J. oflnequal. & Appl., 2002, Vol. 7(2), pp. 169-177 Taylor & Francis Taylor & Francis Group Two Kantorovich-type Inequalities and Efficiency Comparisons Between the OLSE and BLUE SHUANGZHE LIU a,. and MAXWELL L. KING b,t aSchool of Mathematical Sciences, Australian National University, Canberra ACT 02.00, Australia; bDepartment of Econometrics and Business Statistics, Monash University, Clayton, Victoria 3168, Australia (Received 18 February 2000; In final form 3 July 2000) We first establish two matrix-determinant Kantorovich-type inequalities. Then we in- troduce two efficiency criteria and make efficiency comparisons between the ordinary least squares estimator and best linear unbiased estimator in linear models. Keywords: General linear model; Efficiency comparisons; Matrix differential; Determi- nant; Trace AMS Subject Classifications: 15A45, 62J05 1. INTRODUCTION Based on matrix-trace (and vector) Kantorovich-type inequalities, Bloomfield and Watson (1975); Styan (1983) and Rao (1985) have studied efficiency comparisons between the ordinary least squares estimator (OLSE) and best linear unbiased estimator (BLUE) in linear models. In addition, Hillicr and King (1987); Liu (1995, 2000); Rao and Rao (1998); Liu and Ncudcckcr (1999) and Alpargu and Styan (2000) have reported further studies and statistical applications of Kantorovich-typc inequalities. * Corresponding author, e-mail: shuangzhe.liu@maths.anu.edu.au e-mail: Max.King@BusEco.monash.edu.au ISSN: 1025-5834. Online ISSN: 1029-242X. (C) 2002 Taylor & Francis Ltd DOI: 10.1080/10255830290019753