High quantile regression for extreme events
Abstract
For extreme events, estimation of high conditional quantiles for heavy tailed distributions is an important problem. Quantile regression is a useful method in this field with many applications. Quantile regression uses an L _{1}loss function, and an optimal solution by means of linear programming. In this paper, we propose a weighted quantile regression method. Monte Carlo simulations are performed to compare the proposed method with existing methods for estimating high conditional quantiles. We also investigate two realworld examples by using the proposed weighted method. The Monte Carlo simulation and two realworld examples show the proposed method is an improvement of the existing method.
Keywords
Bivariate Pareto distribution Conditional quantile Extreme value distribution Generalized Pareto distribution Linear programming Weighted loss functionAMS 2010 Subject Classifications
primary: 62G32 secondary: 62J05Introduction
Extreme value events are highly unusual events that can cause severe harm to people and costly damage to the environment. Examples of such harmful events are stock market crashes, equity risks, pipeline failures, large flooding, wildfires, pollution and hurricanes. The response variable, y, of an extreme event is usually distributed according to a heavytailed distribution. It is important to estimate high conditional quantiles of a random variable y given a variable vector x=(1,x _{1},x _{2},…,x _{ k })^{ T }∈R ^{ p } and p=k+1.
that is, \(\widehat {\boldsymbol {\beta }}_{LS}\) is obtained by minimizing the L _{2}distance.
The mean linear regression provides the mean relationship between a response variable and explanatory variables (Yu et al. 2003). However, there are limitations present in the conditional mean models. Outliers significantly affect the conditional mean models and as a result, it affects the measurement of the central location, which may be misleading. Also, when analyzing extreme value events, where the response variable y has a heavytailed distribution, the mean linear regression cannot be extended to noncentral locations (Hao and Naiman 2007). Therefore, it cannot provide insightful information for extreme events. Quantile regression offers a more complete statistical model by specifying the changes in the high conditional quantiles and it will be used to estimate values of extreme events (Yu et al. 2003; Hao and Naiman 2007). We will study two real world examples in the following sections.
1.1 Snowfall in Buffalo (19942015)
Large snowstorms can be very hazardous to people’s safety, communities and their properties. They can significantly reduce visibility in an area, which makes it very dangerous for densely populated areas where major car accidents can happen on the road or accidents while flying can occur. A significant amount of snow, such as 12 inches (30 cm) or more, can cave in roofs of homes and buildings, standing trees can fall down on homes and cause the loss of electricity. There have been cases of deaths due to hypothermia, infections brought on by frostbite, car accidents caused due to slippery roads, heart attacks by overexertion while shoveling heavy snow and carbon monoxide poisoning from a power outage.
The top 10 largest daily snowfalls with the maximum temperature in Buffalo, New York, from January 1994 to January 2015
Date  Snowfall (cm)  Maximum temperature (∘C) 

December 10, 1995  86.11  –9.44 
December 28, 2001  66.55  –2.78 
November 20, 2000  63.25  1.67 
December 27, 2001  55.63  –4.44 
December 24, 2001  52.07  2.22 
March 16, 2004  36.30  –2.78 
October 13, 2006  35.61  9.39 
March 12, 2014  35.10  3.89 
March 8, 2008  33.30  –2.22 
January 4, 1999  31.50  –6.67 
where y represents the total snowfall (cm) and x represents the maximum temperature (°C). The red curve represents the least squares (LS) curve \(\mu _{LS}= \widehat {\mu }_{yx}=8.98790.2144x+0.0040x^{2}\) which was obtained by using (1) and the model (2) to estimate the mean of daily snowfall y for a given maximum temperature x. But, the least squares curve does not provide information about extreme heavy snowfalls that may cause damage. The quantile regression method will be able to estimate the high conditional quantiles. We will discuss this example further in Section 5.
1.2 CO_{2} Emission
Climate change is considered to be one of the most important environmental issues as it is transforming life on Earth. It affects all aspects of our natural environment including the air and water quality, health and conservation of species at risk. It has been observed that temperatures and sea levels are rising, there are stronger storms and increased damage, and increased risk of drought, fire and floods. Climate change will rapidly alter the lands and waters that we depend upon for survival and we will no longer be able to preserve our environment for our social and economic wellbeing.
The top 10 countries with the largest CO_{2} emissions per capita with their GDP and Electricity Consumption (E.C.) in 2010
Country  CO_{2}Emission  GDP  Electricity 

