Difference between revisions of "Herb References"

From ShotStat
Jump to: navigation, search
 
(16 intermediate revisions by the same user not shown)
Line 4: Line 4:
 
= References =
 
= References =
  
* <div id="Blischke1966"></div>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
+
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.
* <div id="Chew1962"></div>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
+
 
* <div id="Culpepper1978"></div>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. http://handle.dtic.mil/100.2/ADA059117
+
* <div id="Blischke1966"></div>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| (Abstract @ http://www.jstor.org/stable/2282775)]
 +
 
 +
* <div id="Chew1962"></div>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 (Abstract @ http://www.jstor.org/stable/1266181)]
 +
 
 +
* <div id="Culpepper1978"></div>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. [http://handle.dtic.mil/100.2/ADA059117 (PDF @ http://handle.dtic.mil/100.2/ADA059117)]
 +
 
 
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2
 
* <div id="Davies1980"></div>Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of <math>\chi^{2
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333.
+
}</math> random variables. Journal of the Royal Statistical Society, C , 29 , 323–333. [http://www.jstor.org/stable/2346911?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2346911?seq=1#page_scan_tab_contents)] [http://www.robertnz.net/pdf/lc_chisq.pdf (PDF @ http://www.robertnz.net/pdf/lc_chisq.pdf)]
* <div id="DiDonato1988"></div>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
+
 
* <div id="DiDonato2004"></div>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
+
* <div id="DiDonato1988"></div>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 (PDF @ http://www.dtic.mil/dtic/tr/fulltext/u2/a198129.pdf)]
* <div id="DiDonato2007"></div>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
+
 
* <div id="DiDonato1961a"></div>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
+
* <div id="DiDonato2004"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA476368)]
 +
 
 +
* <div id="DiDonato2007"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA476368)]
 +
 
 +
* <div id="DiDonato1961a"></div>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 (READ @ http://www.jstor.org/stable/2003026)] [http://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0129116-8/ (PDF @ http://www.ams.org/journals/mcom/1961-15-076/S0025-5718-1961-0129116-8/)]
 +
 
 
* <div id="DiDonato1961b"></div>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.
 
* <div id="DiDonato1961b"></div>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.
* <div id="DiDonato1962a"></div>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
+
 
* <div id="DiDonato1962b"></div>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.
+
* <div id="DiDonato1962a"></div>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 (READ @ http://www.jstor.org/stable/2004054)]
* <div id="Duchesne2010"></div>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.
+
 
* <div id="Elder1986"></div>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
+
* <div id="DiDonato1962b"></div>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. [http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0270739 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=AD0270739)]
 +
 
 +
* <div id="Duchesne2010"></div>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. [http://www.sciencedirect.com/science/article/pii/S0167947309004381 (Abstract @ http://www.sciencedirect.com/science/article/pii/S0167947309004381)]
 +
 
 +
* <div id="Elder1986"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA172498)]
 +
 
 
* <div id="Ethridge1983"></div>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.
 
* <div id="Ethridge1983"></div>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.
* <div id="Evans1985"></div>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
+
 
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339.
+
* <div id="Evans1985"></div>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 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA163257)]
* <div id="Farebrother1990"></div>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.
+
 
* <div id="Greenwalt1962"></div>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
+
* <div id="Farebrother1984"></div>Farebrother, R. W. (1984). Algorithm AS 204: The distribution of a positive linear combination of <math>\chi^{2}</math> random variables. Journal of the Royal Statistical Society, C, 33, 332–339. [http://www.jstor.org/stable/2347721?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2347721?seq=1#page_scan_tab_contents)] [http://www.robertnz.net/pdf/lc_chisq.pdf (PDF @ http://www.robertnz.net/pdf/lc_chisq.pdf)]
* <div id="Grubbs1944"></div>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
+
 
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. http://www.jstor.org/stable/167752
+
* <div id="Farebrother1990"></div>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. [http://www.jstor.org/stable/2347778?seq=1#page_scan_tab_contents (Read @ http://www.jstor.org/stable/2347778?seq=1#page_scan_tab_contents)]
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| private monograph]].
+
 
