In statistics, a variational series is a non-decreasing sequence X ( 1 ) ⩽ X ( 2 ) ⩽ ⋯ ⩽ X ( n − 1 ) ⩽ X ( n ) {\displaystyle X_{(1)}\leqslant X_{(2)}\leqslant \cdots \leqslant X_{(n-1)}\leqslant X_{(n)}} composed from an initial series of independent and identically distributed random variables X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} . The members of the variational series form order statistics, which form the basis for nonparametric statistical methods.
X ( k ) {\displaystyle X_{(k)}} is called the kth order statistic, while the values X ( 1 ) = min 1 ≤ k ≤ n X k {\displaystyle X_{(1)}=\min _{1\leq k\leq n}{X_{k}}} and X ( n ) = max 1 ≤ k ≤ n X k {\displaystyle X_{(n)}=\max _{1\leq k\leq n}{X_{k}}} (the 1st and n {\displaystyle n} th order statistics, respectively) are referred to as the extremal terms. The sample range is given by R n = X ( n ) − X ( 1 ) {\displaystyle R_{n}=X_{(n)}-X_{(1)}} , and the sample median by X ( m + 1 ) {\displaystyle X_{(m+1)}} when n = 2 m + 1 {\displaystyle n=2m+1} is odd and ( X ( m + 1 ) + X ( m ) ) / 2 {\displaystyle (X_{(m+1)}+X_{(m)})/2} when n = 2 m {\displaystyle n=2m} is even. The variational series serves to construct the empirical distribution function F ^ ( x ) = μ ( x ) / n {\displaystyle {\hat {F}}(x)=\mu (x)/n} , where μ ( x ) {\displaystyle \mu (x)} is the number of members of the series which are less than x {\displaystyle x} . The empirical distribution F ^ ( x ) {\displaystyle {\hat {F}}(x)} serves as an estimate of the true distribution F ( x ) {\displaystyle F(x)} of the random variables X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} , and according to the Glivenko–Cantelli theorem converges almost surely to F ( x ) {\displaystyle F(x)} .
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