Assessing circular error probable when the errors are elliptical normal

An important problem of continuing interest to engineers is the need to assess the circular error probable (CEP), a measure of the impact accuracy of a projectile or a measure of GPS point positioning accuracy. One of the challenges in addressing this problem is to construct some accurate confidence bounds or intervals for CEP in the small sample settings, where certain amount of systematic biases exist in testing experiments. Currently there is no general method available to deal with this challenge due to the intractability of the distributions of the existing CEP estimators. In this paper, in order to meet this challenge, several new approximate formulas are derived for calculating CEP, which are more accurate than the existing ones but still simple to use. Both exact and empirical expressions for the derivatives of CEP with respect to the population means and variances are also given. Using these formulas, three kinds of confidence bounds or intervals for CEP are proposed, which are based on the parametric bootstrap, the asymptotic distribution, and the Cornish-Fisher expansion, respectively. Moreover, a bias-corrected estimator of CEP is provided. The performances of these procedures are evaluated based on some Monte Carlo simulation studies. Both the theoretical and simulation results show that the Cornish-Fisher expansion-based procedure performs slightly better than the other two procedures when the downrange and cross-range variances are assumed the same. However, when these two variances are different, the simulation demonstrates that the bootstrap approach can be superior to the Cornish-Fisher for the small samples (say n = 10), and vice versa for the moderate samples (say n = 20).