* <div id="Guenther1964"></div>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
+
* <div id="Greenwalt1962"></div>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 (PDF @ http://earth-info.nga.mil/GandG/publications/tr96.pdf]
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. http://www.jstor.org/stable/2281595
+
 
* <div id="Hogg1967"></div>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
+
* <div id="Grubbs1944"></div>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 (PDF @ https://projecteuclid.org/euclid.aoms/1177731316)]
* <div id="Hoover1984"></div>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
+
 
* <div id="Hoyt1947"></div>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
+
* <div id="Grubbs1964"></div>Grubbs, F. E. (1964). Approximate circular and noncircular offset probabilities of hitting. Operations Research, 12 (1), 51–62. [http://www.jstor.org/stable/167752 (Abstract @ http://www.jstor.org/stable/167752)]
* <div id="Imhof1961"></div>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
+
 
* <div id="Kamat1962"></div>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
+
* <div id="Grubbs1964_B"></div>Grubbs, F. E. (1964). Statistical Measures of Accuracy for Riflemen and Missile Engineers. [[Media:Statistical Measures for Riflemen and Missile Engineers - Grubbs 1964.pdf| (Cached private monograph)]].  
* <div id="Liu2009"></div>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.
+
 
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. http://statshooting.com/
+
* <div id="Guenther1964"></div>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 (PDF @ http://projecteuclid.org/euclid.aoms/1177703747)]
* <div id="Moranda1959"></div>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
+
 
* <div id="Moranda1960"></div>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
+
* <div id="Harter1960"></div>Harter, H. L. (1960). Circular error probabilities. Journal of the American Statistical Association, 55 (292), 723–731. [http://www.jstor.org/stable/2281595 (Abstract @ http://www.jstor.org/stable/2281595)]
* <div id="Nelson1988"></div>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
+
 
* <div id="Nuttall1975"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96.
+
* <div id="Harter1960b"></div>Harter, H. Leon (1960). Tables of Range and Studentized Range. Ann. Math. Statist., 31(4), 1122-1147. [http://projecteuclid.org/euclid.aoms/1177705684 (PDF @ http://projecteuclid.org/euclid.aoms/1177705684)]
* <div id="Paris2009"></div>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
+
 
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. http://www.jstor.org/stable/2332542
+
* <div id="Hogg1967"></div>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 (Abstract @ http://www.jstor.org/stable/2283768)]
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. http://www.jstor.org/stable/2333533
+
 
 +
* <div id="Hoover1984"></div>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 (PDF @ http://www.ngs.noaa.gov/PUBS_LIB/Brunswick/NOAATRNOS107CGS3.pdf)]
 +
 
 +
* <div id="Hoyt1947"></div>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 (PDF @ https://archive.org/details/bstj26-2-318)]
 +
 
 +
* <div id="Imhof1961"></div>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 (Read @ http://www.jstor.org/stable/2332763)]
 +
 
 +
* <div id="Kamat1962"></div>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 (Abstract @ http://www.jstor.org/stable/2282450)]
 +
 
 +
* <div id="Leslie_1993"></div>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.) [[Media:Is_Group_Size_the_Best_Measure_of_Accuracy_by_J.E._Leslie_III.pdf|(cached PDF)]] [[Leslie_1993 | (Ballistipedia Notes)]]
 +
 
 +
* <div id="Litz2011"></div>Litz,Bryan. (2011). Applied Ballistics for Long Range Shooting. Applied Ballistics, LLC. 2nd Edition. ISBN-13: 978-0615452562
 +
 
 +
* <div id="Liu2009"></div>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. [http://www.sciencedirect.com/science/article/pii/S0167947308005653 (Abstract @ http://www.sciencedirect.com/science/article/pii/S0167947308005653)] [http://www4.stat.ncsu.edu/~hzhang2/paper/chisq.pdf (PDF @ http://www4.stat.ncsu.edu/~hzhang2/paper/chisq.pdf)]
 +
 
