Editing Skewness (section)

===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.<ref>Szekely, G.J. (2000).  "Pre-limit and post-limit theorems for statistics", In: ''Statistics for the 21st Century'' (eds. [[C. R. Rao]] and G. J. Szekely), Dekker, New York, pp. 411–422.</ref> It is called '''distance skewness''' and denoted by {{math|dSkew}}. If ''X'' is a random variable taking values in the {{mvar|d}}-dimensional Euclidean space, {{mvar|X}} has finite expectation, {{math|''X''{{'}}}} is an independent identically distributed copy of {{mvar|X}}, and <math>\|\cdot\|</math> denotes the norm in the Euclidean space, then a simple ''measure of asymmetry'' with respect to location parameter {{mvar|θ}} is
<math display="block">
\operatorname{dSkew}(X) := 1 - \frac{\operatorname{E}\|X-X'\|}{\operatorname{E}\|X+X'-2 \theta\|} \text{ if } \Pr(X=\theta)\ne 1
</math>
and {{math|1=dSkew(''X'') := 0}} for {{math|1=''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 {{mvar|θ}} ({{mvar|X}} and {{math|2''θ'' − ''X''}} have the same probability distribution) and equals 1 if and only if ''X'' is a constant ''c'' (<math>c \neq \theta</math>) with probability one.<ref>Szekely,&nbsp;G.&nbsp;J. and Mori,&nbsp;T.&nbsp;F. (2001) "A characteristic measure of asymmetry and its application for testing diagonal symmetry", ''Communications in Statistics – Theory and Methods'' 30/8&9, 1633–1639.</ref> Thus there is a simple consistent [[statistical test]] of diagonal symmetry based on the '''sample distance skewness''':
<math display="block">
\operatorname{dSkew}_n(X) := 1 - \frac{\sum_{i,j} \|x_i-x_j\| }{\sum_{i,j} \|x_i+x_j-2\theta \|}.
</math>