# References

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.

• 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. Abstract @ 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. Abstract @ http://www.jstor.org/stable/1266181
• Culpepper, G. A. (1978). Statistical analysis of radial error in two dimensions (Tech. Rep.). White Sands Missile Range, NM: U.S. Army Material Test and Evaluation Directorate. PDF @ http://handle.dtic.mil/100.2/ADA059117
• Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of $$\chi^{2 }$$ random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.
• DiDonato, A. R. (1988). Integration of the trivariate normal distribution over an offset spehere and an inverse problem (Tech. Rep. No. NSWC TR 87-27). Dahlgren, VA: U.S. Naval Surface Weapons Center Dahlgren Division. http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf
• DiDonato, A. (2004). An inverse of the generalized circular error function (Tech. Rep. No. NSWCDD/TR-04/43). Dahlgren, VA: U.S. 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: U.S. Naval Surface Weapons Center Dahlgren Division. http://handle.dtic.mil/100.2/ADA476368
• DiDonato, A. R., & Jarnagin, M. P. (1961a). Integration of the general bivariate Gaussian distribution over an offset circle. Mathematics of Computation, 15 (76), 375–382. http://www.jstor.org/stable/2003026
• 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.
• DiDonato, A. R., & Jarnagin, M. P. (1962a). A method for computing the circular coverage function. Mathematics of Computation, 16 (79), 347–355. http://www.jstor.org/stable/2004054
• DiDonato, A. R., & Jarnagin, M. P. (1962b). A method for computing the generalized circular error function and the circular coverage function (Tech. Rep. No. NWL TR 1786). Dahlgren, VA: U.S. Naval Weapons Laboratory.
• Duchesne, P., & Lafaye de Micheaux, P. (2010). Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Computational Statistics and Data Analysis, 54 (4), 858–862.
• 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. 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.
• Evans, M. J., Govindarajulu, Z., & Barthoulot, J. (1985). Estimates of circular error probabilities (Tech. Rep. No. TR 367). Arlington, VA: U.S. Office of Naval Research. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257
• Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of $$\chi^{2}$$ random variables. Journal of the Royal Statistical Society, C, 33, 332–339.
• Farebrother, R. W. (1990). Algorithm AS 256: The distribution of a quadratic form in normal variables. Journal of the Royal Statistical Society, C, 39, 394–309.
• Greenwalt, C. R., & Shultz, M. E. (1962). Principles of Error Theory and Cartographic Applications (Tech. Rep. No. ACIC TR-96). St. Louis, MO: U.S. 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
• Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. private monograph.
• Guenther, W. C., & Terragno, P. J. (1964). A Review of the Literature on a Class of Coverage Problems. Annals of Mathematical Statistics 35 (1), 232-260. http://projecteuclid.org/euclid.aoms/1177703747
• Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595
• Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [1]
• 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: U.S. 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. https://archive.org/details/bstj26-2-318
• Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48 (3–4), 419–426. http://www.jstor.org/stable/2332763
• 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
• 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 copy) (Ballistipedia Notes)
• Liu, H., Tang, Y., & Zhang, H. H. (2009). A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables. Computational Statistics & Data Analysis, 53 , 853–856.
• 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
• Nelson, W. (1988). Use of circular error probability in target detection (Tech. Rep. Nos. ESD-TR-88-109, MTR-10293). Bedford, MA: MITRE Corporation. http://handle.dtic.mil/100.2/ADA199190
• 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
• Patnaik, P. B. (1949). The non-central $$\chi^{2}$$- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542
• Pearson, E. S. (1959). Note on an approximation to the distribution of non-central $$\chi^{2}$$. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533
• 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: U.S. 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. U.S. Naval Research Logistics (NRL), 38 (1), 33–40.
• Saxena, S., & Singh, H. P. (2005). Some estimators of the dispersion parameter of a chi-distributed radial error with applications to target analysis. Austrial Journal of Statistics, 34 (1), 51–63. http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf
• Sheil, J., & O’Muircheartaigh, I. (1977). Algorithm as 106. The distribution of non-negative quadratic forms in normal variables. Applied Statistics, 26 (1), 92–98.
• 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
• Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html
• Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.
• 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. URL 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. 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. http://handle.dtic.mil/100.2/ADA153828
• Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf
• Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Journal of Naval Research Logistics Quarterly 22 (4), 1713-1719. http://dx.doi.org/10.1002/nav.3800220407
• 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. URL 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. 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: U.S. 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. http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797

# Groups of Publications

## CEP

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.

### 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.