 +
* <div id="McCoy2011"></div>McCoy, Robert. (2012). Modern Exterior Ballistics, Schiffer Publishing, Ltd.; 2nd edition edition. ISBN-13: 978-0764338250.
 +
 
 +
* <div id="McMillan2008"></div>McMillan, C., & McMillan, P. (2008). Characterizing rifle performance using circular error probable measured via a flatbed scanner. (Version 1.01) [http://statshooting.com/papers/measuring-cep-mcmillan2008.pdf (PDF @ http://statshooting.com/papers/measuring-cep-mcmillan2008.pdf)]
 +
 
 +
* <div id="Molon2006">Molon (2006). The Trouble With 3-Shot Groups. [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 (Webpage @ http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218)] (ID of poster??)
 +
 
 +
* <div id="Moranda1959"></div>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 (Abstract @ http://www.jstor.org/stable/2282503)]
 +
 
 +
* <div id="Moranda1960"></div>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 (Abstract @ http://www.jstor.org/stable/2281596)]
 +
 
 +
* <div id="Nelson1988"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA199190)]
 +
 
 +
* <div id="Nuttall1975a"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. Report: AD0779846 Naval Underwater Systems Center, New London Laboratory, New London. Connecticut [http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0779846 (PDF @ http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0779846)]
 +
 
 +
* <div id="Nuttall1975b"></div>Nuttall, A. H. (1975). Some integrals involving the Q-M function. IEEE Transactions on Information Theory, 21 (1), 95-96. [http://dx.doi.org/10.1109/TIT.1975.1055327 (Abstract @ http://dx.doi.org/10.1109/TIT.1975.1055327)]
 +
 
 +
* <div id="Paris2009"></div>Paris, J. F. (2009). Nakagami-q (Hoyt) distribution function with applications. Electronics Letters, 45 (4), 210–211. [http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4784312 (Abstract @ http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4784312)] [http://dx.doi.org/10.1049/el.2009.0828 (Erratum Notice, Electronics Letters, 45 (8), 432. @ http://dx.doi.org/10.1049/el.2009.0828)]
 +
 
 +
* <div id="Patnaik1949"></div>Patnaik, P. B. (1949). The non-central <math>\chi^{2}</math>- and F-distributions and their applications. Biometrika, 36 (1–2), 202–232. [http://www.jstor.org/stable/2332542 (Read @ http://www.jstor.org/stable/2332542)]
 +
 
 +
* <div id="Pearson1959"></div>Pearson, E. S. (1959). Note on an approximation to the distribution of non-central <math>\chi^{2}</math>. Biometrika, 46 (3–4), 364. [http://www.jstor.org/stable/2333533 (Read @ http://www.jstor.org/stable/2333533)]
 +
 
 
* <div id="Pesapane1977"></div>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.
 
* <div id="Pesapane1977"></div>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.
* <div id="Puhek1992"></div>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
+
 
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024.
+
* <div id="Puhek1992"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA248105)]
* <div id="RAND1952"></div>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
+
 
 +
* <div id="Pyati1993"></div>Pyati, V. P. (1993). Computation of the circular error probability (CEP) integral. IEEE Transactions on Aerospace and Electronic Systems, 29 (3), 1023–1024. [http://dx.doi.org/10.1109/7.220962 (Abstract @ http://dx.doi.org/10.1109/7.220962)]
 +
 
 +
* <div id="RAND1952"></div>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 (PDF @ http://www.rand.org/pubs/reports/2008/R234.pdf)]
 +
 
 
* <div id="Sathe1991"></div>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.
 
* <div id="Sathe1991"></div>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.
* <div id="Saxena2005"></div>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
+
 
* <div id="Sheil1977"></div>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.
+
* <div id="Saxena2005"></div>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 (PDF @ http://www.stat.tugraz.at/AJS/ausg051/051Saxena&Singh.pdf)]
 +
 
 +
* <div id="Sheil1977"></div>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. [http://www.jstor.org/stable/2346884 (Read @ http://www.jstor.org/stable/2346884)]
 +
 
 
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.
 
