Expectile
In the mathematical theory of probability, the expectiles of a probability distribution are related to the expected value of the distribution in a way analogous to that in which the quantiles of the distribution are related to the median. For τ ∈ ( 0 , 1 ) {\textstyle \tau \in (0,1)} , the expectile t {\textstyle t} at level τ {\textstyle \tau } of the probability distribution with cumulative distribution function F {\textstyle F} is uniquely 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 ) .