Mathematics (Sep 2024)

Generalizations of the Kantorovich and Wielandt Inequalities with Applications to Statistics

  • Yunzhi Zhang,
  • Xiaotian Guo,
  • Jianzhong Liu,
  • Xueping Chen

DOI
https://doi.org/10.3390/math12182860
Journal volume & issue
Vol. 12, no. 18
p. 2860

Abstract

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By utilizing the properties of positive definite matrices, mathematical expectations, and positive linear functionals in matrix space, the Kantorovich inequality and Wielandt inequality for positive definite matrices and random variables are obtained. Some novel Kantorovich type inequalities pertaining to matrix ordinary products, Hadamard products, and mathematical expectations of random variables are provided. Furthermore, several interesting unified and generalized forms of the Wielandt inequality for positive definite matrices are also studied. These derived inequalities are then exploited to establish an inequality regarding various correlation coefficients and study some applications in the relative efficiency of parameter estimation of linear statistical models.

Keywords