* <div id="Shnidman1995"></div>Shnidman, D. A. (1995). Efficient computation of the circular error probability (CEP) integral. IEEE Transactions on Automatic Control, 40 (8), 1472–1474.
* <div id="Singh1992"></div>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
+
 
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html
+
* <div id="Siddiqui1961"></div>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. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|(Cached PDF)]]
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its stimation. Metrika, 15 (1), 149–163.
+
 
* <div id="Spall1992"></div>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
+
* <div id="Siddiqui1964"></div>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.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|(Cached PDF)]]
* <div id="Taub1983a"></div>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
+
 
* <div id="Taub1983b"></div>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
+
* <div id="Singh1992"></div>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 (Abstract @ http://www.jstor.org/stable/25052751)]
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf  
+
 
* <div id="Taylor1975b"></div>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
+
* <div id="Singh1962"></div>Singh, N. (1962). Spherical probable error. Nature, 193 (4815), 605. [http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html (HTML Fulltext @ http://www.nature.com/nature/journal/v193/n4815/abs/193605a0.html)]
* <div id="Thomas1973"></div>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
+
 
* <div id="Tongue1993"></div>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
+
* <div id="Singh1970"></div>Singh, N. (1970). Spherical probable error (SPE) and its estimation. Metrika, 15 (1), 149–163. [http://link.springer.com/article/10.1007%2FBF02613568 (Abstract @ http://link.springer.com/article/10.1007%2FBF02613568)]
* <div id="Wang2013a"></div>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.
+
 
* <div id="Wang2013b"></div>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.
+
* <div id="Spall1992"></div>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. [http://www.jstor.org/stable/2290205 (Abstract @ http://www.jstor.org/stable/2290205)]
 +
 
 +
* <div id="Taub1983a"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA153828)]
 +
 
 +
* <div id="Taub1983b"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA153828)]
 +
 
 +
* <div id="Taylor1975"></div>Taylor, M. S. & Grubbs, F. E. (1975), Approximate Probability Distributions for the Extreme Spread. Memorandum Rept.  ADA006586, Army Ballistic Research Lab, Aberdeen Proving Ground [http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf (PDF @ http://www.dtic.mil/dtic/tr/fulltext/u2/a006586.pdf)]
 +
 
 +
* <div id="Taylor1975b"></div>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 Abstract @ http://dx.doi.org/10.1002/nav.3800220407] [https://archive.org/details/navalresearchlog2241975offi (PDF of Naval Logistics Quarterly issue @ https://archive.org/details/navalresearchlog2241975offi0]
 +
 
 +
* <div id="Thomas1973"></div>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. [http://handle.dtic.mil/100.2/AD0759989 (PDF @ http://handle.dtic.mil/100.2/AD0759989)]
 +
 
 +
* <div id="Tongue1993"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA266528)]
 +
 
 +
* <div id="Wang2013a"></div>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. [http://www.scientific.net/AMM.341-342.955 (Abstract @ http://www.scientific.net/AMM.341-342.955)]
 +
 
 +
* <div id="Wang2013b"></div>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. [http://www.scientific.net/AMR.765-767.2224 (Abstract @ http://www.scientific.net/AMR.765-767.2224)]
 +
 
 
* <div id="Wang2014"></div>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.
 
* <div id="Wang2014"></div>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.
* <div id="Williams1997"></div>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
 
* <div id="Zhang2012"></div>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.
 
  
 +
* <div id="Williams1997"></div>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 (PDF @ http://handle.dtic.mil/100.2/ADA324337)]
 +
 +
* <div id="Zhang2012"></div>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 (Abstract @ http://www.tandfonline.com/doi/abs/10.1080/00949655.2010.546797)]
 +
 +
= Groups of Publications =
  
= Groups of CEP publications =
+
== CEP ==
  
 
The literature on the [[Circular_Error_Probable|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.
 
