Expectile

In the mathematical theory of <a href="/facts/Probability/fgYvMhwb">probability</a>, the expectiles of a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> are related to the <a href="/facts/Expected_value/1XV0JKL8">expected value</a> of the distribution in a way analogous to that in which the <a href="/facts/Quantile/oWr1MC3s">quantiles</a> of the distribution are related to the <a href="/facts/Median/YwXGWTyr">median</a>.
For 
 
 
 
 τ
 ∈
 (
 0
 ,
 1
 )
 
 
 {\textstyle \tau \in (0,1)}
 
, the expectile of the probability distribution with cumulative distribution function 
 
 
 
 F
 
 
 {\textstyle F}
 
 is characterized by any of the following equivalent conditions:

(
                1
                −
                τ
                )
                
                  ∫
                  
                    −
                    ∞
                  
                  
                    t
                  
                
                (
                t
                −
                x
                )
                
                d
                F
                (
                x
                )
                =
                τ
                
                  ∫
                  
                    t
                  
                  
                    ∞
                  
                
                (
                x
                −
                t
                )
                
                d
                F
                (
                x
                )
              
            
            
              
              
                
                
                  ∫
                  
                    −
                    ∞
                  
                  
                    t
                  
                
                
                  |
                
                t
                −
                x
                
                  |
                
                
                d
                F
                (
                x
                )
                =
                τ
                
                  ∫
                  
                    −
                    ∞
                  
                  
                    ∞
                  
                
                
                  |
                
                x
                −
                t
                
                  |
                
                
                d
                F
                (
                x
                )
              
            
            
              
              
                t
                −
                E
                ⁡
                [
                X
                ]
                =
                
                  
                    
                      2
                      τ
                      −
                      1
                    
                    
                      1
                      −
                      τ
                    
                  
                
                
                  ∫
                  
                    t
                  
                  
                    ∞
                  
                
                (
                x
                −
                t
                )
                
                d
                F
                (
                x
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&(1-\tau )\int _{-\infty }^{t}(t-x)\,dF(x)=\tau \int _{t}^{\infty }(x-t)\,dF(x)\\[5pt]&\int _{-\infty }^{t}|t-x|\,dF(x)=\tau \int _{-\infty }^{\infty }|x-t|\,dF(x)\\[5pt]&t-\operatorname {E} [X]={\frac {2\tau -1}{1-\tau }}\int _{t}^{\infty }(x-t)\,dF(x)\end{aligned}}}

<a href="/facts/Quantile_regression/5J05lVIx">Quantile regression</a> minimizes an asymmetric 
 
 
 
 
 L
 
 1
 
 
 
 
 {\displaystyle L_{1}}
 
 loss (see <a href="/facts/Least_absolute_deviations/fcqbHnEt">least absolute deviations</a>).
Analogously, expectile regression minimizes an asymmetric 
 
 
 
 
 L
 
 2
 
 
 
 
 {\displaystyle L_{2}}
 
 loss (see <a href="/facts/Ordinary_least_squares/q7H9k5vM">ordinary least squares</a>):

quantile
                ⁡
                (
                τ
                )
              
              
                
                ∈
                
                  argmin
                  
                    t
                    ∈
                    
                      R
                    
                  
                
                ⁡
                E
                ⁡
                [
                
                  |
                
                X
                −
                t
                
                  |
                
                
                  |
                
                τ
                −
                H
                (
                t
                −
                X
                )
                
                  |
                
                ]
              
            
            
              
                expectile
                ⁡
                (
                τ
                )
              
              
                
                ∈
                
                  argmin
                  
                    t
                    ∈
                    
                      R
                    
                  
                
                ⁡
                E
                ⁡
                [
                
                  |
                
                X
                −
                t
                
                  
                    |
                  
                  
                    2
                  
                
                
                  |
                
                τ
                −
                H
                (
                t
                −
                X
                )
                
                  |
                
                ]
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\operatorname {quantile} (\tau )&\in \operatorname {argmin} _{t\in \mathbb {R} }\operatorname {E} [|X-t||\tau -H(t-X)|]\\\operatorname {expectile} (\tau )&\in \operatorname {argmin} _{t\in \mathbb {R} }\operatorname {E} [|X-t|^{2}|\tau -H(t-X)|]\end{aligned}}}

where 
 
 
 
 H
 
 
 {\displaystyle H}
 
 is the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>.

Expectile open-in-new

Expectile