CEP literature

The literature on the circular error probable (CEP) is extensive and diverse: Applications for CEP are found in areas such as target shooting, missile ballistics, or positional accuracy of navigation and guidance systems like GPS. The statistical foundation also has applications in (wireless) signal processing.

Groups of CEP publications

The following list is by no means intended to be complete. In particular, it contains no references to the 3D-generalization to the spherical error probable (SEP). Beware that the quality of the listed publications is mixed. The relevant publications may be roughly categorized into four groups:

• Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 (RAND Corporation, 1952), modified RAND-234 (Pesapane & Irvine, 1977), Ethridge (1983; Hogg, 1967), Grubbs (1964), Rayleigh (Grubbs, 1944; Singh, 1992). Some articles focus on the confidence intervals for CEP (DiDonato, 2007; Sathe, Joshi, & Nabar, 1991; Taub & Thomas, 1983b; Zhang & An, 2012).
• Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios (Blischke & Halpin, 1966; Elder, 1986; Kamat, 1962; McMillan & McMillan, 2008; Moranda, 1959, 1960; Puhek, 1992; Tongue, 1993; Taub & Thomas, 1983a; Wang, Jia, Yang, & Wang, 2013; Wang, Yang, Jia, & Wang, 2013; Wang, Yang, Yan, Wang, & Song, 2014; Williams, 1997).
• Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle (Chew & Boyce, 1962; Greenwalt & Shultz, 1962; Harter, 1960; Hoover, 1984; Hoyt, 1947). The distribution of the radius is known as the Hoyt (1947) distribution. The closed form expression for its cumulative distribution function has only recently been identified as the symmetric difference between two Marcum Q-functions (Paris, 2009). The latter are special cases of the non-central \(\chi^2\)-distribution (Nuttall, 1975).
• A number of articles present algorithms for the efficient numerical calculation of the Hoyt cumulative distribution function (cdf), as well as for its inverse, the quantile function (DiDonato, 2004, 2007; Pyati, 1993; Shnidman, 1995). Note that different techniques may be applicable following the identification of the closed form expression for the cdf by Paris (2009).

List of references

• Blischke, W. R., & Halpin, A. H. (1966). Asymptotic properties of some estimators of quantiles of circular error. Journal of the American Statistical Association, 61 (315), 618-632. http://www.jstor.org/stable/2282775
• Chew, V., & Boyce, R. (1962). Distribution of radial error in bivariate elliptical normal distributions. Technometrics, 4 (1), 138–140. http://www.jstor.org/stable/1266181
• DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368
• DiDonato, A. (2007). Computation of the Circular Error Probable (CEP) and Confidence Intervals in Bombing Tests (Tech. Rep. No. NSWCDD/TR-07/13). Dahlgren, VA: Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368
• Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA172498
• Ethridge, R. A. (1983). Robust estimation of circular error probable for small samples (Tech. Rep. No. ACSC 83-0690). Maxwell AFB, AL: Air Command and Staff College.
• Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: Aeronautical Chart & Information Center. http://earth-info.nga.mil/GandG/publications/tr96.pdf
• Grubbs, F. E. (1944). On the distribution of the radial standard deviation. Annals of Mathematical Statistics, 15 (1), 75–81. https://projecteuclid.org/euclid.aoms/1177731316
• Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752
• Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595
• Hogg, R. V. (1967). Some observations on robust estimation. Journal of the American Statistical Association, 62 (320), 1179–1186. http://www.jstor.org/stable/2283768
• Hoover, W. E. (1984). Algorithms for confidence circles, and ellipses (Tech. Rep. No. NOAA TR NOS 107 C&GS 3). Rockville, MD: National Oceanic and Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf
• Hoyt, R. S. (1947). Probability functions for the modulus and angle of the normal complex variate. Bell System Technical Journal, 26 (2), 318–359. http://www3.alcatel-lucent.com/bstj/vol26-1947/articles/bstj26-2-318.pdf
• Kamat, A. R. (1962). Some more estimates of circular probable error. Journal of the American Statistical Association, 57 (297), 191–195. http://www.jstor.org/stable/2282450
• McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/
• Moranda, P. B. (1959). Comparison of estimates of circular probable error. Journal of the American Statistical Association, 54 (288), 794–780. http://www.jstor.org/stable/2282503
• Moranda, P. B. (1960). Effect of bias on estimates of the circular probable error. Journal of the American Statistical Association, 55 (292), 732–735. http://www.jstor.org/stable/2281596
• Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.
• Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. Erratum: http://dx.doi.org/10.1049/el.2009.0828
• Pesapane, J., & Irvine, R. B. (1977). Derivation of CEP formula to approximate RAND-234 tables (Tech. Rep.). Offut AFB, NE: Ballistic Missile Evaluation, HQ SAC.
• Puhek, P. (1992). Sensitivity analysis of circular error probable approximation techniques (Tech. Rep. No. AFIT/GOR/ENS/92M-23). Wright-Patterson AFB, OH: Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA248105
• Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.
• RAND Corporation. (1952). Offset circle probabilities (Tech. Rep. No. RAND-234). Santa Monica, CA: RAND Corporation. http://www.rand.org/pubs/reports/2008/R234.pdf
• Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. Naval Research Logistics (NRL), 38 (1), 33–40.
• Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.
• Singh, H. P. (1992). Estimation of Circular Probable Error. The Indian Journal of Statistics, Series B, 54 (3), 289–305. http://www.jstor.org/stable/25052751
• Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828
• Taub, A. E., & Thomas, M. A. (1983b). Confidence Intervals for CEP When the Errors are Elliptical Normal (Tech. Rep. No. NSWC/TR-83-205). Dahlgren, VA: Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA153828
• Tongue, W. L. (1993). An empirical evaluation of five circular error probable estimation techniques and a method for improving them (Tech. Rep. No. AFIT/GST/ENS/93M-13). Wright-Patterson AFB, OH: Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA266528
• Wang, Y., Jia, X. R., Yang, G., & Wang, Y. M. (2013). Comprehensive CEP evaluation method for calculating positioning precision of navigation systems. Applied Mechanics and Materials, 341–342, 955–960.
• Wang, Y., Yang, G., Jia, X. R., & Wang, Y. M. (2013). Comprehensive TCEP assessment of methods for calculating MUAV navigation position accuracy based on visual measurement. Advanced Materials Research, 765–767, 2224–2228.
• Wang, Y., Yang, G., Yan, D., Wang, Y. M., & Song, X. (2014). Comprehensive assessment algorithm for calculating CEP of positioning accuracy. Measurement, 47 (January), 255–263.
• Williams, C. E. (1997). A comparison of circular error probable estimators for small samples (Tech. Rep. No. AFIT/GOA/ENS/97M-14). Wright-Patterson AFB, OH: Air Force Institute of Technology. http://handle.dtic.mil/100.2/ADA324337
• Zhang, J., & An, W. (2012). Assessing circular error probable when the errors are elliptical normal. Journal of Statistical Computation and Simulation, 82 (4), 565–586.