Simple and Accurate Approximations for Computing Covariance Matrices of Gamma and Weibull Order Statistics

Abstract

In this article, we propose a simple technique that can be used to calculate accurate approximations for the covariances of Weibull and/or Gamma order statistics, given knowledge of the corresponding means and variances of these order statistics. Ratio formulas for relating the covariances and variances of the order statistics for these two probability distribution functions (pdf's) are intuitively derived from a first-order Mosteller approximation, and then refined using the Blom adjustment technique. The necessary Blom coefficients are in turn found by optimizing the well-known recursive formula for mixed moments using standard nonlinear estimation techniques. Analytical and simulation results confirm that very good approximations to the covariance matrices of Gamma and Weibull order statistics can be obtained for a wide range of shape parameters using this methodology. Four examples are provided which demonstrate how this approach can be used to derive accurate o-BLU weights, precise variance approximations for linear functions of order statistics, and precise power calculations for the Gini test statistic.

Keywords:

• Blom technique
• o-BLU estimation
• Moment approximations
• Recursion relations

Mathematics Subject Classification:

• 62G30 (Order statistics; Empirical distribution functions)
• 62E17 (Approximations to distributions; Non-asymptotic)

Notes

Notes (§): Gamma(r = 0.5) correlations estimated using 1 million simulation runs; variance estimates calculated via Sec. 2.1 numerical techniques, using a grid size of 100,000 points.