Herb References

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The following list is by no means intended to be complete. Beware that the quality of the listed items is not uniformly high. On the right, in the Table of Contents, the some of the representative publications are roughly categorized into different topics.

  • 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: U.S. Naval Surface Weapons Center Dahlgren Division. (PDF @ http://handle.dtic.mil/100.2/ADA476368)
  • DiDonato, A. R., & Jarnagin, M. P. (1961b). Integration of the general bivariate Gaussian distribution over an offset ellipse (Tech. Rep. No. NWL TR 1710). Dahlgren, VA: U.S. Naval Weapons Laboratory.
  • Elder, R. L. (1986). An examination of circular error probable approximation techniques (Tech. Rep. No. AFIT/GST/ENS/86M-6). Wright-Patterson AFB, OH: U.S. Air Force Institute of Technology. (PDF @ 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: U.S. Air Command and Staff College.
  • Leslie, John E. III (1993). Is "Group Size" the Best Measure of Accuracy? (originally published as "Is 'Group Size' the Best Measure of Accuracy?", The Canadian Marksman 129 (1), (Autumn 1994): p46-8.) (cached PDF) (Ballistipedia Notes)
  • Litz,Bryan. (2011). Applied Ballistics for Long Range Shooting. Applied Ballistics, LLC. 2nd Edition. ISBN-13: 978-0615452562
  • McCoy, Robert. (2012). Modern Exterior Ballistics, Schiffer Publishing, Ltd.; 2nd edition edition. ISBN-13: 978-0764338250.
  • 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.
  • Sathe, Y. S., Joshi, S. M., & Nabar, S. P. (1991). Bounds for circular error probabilities. U.S. 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.
  • Siddiqui, M. M. (1961). Some Problems Connected With Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Science, Vol. 68D, No. 9. (Cached PDF)
  • Siddiqui, M. M. (1964). Statistical Inference for Rayleigh Distributions. The Journal of Research of the National Bureau of Standards, Sec. D: Radio Propagation, Vol. 66D, No. 2. (Summarizes and extends Siddiqui, 1961.) (Cached PDF)
  • Spall, J. C., & Maryak, J. L. (1992). A feasible Bayesian estimator of quantiles for projectile accuracy from non-iid data. Journal of the American Statistical Association , 87 (419), 676–681. (Abstract @ http://www.jstor.org/stable/2290205)
  • Taub, A. E., & Thomas, M. A. (1983a). Comparison of CEP estimators for elliptical normal errors (Tech. Rep. No. ADP001580). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. (PDF @ 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: U.S. Naval Surface Weapons Center Dahlgren Division. (PDF @ http://handle.dtic.mil/100.2/ADA153828)
  • Thomas, M. A., Crigler, J. R., Gemmill, G. W., & Taub, A. E. (1973). Tolerance limits for the Rayleigh (radial normal) distribution with emphasis on the CEP (Tech. Rep. No. NWL TR 2946). Dahlgren, VA: U.S. Naval Weapons Laboratory. (PDF @ http://handle.dtic.mil/100.2/AD0759989)
  • 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: U.S. Air Force Institute of Technology. (PDF @ http://handle.dtic.mil/100.2/ADA266528)
  • 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. (Abstract @ http://www.scientific.net/AMR.765-767.2224)
  • 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: U.S. Air Force Institute of Technology. (PDF @ http://handle.dtic.mil/100.2/ADA324337)

Groups of Publications


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 foundations in quadratic forms of normal variables are important for analyzing the power of inference tests. The Hoyt and Rayleigh distribution have applications in (wireless) signal processing.

Develop CEP Estimator

Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 (RAND Corporation, 1952), modified RAND-234 (Pesapane & Irvine, 1977), Grubbs (1964), Rayleigh (Culpepper, 1978; Saxena & Singh, 2005; Singh, 1992), Ethridge (1983; Hogg, 1967), Spall & Maryak (1992) general bivariate normal (DiDonato & Jarnagin, 1961a; Evans, Govindarajulu, & Barthoulot, 1985). Some articles focus on the confidence intervals for CEP (DiDonato, 2007; Sathe, Joshi, & Nabar, 1991; Taub & Thomas, 1983b; Thomas, Crigler, Gemmill & Taub, 1973; Zhang & An, 2012).

Comparing CEP Estimators

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; Nelson, 1988; 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).

CEP in polar Coordinates

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 is the complement (with respect to unity) of a special case of the non-central \(\chi^2\)-distribution (Nuttall, 1975). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in Guenther & Terragno (1964).

CEP with Bias

DiDonato and Jarnagin (1961a, 1961b, 1962a, 1962b) as well as Evans et al. (1985) develop methods to use the correlated bivariate normal distribution for CEP estimation when systematic accuracy bias must be taken into account. This requires integrating the distribution over a disc that is not centered on the true mean of the shot group but on the point of aim. This so-called offset circle probability is the probability of a quadratic form of a normal variable The exact distribution of quadratic forms is a weighted average of non-central \(\chi^{2}\)-distributions and difficult to calculate without numerical tools. Therefore, the Patnaik (1949) two-moment central \(\chi^{2}\)-approximation or the Pearson (Imhof, 1961; Pearson, 1959) three-moment central \(\chi^{2}\)-approximation are often used. Recently, Liu, Tang & Zhang (2009) proposed a four-moment non-central \(\chi^{2}\)-approximation.

Hoyt Distribution Algorithms

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). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by Davies (1980), Farebrother (1984, 1990), Imhof (1961), Sheil & O'Muircheartaigh (1977). A comparison and implementation can be found in Duchesne and Lafaye de Micheaux (2010).

Spherical Error Probable

DiDonato (1988) and Singh (1962, 1970) treat the Spherical Error Probable SEP.

Extreme Spread

Monte Carlo Simulation

circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a study and the second in Naval Research Quarterly.

Sampling Problems

Through an extended forum thread Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.

Advocating Conversion From

Rayleigh Distribution

Siddiqui had a two part series on the Rayleigh distribution ( 1961 and 1964).


Sample Range

Harter (1964) gives tables for percentiles of the studentized range, as well as values for the mean and variance of samples from the studentized range.

Reference Data

  • File:Sigma1RangeStatistics.xls: Simulated median, 50%, 80%, and 95% quantiles, plus first four sample moments, for shot groups containing 2 to 100 shots, of: Extreme Spread, Diagonal, Figure of Merit.