stylistic:style_transfer
Distance skewness

A value of skewness equal to zero does not imply that the probability distribution is symmetric. Thus there is a need for another measure of asymmetry that has this property: such a measure was introduced in 2000. It is called distance skewness and denoted by dSkew. If X is a random variable taking values in the d -dimensional Euclidean space, X has finite expectation, X ' is an independent identically distributed copy of X, and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denotes the norm in the Euclidean space, then a simple measure of asymmetry with respect to location parameter θ is dSkew (X):= 1 − E ‖ X − X ′ ‖ E ‖ X + X ′ − 2 θ ‖ if Pr (X = θ) ≠ 1 {\displaystyle \operatorname {dSkew} (X):=1-{\frac {\operatorname {E} \|X-X'\|}{\operatorname {E} \|X+X'-2\theta \|}}{\text{ if }}\Pr(X=\theta)\neq 1} and dSkew(X) := 0 for X = θ (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if X is diagonally symmetric with respect to θ (X and 2 θ − X have the same probability distribution) and equals 1 if and only if X is a constant c (c ≠ θ {\displaystyle c\neq \theta }) with probability one. Thus there is a simple consistent statistical test of diagonal symmetry based on the sample distance skewness: dSkew n (X):= 1 − ∑ i, j ‖ x i − x j ‖ ∑ i, j ‖ x i + x j − 2 θ ‖. {\displaystyle \operatorname {dSkew} _{n}(X):=1-{\frac {\sum _{i,j}\|x_{i}-x_{j}\|}{\sum _{i,j}\|x_{i}+x_{j}-2\theta \|}}.}