The literature on the [[Circular_Error_Probable|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.
  
The following list is by no means intended to be complete. Beware that the quality of the listed publications is not uniformly high. The relevant publications may be roughly categorized into different groups:
+
=== Develop CEP Estimator ===
 
 
== Develop CEP Estimator ==
 
  
 
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).
 
Articles that develop a CEP estimator or the modification of one – e. g., RAND-234 ([[#RAND1952|RAND Corporation, 1952]]), modified RAND-234 ([[#Pesapane1977|Pesapane & Irvine, 1977]]), [[#Grubbs1964|Grubbs (1964)]], Rayleigh ([[#Culpepper1978|Culpepper, 1978]]; [[#Saxena2005|Saxena & Singh, 2005]]; [[#Singh1992|Singh, 1992]]), [[#Ethridge1983|Ethridge (1983]]; [[#Hogg1967|Hogg, 1967]]), [[#Spall1992|Spall & Maryak (1992)]] general bivariate normal ([[#DiDonato1961a|DiDonato & Jarnagin, 1961a]]; [[#Evans1985|Evans, Govindarajulu, & Barthoulot, 1985]]). Some articles focus on the confidence intervals for CEP ([[#DiDonato2007|DiDonato, 2007]]; [[#Sathe1991|Sathe, Joshi, & Nabar, 1991]]; [[#Taub1983b|Taub & Thomas, 1983b]]; [[#Thomas1973|Thomas, Crigler, Gemmill & Taub, 1973]]; [[#Zhang2012|Zhang & An, 2012]]).
  
== Comparing CEP Estimators ==
+
=== Comparing CEP Estimators ===
  
 
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).
 
<div id="compStudies"></div>Articles or Master’s theses comparing the characteristics of CEP estimators in different scenarios ([[#Blischke1966|Blischke & Halpin, 1966]]; [[#Elder1986|Elder, 1986]]; [[#Kamat1962|Kamat, 1962]]; [[#McMillan2008|McMillan & McMillan, 2008]]; [[#Moranda1959|Moranda, 1959]], [[#Moranda1960|1960]]; [[#Nelson1988|Nelson, 1988]]; [[#Puhek1992|Puhek, 1992]]; [[#Tongue1993|Tongue, 1993]]; [[#Taub1983a|Taub & Thomas, 1983a]]; [[#Wang2013a|Wang, Jia, Yang, & Wang, 2013]]; [[#Wang2013b|Wang, Yang, Jia, & Wang, 2013]]; [[#Wang2014|Wang, Yang, Yan, Wang, & Song, 2014]]; [[#Williams1997|Williams, 1997]]).
  
== CEP in polar Coordinates ==
+
=== CEP in polar Coordinates ===
  
 
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|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 ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].
 
Publications studying the correlated bivariate normal distribution re-written in polar coordinates radius and angle ([[#Chew1962|Chew & Boyce, 1962]]; [[#Greenwalt1962|Greenwalt & Shultz, 1962]]; [[#Harter1960|Harter, 1960]]; [[#Hoover1984|Hoover, 1984]]; [[#Hoyt1947|Hoyt, 1947]]). The distribution of the radius is known as the Hoyt ([[#Hoyt1947|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 ([[#Paris2009|Paris, 2009]]). The latter is the complement (with respect to unity) of a special case of the non-central <math>\chi^2</math>-distribution ([[#Nuttall1975|Nuttall, 1975]]). The statistical literature on coverage problems in the multivariate normal distribution is reviewed in [[#Guenther1964|Guenther & Terragno (1964)]].
  
== CEP with Bias ==
+
=== CEP with Bias ===
  
 
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|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 <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.
 