per Capita  per Capita  Consumption (E.C.)  
(tonnes)  (US $)  per Capita (kilowatts)  
Qatar  40.31  71,510.16  86.01 
Trinidad and Tobago  38.16  15,630.05  1657.02 
Kuwait  31.32  38,584.48  913.04 
Brunei  22.87  30,880.34  239.40 
Aruba  22.85  24,289.14  3,262.30 
Luxembourg  21.36  102,856.97  2751.26 
Oman  20.41  20,922.66  1562.59 
United Arab Emitrates  19.85  33,885.93  9007.35 
Bahrain  19.34  20,545.97  10,142.73 
United States  17.56  48,377.39  7588.42 
Since the mean regression provides only the mean relationship between CO _{2} emission per capita and GDP or E.C., it cannot provide estimation for high conditional quantiles of CO_{2} emission. But the quantile regression method can estimate high CO_{2} emission quantile curves. We will discuss this example further in Section 5.
1.3 Main methods and results
 1.
The theoretical approach will be investigated.
 2.
Monte Carlo simulations will be performed to show the efficiency of the new weighted method relative to the existing methods.
 3.
The new proposed method will be applied to realworld examples on extreme events and compared to mean regression and classical quantile regression.
In Section 2, we review some notation. In Section 3, we propose an optimal weighted quantile regression method, and give its good asymptotic properties for any uniformly bounded positive weight independent of response variable y, with conditional density as the weight. In Section 4, the results of Monte Carlo simulations generated from the bivariate Pareto distribution show that the proposed weighted method produces high efficiencies relative to existing methods. In Section 5, the three regression methods: mean regression, classic quantile regression and proposed weighted quantile regression, are applied to the reallife examples: the Buffalo snowfall (Subsection 1.1) and CO_{2} emission (Subsection 1.2). Three goodnessoffit tests are used to assess the distributions of the data. Studies of the examples illustrate that the proposed weighted quantile regression model fits better to the datasets than the existing quantile regression method.
Notation
Pickands (1975) first introduced the Generalized Pareto Distribution (GPD). (Also see de Haan and Ferreira 2006).
Definition 2.1
Definition 2.2
Definition 2.3
Proposed weighted quantile regression
3.1 Proposed weighted quantile regression
where w _{ i }(x _{ i },τ) is any uniformly bounded positive weight function independent of y _{ i }, i=1,…,n, for x _{ i }=(1,x _{ i1},x _{ i2},…,x _{ ik })^{ T }.
where f _{ i }(ξ _{ i }(τ,x _{ i })) is uniformly bounded at the quantile points ξ _{ i }(τ,,x _{ i }).
 (1)
Koenker (2005, Chapter 5, Subsection 5.3) suggested that when the conditional densities of the response are heterogeneous, it is natural to consider whether weighted quantile regression might lead to efficiency improvements.
 (2)
The error function \(\rho _{\tau }(y_{i}\mathbf {x}_{i}^{T}\boldsymbol {\beta } (\tau))\) in (6)is an absolute error measure between y _{ i } and the τth conditional quantile ξ _{ i }(τ,x _{ i }) at the i th sample point (y _{ i },x _{ i }), i=1,2,…,n. f _{ i }(ξ _{ i }(τ,x _{ i })) can be interpreted as providing the local relative likelihood that the response varaibale y takes values in a neighborhood of y=ξ _{ i }(τ,x _{ i }):(ξ _{ i }(τ,x _{ i })−ε, ξ _{ i }(τ,x _{ i })+ε), for small ε>0. Giving the weight f _{ i }(ξ _{ i }(τ,x _{ i })) on \(\rho _{\tau }(y_{i}\mathbf {x}_{i}^{T} \boldsymbol {\beta }(\tau))\) will make the total error \(\sum _{i=1}^{n}w_{i}(\mathbf {x}_{i},\tau)\rho _{\tau }(y_{i}\mathbf {x}_{i}^{T}\boldsymbol {\beta } (\tau))\) more reasonable.
 (3)
The weighted estimator \(\widehat {\boldsymbol {\beta }}_{w}(\tau)\) in (6)using weight (7)has good properties, which we will discuss in Subsection 3.2 below.
 (4)
Is weight (7) an optimal weight? It is a difficult problem in the field as described in Chapter 5, Subsection 5.3 in Koenker (2005). Also, f _{ i }(ξ _{ i }(τ,x _{ i })) is difficult to estimate. In this paper, we explore these two difficulties, we estimate the f _{ i }(ξ _{ i }(τ,x _{ i })), then obtain positive results by using weight (7).
where w _{ i }(x _{ i },τ)∈ [ 0,1] and \( \sum \limits _{i=1}^{n}w_{i}(\mathbf {x}_{i},\tau)=1,\;i=1,\ldots,n, \left \vert \left \vert \mathbf {x}_{i}\right \vert \right \vert =\sqrt { x_{i1}^{2}+x_{i2}^{2}+\cdot \cdot \cdot +x_{ik}^{2}},\) k is the number of regressors.
In this paper, we are looking for improvement of efficiency by using weights (7) in Section 4 simulations, and applications of the Buffalo snowfall and CO_{2} emission examples in Section 5.
3.2 Properties of weighted quantile regression
The following regularity conditions are necessary in deriving the asymptotic distribution of \(\widehat {\boldsymbol {\beta }}_{n(w)}(\tau)\) in (6)with weight w _{ i }(x _{ i },τ)=f _{ i }(ξ _{ i }(τ,x _{ i })), i=1,2,…, 0<τ<1, in (7). Let Y _{1},Y _{2},… be independent random variables with distribution function F _{1},F _{2},….
Condition 1
(C1). The F _{ i }’s are absolutely continuous, with continuous densities f _{ i }(ξ) uniformly bounded away from 0 and ∞ at the quantile points ξ _{ i }(τ,x _{ i }),i=1,2,….
Condition 2

