1 β 2 β 3 β 1 β 2 β 3
0.10 ESIM 0.0195543 0.0147065 0.0350343 0.0462157 0.0091205 0.0400359
0.10 BESIM 0.0263048 0.0181031 0.0200267 0.0382094 0.0276661 0.0088778
0.20 ESIM 0.0213920 0.0170607 0.0144794 0.0187969 0.0431808 0.0427004
0.20 BESIM 0.0206956 0.0167130 0.0158080 0.0065181 0.0360150 0.0328985
0.30 ESIM 0.0323232 0.0316921 0.0358028 0.0304921 0.0359099 0.0416645
0.30 BESIM 0.0384871 0.0459071 0.0180494 0.0126898 0.0031730 0.0167676
0.40 ESIM 0.0368509 0.0341720 0.0431401 0.0460528 0.0293005 0.0366979
0.40 BESIM 0.0034700 0.0044148 0.0470265 0.0030763 0.0442098 0.0328749
0.50 ESIM 0.0350871 0.0415319 0.0367816 0.0481550 0.0443593 0.0085951
0.50 BESIM 0.0017297 0.0386636 0.0176232 0.0442574 0.0122571 0.0295646
0.60 ESIM 0.0133486 0.0161901 0.0031393 0.0488498 0.0079065 0.0070142
0.60 BESIM 0.0124804 0.0332511 0.0073802 0.0203862 0.0494447 0.0395940
0.70 ESIM 0.0091733 0.0226894 0.0080150 0.0022625 0.0355585 0.0243689
0.70 BESIM 0.0253354 0.0371858 0.0027166 0.0086380 0.0480610 0.0149382
0.80 ESIM 0.0363234 0.0051813 0.0354510 0.0325703 0.0204459 0.0225520
0.80 BESIM 0.0371311 0.0235138 0.0393036 0.0218888 0.0047231 0.0315689
0.90 ESIM 0.0463590 0.0071871 0.0494710 0.0213533 0.0262922 0.0169395
0.90 BESIM 0.0440110 0.0481442 0.0460268 0.0072926 0.0305382 0.0360484
Comparison of MSE results in Simulation 1 for BESIM and ESIM based on <inline-formula> <mml:math display="inline"> <mml:mn>150</mml:mn> </mml:math> </inline-formula> replications when <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>250</mml:mn> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>500</mml:mn> </mml:math> </inline-formula>.
n = 250 n = 500
τ Methods β 1 β 2 β 3 β 1 β 2 β 3
0.10 ESIM 0.0023184 0.0018601 0.0160216 0.0159654 0.0016879 0.0168082
0.10 BESIM 0.0079913 0.0161750 0.0157870 0.0195653 0.0146954 0.0055722
0.20 ESIM 0.0159583 0.0151175 0.0186634 0.0111082 0.0030300 0.0183915
0.20 BESIM 0.0044731 0.0052732 0.0043108 0.0126903 0.0101296 0.0134150
0.30 ESIM 0.0100353 0.0160354 0.0066963 0.0165764 0.0114315 0.0162483
0.30 BESIM 0.0101673 0.0017985 0.0104027 0.0147564 0.0170205 0.0065727
0.40 ESIM 0.0125452 0.0126779 0.0197187 0.0097915 0.0116293 0.0095833
0.40 BESIM 0.0098691 0.0020233 0.0189907 0.0141596 0.0117513 0.0149975
0.50 ESIM 0.0022255 0.0184194 0.0032427 0.0015221 0.0013674 0.0045875
0.50 BESIM 0.0145696 0.0016308 0.0183293 0.0165074 0.0127137 0.0040005
0.60 ESIM 0.0177884 0.0027987 0.0158476 0.0030793 0.0110199 0.0069175
0.60 BESIM 0.0044845 0.0135674 0.0072586 0.0103915 0.0072386 0.0132245
0.70 ESIM 0.0036283 0.0079414 0.0104264 0.0101376 0.0184989 0.0047943
0.70 BESIM 0.0072167 0.0035641 0.0036285 0.0013724 0.0038420 0.0131117
0.80 ESIM 0.0149326 0.0028020 0.0024689 0.0067298 0.0170474 0.0179972
0.80 BESIM 0.0091541 0.0188816 0.0172111 0.0041089 0.0141367 0.0104179
0.90 ESIM 0.0040758 0.0166996 0.0122218 0.0084545 0.0125620 0.0037598
0.90 BESIM 0.0164282 0.0095112 0.0040479 0.0149395 0.0064739 0.0199554
Summarizing estimators of <inline-formula> <mml:math display="inline"> <mml:mi>β</mml:mi> <mml:mspace width="0.25em"/> </mml:math> </inline-formula>for <inline-formula> <mml:math display="inline"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>50</mml:mn> <mml:mo>,</mml:mo> <mml:mn>150</mml:mn> </mml:math> </inline-formula> in Simulation 1. ‘BESIM <inline-formula> <mml:math display="inline"> <mml:mn>50</mml:mn> </mml:math> </inline-formula>’ denotes BESIM with <inline-formula> <mml:math display="inline"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>50</mml:mn> </mml:math> </inline-formula>, for example.

