Student's T-Distribution: Improved Formulae for Moments of Order Statistics

• Activity Number: 317
• Type: Contributed
• Date/Time: Tuesday, August 11, 2015 : 8:30 AM to 10:20 AM
• Sponsor: SSC
• Author(s): David Vaughan*
• Companies: Wilfrid Laurier University
• Keywords: Student's t-distribution; order statistics; estimation

Abstract

Student's t with low degrees of freedom (including the Cauchy, t-2, and others) are used as models of non-normal but symmetric data or in robustness studies. Estimators of parameters and their efficiencies can depend on expected values, variances and covariances of order statistics (e.g., Lloyd (1952), Barnett (1966), Tiku and Suresh (1991)). Low order moments of order statistics from the Cauchy (Barnett (1966) and Vaughan (1994)), t-2 (Vaughan (1991)) and t with greater degrees of freedom (Tiku and Kumra (1981) have been tabulated, typically with restricted sample sizes. Simplified formulae for pure moments of order statistics, especially the t-2, have been developed by Jones (2002). In this work, a general framework for the efficient evaluation of pure and mixed moments is developed and applied to Student's t for low degrees of freedom, and a greater range of sample sizes.