\({\lim }_{n\rightarrow \infty }n^{1}\sum f_{i}^{2}(\xi _{i}(\tau, \mathbf {x}_{i}))\mathbf {x}_{i}\mathbf {x}_{i}^{T}=\mathbf {D}_{0}(\tau),\) and

\(\lim \limits _{i=1,..,n}\left \Vert f_{i}(\xi _{i}(\tau,\mathbf {x} _{i}))\right \Vert \)/\(\sqrt {n}\rightarrow 0.\)
We have the main asymptotic results for \(\widehat {\mathbf {\beta }} _{n(w)}(\tau)\) in (6) using weight (7). In this case, we let \(\widehat {\boldsymbol {\beta }}_{n(w)}(\tau)=\widehat {\boldsymbol {\beta }}_{n(f)}(\tau)\) in the following theorem.
Theorem 3.1
The proof of Theorem 3.1 is similar as the proof has been provided in Huang et al. (2015).
3.3 Comparison of quantile regression models
Simulations
 The regular quantile regression Q _{ R }(τx) estimation based on (5),$$ Q_{R}(\tau x)=\widehat{\beta }_{0}(\tau)+\widehat{\beta }_{1}(\tau)x $$(13)
 The weighted quantile regression Q _{ W }(τx) estimation based on (6)$$ Q_{W}(\tau x)=\widehat{\beta }_{w0}(\tau)+\widehat{\beta }_{w1}(\tau)x. $$(14)For each method, we generate size n=300,m=1,000 samples. Q _{ R,i }(τx) or Q _{ W,i }(τx), i=1,…m, are estimated in the i th sample. Let α=3 in (12), then the weights in (7) are$$ w_{i}(\mathbf{x}_{i},\tau)=\frac{4(1\tau)^{(5/4)}}{x_{i}},\;x_{i}>1,\;i=1,2,\ldots,n. $$(15)
Simulation mean square errors (SMSE) and efficiencies (SEFF) of estimating Q _{ y }(τx),m=1000,n=300,N=1000
τ  0.95  0.96  0.97  0.98  0.99 

S M S E(Q _{ R }(τ))  2.0533 ×10^{8}  2.0103 ×10^{8}  2.7731 ×10^{8}  5.1199 ×10 ^{8}  1.7458 ×10^{9} 
S M S E(Q _{ W(f)}(τ))  1.4939 ×10^{8}  1.4146 ×10^{8}  2.2462 ×10^{8}  4.5520 ×10^{8}  1.7161 ×10^{9} 
S E F F(Q _{ W(f)}(τ))  1.6770  1.4211  1.2346  1.1248  1.0174 
Simulation efficiency (SEFF) of estimating Q _{ y }(τx),m=1000,n=300,N=1000
τ  0.95  0.96  0.97  0.98  0.99 

S E F F(Q _{ W(1)}(τ))  1.0783  1.0729  1.0504  1.0008  1.0086 
S E F F(Q _{ W(f)}(τ))  1.6770  1.4211  1.2346  1.1248  1.0174 

Table 3, Figs. 5 and 6 show that for τ=0.95,…,0.99, the proposed weighted regression Q _{ W(f)}(τx) with the weight (15) is more efficient relative to the regular regression Q _{ R }(τx).