500 and all nine expectiles. MSE results show that the proposed method generally behaves better than the ESIM method in terms of the MMAD.

 Summarizing estimators of <inline-formula> <mml:math display="inline"> <mml:mi>β</mml:mi> </mml:math> </inline-formula> for <inline-formula> <mml:math display="inline"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>250</mml:mn> <mml:mo>,</mml:mo> <mml:mn>500</mml:mn> </mml:math> </inline-formula> in Simulation 1. ‘BESIM <inline-formula> <mml:math display="inline"> <mml:mn>250</mml:mn> </mml:math> </inline-formula>’ denotes BESIM with <inline-formula> <mml:math display="inline"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>250</mml:mn> </mml:math> </inline-formula>, for example. 3.2 Simulation 2

In this simulation study, we simulate data from model (16), where β = ( 1 , 1 , 1 ) / 3 and σ ( x i ′ β ) = 1 + sin ( x i ′ β ) . Mean squared errors (MSE) of the estimates based on the 100 replications in each case are shown in Tables 3 and 4 for four sample sizes n = 50 , 150 , 250 and 500 and all nine expectiles. Again, MSE results show that the proposed method generally behaves better than the ESIM method in terms of the MSE.

Comparison of MSE results in Simulation 2 for BESIM and ESIM based on <inline-formula> <mml:math display="inline"> <mml:mn>150</mml:mn> </mml:math> </inline-formula> replications when <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>50</mml:mn> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>150</mml:mn> </mml:math> </inline-formula>.
n = 50 n = 150
τ Methods β 1 β 2 β 3 β 1 β 2 β 3
0.10 ESIM 0.0107191 0.0180942 0.0194673 0.0175731 0.0155628 0.0063909
0.10 BESIM 0.0245536 0.0102368 0.0204030 0.0240220 0.0154682 0.0035139
0.20 ESIM 0.0231226 0.0072586 0.0189763 0.0061164 0.0018900 0.0075795
0.20 BESIM 0.0127013 0.0232348 0.0072481 0.0059392 0.0160106 0.0013501
0.30 ESIM 0.0117925 0.0122501 0.0109938 0.0181383 0.0151146 0.0077859
0.30 BESIM 0.0204627 0.0240346 0.0167902 0.0168055 0.0094831 0.0111475
0.40 ESIM 0.0234465 0.0021388 0.0167361 0.0099125 0.0242583 0.0018942
0.40 BESIM 0.0037376 0.0049627 0.0183407 0.0049792 0.0217180 0.0238382
0.50 ESIM 0.0184912 0.0019595 0.0114310 0.0163468 0.0084606 0.0191856
0.50 BESIM 0.0156928 0.0132497 0.0158988 0.0136745 0.0229315 0.0227252
0.60 ESIM 0.0016549 0.0216356 0.0030284 0.0153109 0.0218318 0.0242549
0.60 BESIM 0.0073266 0.0215684 0.0159264 0.0199063 0.0069357 0.0141723
0.70 ESIM 0.0151522 0.0102854 0.0222214 0.0124963 0.0045482 0.0173132
0.70 BESIM 0.0126653 0.0068737 0.0053582 0.0053046 0.0069645 0.0223847
0.80 ESIM 0.0249352 0.0028472 0.0135214 0.0220645 0.0126694 0.0234537
0.80 BESIM 0.0152664 0.0064641 0.0151681 0.0194387 0.0052556 0.0020447
0.90 ESIM 0.0100017 0.0205785 0.0157309 0.0045215 0.0080002 0.0154013
0.90 BESIM 0.0056914 0.0147402 0.0067066 0.0078914 0.0214991 0.0227348
Comparison of MSE results in Simulation 2 for BESIM and ESIM based on <inline-formula> <mml:math display="inline"> <mml:mn>150</mml:mn> </mml:math> </inline-formula> replications when <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>250</mml:mn> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mi mathvariant="bold-italic">n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>500</mml:mn> </mml:math> </inline-formula>.
n = 250 n = 500
τ Methods β 1 β 2 β 3 β 1 β 2 β 3
0.10 ESIM 0.0035559 0.0060365 0.0092645 0.0040971 0.0023224 0.0074677
0.10 BESIM 0.0072896 0.0030640 0.0059215 0.0029716 0.0019371 0.0031230
0.20 ESIM 0.0091081 0.0018402 0.0054281 0.0089195 0.0095040 0.0096261
0.20 BESIM 0.0074059 0.0087256 0.0075169 0.0022337 0.0012525 0.0078955
0.30 ESIM 0.0099915 0.0049505 0.0088972 0.0021822 0.0085752 0.0096027
0.30 BESIM 0.0073508 0.0032648 0.0090581 0.0068663 0.0038764 0.0060716
0.40 ESIM 0.0078945 0.0029055 0.0065168 0.0087705 0.0017163 0.0084206
0.40 BESIM 0.0086904 0.0068451 0.0020872 0.0097546 0.0031230 0.0025890
0.50 ESIM 0.0071910 0.0029777 0.0058355 0.0085510 0.0016177 0.0033637
0.50 BESIM 0.0010848 0.0023306 0.0045046 0.0049204 0.0033204 0.0098460
0.60 ESIM 0.0020000 0.0027075 0.0017791 0.0035235 0.0059186 0.0024654
0.60 BESIM 0.0035538 0.0035468 0.0095305 0.0047394 0.0077648 0.0060467
0.70 ESIM 0.0077003 0.0082037 0.0055475 0.0020283 0.0044568 0.0049402
0.70 BESIM 0.0035222 0.0033758 0.0043629 0.0017790 0.0050371 0.0083322
0.80 ESIM 0.0024182 0.0021317 0.0063671 0.0056524 0.0016956 0.0046949
0.80 BESIM 0.0045966 0.0031001 0.0023693 0.0055960 0.0085606 0.0049265
0.90 ESIM 0.0097013 0.0016838 0.0098522 0.0029054 0.0098691 0.0033133
0.90 BESIM 0.0090078 0.0079991 0.0064896 0.0056303 0.0067910 0.0030292
4. Boston housing data

