极值的r阶矩与底分布的关系

Relation Between the rth-Order Moment of Extreme value and the Base Distribution Function

设X_n^(1)≤X_2^(2)≤…≤X_n^(n)是n个具有公共分布函数共场所F(X)的独立随机变量的顺序统计量,而Y_n^(1)≤Y_n^(2)≤…≤Y_n^(n)是n个具有公共分布函数G(x)的独立随机变量的顺序统计量,0≤r≤1,integral from (-∞) to (+∞)|x|~r(dF(x))<+∞,integral from (-∞) to (+∞) |x|~r(dG(x))<+∞, 在0<r<1时,本文首次给出了对任一n≥1,都有E(|x_n^(n)|~r)=E(|Y_n^(n)|~r)的充分条件,并证明了条件“对任一n≥1,有E(|X_n^(n)|~r)=E(|Y_n^(n)|~r)”与条件“对任一n≥1,存在k_n:1≤k_n≤n,使E(|X_n^(k_n)|~r)=E(|Y_n^(k_n)|~r)”的等价性.文中证明“对任一n≥1,有 E(|X_n^(n)|~r)=E(|Y_n^(n)|~r)的必要条件是F(x)≡G(x)的方法弥补[1]中的不足,应用本文结论还可改进有关文献的结果.

Let Xa(1)≤Xa(2)≤…≤Xa(n)be an ordered statistic of size n from a common distribution function F(x):Ya(1)≤Ya(2)≤…≤Ya(x) be an ordered statintio of size n from a sommon distributionfunction G(x) on the independent random variables, 0 < r≤1, In this paper we can get a necessary and sufficient condition of E(|X | )=E( ) for every positive integer n and prove that there is kn for every positive integer:≤n, so E( ) = E( ) only if E( ) = E( ) for every positive integer n.