Table 4 and Fig. 7 show that for τ=0.95,…,0.99,Q _{ W(f)}(τx) with the proposed weight (15)is more efficient relative to Q _{ W(1)}(τx) with \(w_{i}(\mathbf {x}_{i},\tau)=\left \Vert \mathbf {x} _{i}\right \Vert ^{1}/\sum _{i=1}^{n}\left \Vert \mathbf {x}_{i}\right \Vert ^{1}\) in (8).
Real examples of applications

The traditional mean linear regression (LS) estimator \(\widehat {\boldsymbol {\beta } } _{LS}\) in (1);

The regular quantile regression Q _{ R } estimator \(\widehat {\boldsymbol \beta }(\tau) \) in (5);

The proposed weight quantile regression Q _{ W } estimator \(\widehat {\boldsymbol \beta }_{W}(\tau)\) in(6)with weight w _{ i }(x _{ i },τ)=f _{ i }(ξ _{ i }(τ,x _{ i })) in (7).
5.1 Buffalo snowfall example
In order to fit the Buffalo snowfall data to the GPD model (4), the data was transformed to \(y=\frac {x\mu }{\sigma },\) where μ=5 cm, and then, we used the maximum likelihood estimates (MLEs) of the parameters, \(\widehat {\sigma }_{MLE}=5.1552, \widehat {\gamma }_{MLE}=0.2636,\) for the 2parameter GPD model from the Buffalo snowfall data.
where F(y) is the true but unknown distribution function of the sample and F ^{∗}(y) is the theoretical distribution function, GPD in (4).
The goodness of fit tests for Buffalo snowfall example
K−S  A−D  C−v−M  

Test statistic  pvalue  Test statistic  pvalue  Test statistic  pvalue 
0.0390  0.6055  0.6631  0.5905  0.0645  0.7856 
Buffalo daily snowfalls (cm) at high quantile using Q _{ R } and Q _{ W }
τ=0.95  τ=0.97  

Temperature (°C)  Q _{ R }  Q _{ W }  Q _{ R }  Q _{ W } 
–15  24.72  41.57  37.38  44.38 
–10  25.65  34.37  33.19  35.42 
–5  26.00  28.74  30.98  30.15 
0  25.76  24.69  30.73  28.58 
5  24.93  22.21  32.47  30.70 
10  23.52  21.30  36.17  36.52 
Relative R(τ) values for the Buffalo snowfall example
τ=0.95  τ=0.96  τ=0.97  τ=0.98  τ=0.99  

Relative R(τ)  0.1824  0.1851  0.1919  0.1673  0.0595 
Coefficients of the Q _{ R },Q _{ W } and LS μ _{ LS } regression for buffalo snowfall example
τ  Weight  \(\widehat {\beta }_{0}{ (\tau)}\)  \(\widehat {\beta }_{1}{ (\tau)}\)  \(\widehat {\beta }_{2}{(\tau)}\) 

LS  −  11.5280  –0.1777  0.0053 
0.95  Q _{ R }  25.7589  –10.1068  –0.0117 
Q _{ W }  24.6887  –0.6538  0.0315  
0.96  Q _{ R }  28.7869  0.1543  0.0610 
Q _{ W }  24.8614  –0.7200  0.0352  
0.97  Q _{ R }  30.7341  0.1488  0.0395 
Q _{ W }  28.5776  0.0551  0.0739  
0.98  Q _{ R }  35.5582  –0.2039  0.0223 
Q _{ W }  25.8718  –2.6593  0.3937  
0.99  Q _{ R }  57.8614  –2.6793  0.0330 
Q _{ R }  48.5464  0.5261  0.4768 
The proposed weighted quantile regression Q _{ W } predicts that for moderate temperatures, such as 5 °C to 10 °C, it is likely to have small snowfalls in Buffalo, and for every low temperatures, such as −15 °C to 0 °C, it is more likely to have heavy snowfalls that may cause damage. Predicting heavy snowfalls is related to cold weather forecasts. Quantile regression is useful for predicting extreme heavy snowfalls.
5.2 CO_{2} emission example
The quantile regression method can estimate high conditional quantile curves and will be shown in detail in this Section.
The goodness of fit tests for CO_{2} emission example
K−S  A−D  C−v−M  