We examine the proposed method using the Boston Housing data (BHD). BHD was collected by 9 in a study regarding the impact of clean air on housing prices, which is available in the R package spdep. BHD contains information collected from a random sample of size 506 population census areas in the Boston city. The description of the variables is summarized in Table 5.

Description of covariates in the BHD.
Covariate Description
x 1 Crime rate per capita by city
x 2 Percentage of residential land allocated to land exceeding 25,000 square feet.
x 3 Percentage of non-retail space per city
x 4 Fictitious variable for the Charles River
x 5 Nitric oxide concentration
x 6 Average number of rooms per dwelling
x 7 Percentage of owner-occupied dwellings built before 1940
x 8 Weighted distances to five employment centers in Boston
x 9 Radial highway accessibility index
x 10 Full property tax rate per $ 10,000
x 11 Ratio of students to teachers by city
x 12 1000 ( B k − 0 .63 ) 2 where B k is the proportion of Black residents
x 13 Decline in population %
y 0 .50 quantiles of the owner-occupied dwellings
MSE based on <inline-formula> <mml:math display="inline"> <mml:mn>5</mml:mn> </mml:math> </inline-formula>-fold cross-validation results for the BHD.
Method Test error
τ = 0 .1 τ = 0 .2 τ = 0 .3 τ = 0 .4 τ = 0 .5 τ = 0 .6 τ = 0 .7 τ = 0 .8 τ = 0 .9
ESIM 0.053 0.053 0.052 0.051 0.050 0.052 0.052 0.053 0.053
BESIM 0.046 0.046 0.044 0.042 0.040 0.045 0.046 0.045 0.046

Similar as in Tables 1, 2, 3, and 4, we consider nine choices of τ = 0.1 , 0.2 , 0.3 , 0.4 , 0.5 , 0 . 6 , 0 . 7 , 0 . 8   and 0.9. we consider 5-fold cross-validation to evaluate the performance of the both approaches (ESIM and BESIM). It can be seen that the BESIM performs better than its non-Bayesian counterpart, ESIM, uniformly for all expectiles considered.

 5. Conclusion

Single-index expectile regression models provide a flexible semiparametric regression frame-work for high-dimensional covariates, and capture parameter heterogeneity and nonlinearity especially when focusing on different parts of the conditional distribution of the outcome of interest. In this paper, we introduce the Bayesian expectile regression with single-index model. A Bayesian hierarchical formulation is developed for expectile regression with single-index model (BESIM). Simulations and real data studies show that BESIM generally perform better compared with ESIM.

 Ethical considerations

This study does not involve human participants or animals, therefore ethical approval was not required.

 Data availability

The underlying data is available in Zenodo. https://doi.org/10.5281/zenodo.18479534 1

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

 Software availability

The R code supporting the findings of this study is openly available in the Zenodo repository at: https://doi.org/10.5281/zenodo.18681405. 30

This code is also accessible via GitHub at: https://github.com/zena158/Bayesian-Single-Index-Expectile-Regression/tree/v1.0.1.

