Kantorovich不等式的推广及其在统计中的应用

EXTENSION OF KANTOROVICH INEQUALITY WITH ITS APPLICATION IN STATISTICS

本文推广了Ｋａｎｔｏｒｏｖｉｃｈ不等式，并把所得结果应用到最小二乘估计的精度和效率，以及广义相关系数中。

Assuming that X is a p×r matrix with rank (X) =r,X'X = I and that ∑is a p×p positive definite matrix with eigenvalues λ1≥…≥λp>0,we prove in §2 two new kinds of Kantorovich type inequalities,i.e.,andfor t=1,2 , … ,r and 2r≤p ,where chi(A),i =1,2, …,r,are the real eigenvalues of symmetric p×p matrix satisfying ch1(A)≥…≥chp(A) .In §3 the application of the conclusion obtained in§2 to the precision and inefficiency of least square estimator and the mearsure of multivariate association are given.