Test statistic  pvalue  Test statistic  pvalue  Test statistic  pvalue 
0.0443  0.8397  0.3619  0.8855  0.0517  0.8662 
CO_{2} emission per capita high quantile given ln(GDP) estimators Q _{ R } and Q _{ W } at 2980.96 Kilowatts of electricity consumed per capita
τ=0.97  

ln of GDP per capita ($)  Q _{ R }  Q _{ W } 
7.5  15.2181  13.0840 
8  18.0437  15.6591 
8.5  20.8693  18.2342 
9  23.6950  20.9093 
9.5  26.5206  23.3844 
10  29.3462  25.9595 
10.5  32.1718  28.5346 
11  34.9975  31.1097 
11.5  37.8231  33.5848 
12  40.6487  36.2599 
CO_{2} Emission per capita high quantile given ln(E.C.) estimators Q _{ R } and Q _{ W } at GDP per capita of $13,359.73
ln of Electricity Consumption per capita (kilowatts)  τ=0.97  

Q _{ R }  Q _{ W }  
0  6.9775  11.4376 
2  11.8632  14.4243 
4  16.7490  17.4110 
6  21.6348  20.3977 
8  26.5206  23.3844 
10  31.4064  26.3711 
12  36.2921  29.3578 
Relative R(τ) values for CO_{2} emission example
τ=0.95  τ=0.96  τ=0.97  τ=0.98  τ=0.99  

Relative R(τ)  0.24425  0.21478  0.16810  0.12459  0.06521 
Coefficients of the Q _{ R },Q _{ W } and LS μ _{ LS } regression for CO_{2} emission example
τ  Weight  \(\widehat {\beta }_{0}{ (\tau)}\)  \(\widehat {\beta }_{1}{ (\tau)}\)  \(\widehat {\beta }_{2}{(\tau)}\) 

LS  −  –22.5009  2.0708  1.2998 
0.95  Q _{ R }  –41.6856  5.8924  0.5527 
Q _{ W }  –29.9131  5.1521  –0.2094  
0.96  Q _{ R }  –44.8147  5.4258  1.9505 
Q _{ W }  –24.5504  4.4470  0.1609  
0.97  Q _{ R }  –46.7095  5.6513  2.4429 
Q _{ W }  –37.4893  5.1502  1.4934  
0.98  Q _{ R }  –47.4004  5.7323  2.4739 
Q _{ W }  –47.4004  5.7323  2.4739  
0.99  Q _{ R }  –51.2657  6.1856  2.6475 
Q _{ W }  –51.2657  6.1856  2.6475 
It can be concluded that countries with higher gross domestic product and higher amounts of electricity produce higher CO_{2} emissions. Since CO _{2} is not destroyed over time, it can remain in the atmosphere for thousands of years due to the very slow process by which carbon is transferred to ocean sediments. As a result, countries should monitor their CO_{2} emissions in order to prevent further damages to the environment. Countries can consider producing more energy from renewable sources, such as wind, solar, hydro and geothermal heat and using fuels with lower carbon content to reduce carbon emissions.
Conclusions

Traditional mean regression are concerned with estimating the conditional mean by using the L_{2}loss function. Quantile regression with a L_{1} loss function overcomes the limitations of traditional mean regression. It gives estimates of τth conditional quantiles besides the measures of central tendency. Estimation of high conditional quantiles is very useful for the analysis of extreme events.

The proposed weighted quantile regression method with the local conditional density as the weight has good mathematical asymptotic properties.

The Monte Carlo computational simulation results show that the proposed weighted quantile regression with the local conditional density as the weight is more efficient relative to the classical quantile regression and some existing weighted quantile regression estimators.

The proposed weighted quantile regression can be used to predict extreme values of snowfall and CO_{2} emission real world examples. In the Buffalo snowfall example, communities can use the information that quantile regression provides to prevent car accidents on roads, overexertion, and collapsing of homes. In the CO_{2} emission example, the countries’ increase in gross domestic product and electricity consumption will likely cause an increase in the CO_{2} emissions. CO_{2} emission levels should be monitored to reduce the amount of carbon dioxide in the atmosphere and its long term effects.

It is difficult to estimate the proposed conditional density weight. The nonparametric kernel density estimation method is successful in this paper. Further studies on optimal weights are suggested.
Notes
Acknowledgements
We apprecaite for the reviewers’ and Editors’ very helpful comments, which helped to improve the paper.
This research is supported bythe Natural Science and Engineering Research Council of Canada (NSERC)grant MLH, RGPIN201404621.
Authors’ contributions
The authors MLH and CN carried out this work and drafted the manuscript together. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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