 References Abdulhasan Z Alhamzawi R : Data for: Bayesian Expectile Regression with Single-Index Models (v1.0).[Data set]. Zenodo. 2026. 10.5281/zenodo.18479534 Alhamzawi R Ali HTM : The bayesian adaptive lasso regression. Math. Biosci. 2018;303:75–82. 29920251 10.1016/j.mbs.2018.06.004 Bellini F Di Bernardino E : Risk management with expectiles. Eur. J. Financ. 2017;23(6):487–506. Chen H Fan K : Tail value-at-risk-based expectiles for extreme risks and their application in distributionally robust portfolio selections. Mathematics. 2022;11(1):91. Choi T Shi JQ Wang B : A gaussian process regression approach to a single-index model. Journal of Nonparametric Statistics. 2011;23(1):21–36. Daouia A Girard S Stupfler G : Estimation of tail risk based on extreme expectiles. Journal of the Royal Statistical Society Series B: Statistical Methodology. 2018;80(2):263–292. Daouia A Girard S Stupfler G : Expecthill estimation, extreme risk and heavy tails. J. Econ. 2021;221(1):97–117. Gramacy RB Lian H : Gaussian process single-index models as emulators for computer experiments. Technometrics. 2012;54(1):30–41. Harrison D Jr Rubinfeld DL : Hedonic housing prices and the demand for clean air. J. Environ. Econ. Manag. 1978;5(1):81–102. Hofner B Boccuto L Göker M : Controlling false discoveries in high-dimensional situations: boosting with stability selection. BMC bioinformatics. 2015;16:1–17. Hu Y Gramacy RB Lian H : Bayesian quantile regression for single-index models. Stat. Comput. 2013;23:437–454. Jiang R Hu X Yu K : Single-index expectile models for estimating conditional value at risk and expected shortfall. J. Financ. Economet. 2022;20(2):345–366. Jiang R Peng Y Deng Y : Variable selection and debiased estimation for single-index expectile model. Aust. N. Z. J. Stat. 2021;63(4):658–673. Kuan C-M Yeh J-H Hsu Y-C : Assessing value at risk with care, the condi-tional autoregressive expectile models. J. Econ. 2009;150(2):261–270. Lin W Kulasekera K : Identifiability of single-index models and additive-index models. Biometrika. 2007;94(2):496–501. Maume-Deschamps V Rulli`ere D Said K : Multivariate extensions of expectiles risk measures. Dependence Modeling. 2017;5(1):20–44. Newey WK Powell JL : Asymmetric least squares estimation and testing. Econometrica. 1987;55(4):819–847. Park T Casella G : The bayesian lasso. J. Am. Stat. Assoc. 2008;103(482):681–686. Pendakur K Pendakur R : Minority earnings disparity across the distribution. Can. Public Policy. 2007;33(1):41–61. Picheny V Moss H Torossian L : Bayesian quantile and expectile optimisation. Uncertainty in Artificial Intelligence. PMLR;2022; pp.1623–1633. Spiegel E Kneib T Otto-Sobotka F : Spatio-temporal expectile regression models. Stat. Model. 2020;20(4):386–409. Sun W Ibrahim JG Zou F : Genomewide multiple-loci mapping in experi-mental crosses by iterative adaptive penalized regression. Genetics. 2010;185(1):349–359. Waldmann E Sobotka F Kneib T : Bayesian regularisation in geoadditive expectile regression. Stat. Comput. 2017;27:1539–1553. Waltrup LS Sobotka F Kneib T : Expectile and quantile regression—david and goliath? Stat. Model. 2015;15(5):433–456. Xing J-J Qian X-Y : Bayesian expectile regression with asymmetric normal distribution. Communications in Statistics-Theory and Methods. 2017;46(9):4545–4555. Xu Q Ding X Jiang C : An elastic-net penalized expectile regression with applications. J. Appl. Stat. 2021;48(12):2205–2230. Yao Q Tong H : Asymmetric least squares regression estimation: a nonparametric approach. Journal of nonparametric statistics. 1996;6(2-3):273–292. Yee TW : Vglms and vgams: an overview for applications in fisheries research. Fish. Res. 2010;101(1-2):116–126. Zhao K Lian H : Bayesian tobit quantile regression with single-index models. J. Stat. Comput. Simul. 2015;85(6):1247–1263. zinahabdulhasan-lgtm, & zena158: zena158/Bayesian-Single-Index-Expectile-Regression: Bayesian Single-Index Expectile Regression - Official Release (v1.0.1). Zenodo. 2026. 10.5281/zenodo.18681405