DiDonato and Jarnagin ([[#DiDonato1961a|1961a]], [[#DiDonato1961b|1961b]], [[#DiDonato1962a|1962a]], [[#DiDonato1962b|1962b]]) as well as [[#Evans1985|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 <math>\chi^{2}</math>-distributions and difficult to calculate without numerical tools. Therefore, the [[#Patnaik1949|Patnaik (1949)]] two-moment central <math>\chi^{2}</math>-approximation or the Pearson ([[#Imhof1961|Imhof, 1961]]; [[#Pearson1959|Pearson, 1959]]) three-moment central <math>\chi^{2}</math>-approximation are often used. Recently, [[#Liu2009|Liu, Tang & Zhang (2009)]] proposed a four-moment non-central <math>\chi^{2}</math>-approximation.
  
== Hoyt Distribution Algorithms ==
+
=== Hoyt Distribution Algorithms ===
  
 
<div id="algos"></div>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, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].
 
<div id="algos"></div>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, [[#DiDonato2004|2004]], [[#DiDonato2007|2007]]; [[#Pyati1993|Pyati, 1993]]; [[#Shnidman1995|Shnidman, 1995]]). Numerical algorithms to efficiently and precisely calculate the distribution of quadratic forms of normal random variables were proposed by [[#Davies1980|Davies (1980)]], Farebrother ([[#Farebrother1984|1984]], [[#Farebrother1990|1990]]), [[#Imhof1961|Imhof (1961)]], [[#Sheil1977|Sheil & O'Muircheartaigh (1977)]]. A comparison and implementation can be found in [[#Duchesne2010|Duchesne and Lafaye de Micheaux (2010)]].
  
== Spherical Error Probable ==
+
=== Spherical Error Probable ===
  
 
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.
 
[[#DiDonato1988|DiDonato (1988)]] and Singh ([[#Singh1962|1962]], [[#Singh1970|1970]]) treat the Spherical Error Probable SEP.
  
= Extreme Spread =
+
== Extreme Spread ==
  
== Monte Carlo Simulation ==
+
=== Monte Carlo Simulation ===
Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].
+
circular groups, no fliers - Talyor and Grubbs wrote two papers that are virtually identify the first was published as a [[#Taylor1975|study]] and the second in [[#Taylor1975b|Naval Research Quarterly]].
  
== Sampling Problems ==
+
=== Sampling Problems ===
  
 
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 forum thread] Molon offers intuitive explanations and illustrations of the problems with Extreme Spread samples.
 
Through an extended [http://www.ar15.com/mobile/topic.html?b=3&f=118&t=279218 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 ([[#Siddiqui1961 | 1961]] and [[#Siddiqui1961 | 1964]]).
 +
 +
=== Derivation ===
 +
 +
== Sample Range ==
 +
[[#Harter1960b | 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:Confidence Interval Convergence.xlsx]]: Shows how precision confidence intervals shrink as sample size increases.
 +
 +
* [[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.
 +
 +
* [[File:SymmetricBivariateSigma1.xls]]: Monte Carlo simulation results validating the [[Closed Form Precision]] math.
  
 
----
 
----
Line 111: Line 206:
  
 
* Danielson, Brent J. (2005).  [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].
 
* Danielson, Brent J. (2005).  [[Prior_Art#Danielson.2C_2005.2C_Testing_loads|'''Testing Loads''' &ndash; ''detailed in Prior Art'']].
 
  
 
* Hogema, Jeroen (2005).  [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].
 
* Hogema, Jeroen (2005).  [[Prior_Art#Hogema.2C_2005.2C_Shot_group_statistics|'''Shot group statistics''' &ndash; ''detailed in Prior Art'']].
Line 118: Line 212:
  
 
* Kolbe, Geoffrey (2010).  [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].
 
* Kolbe, Geoffrey (2010).  [[Prior_Art#Kolbe.2C_2010.2C_Group_Statistics|'''Group Statistics''' &ndash; ''detailed in Prior Art'']].
 
* Leslia, John E. III (1993).  [[Prior_Art#Leslie.2C_1993.2C_Is_.22Group_Size.22_the_Best_Measure_of_Accuracy.3F|'''Is "Group Size" the Best Measure of Accuracy?''' &ndash; ''detailed in Prior Art'']].
 
 
* Molon (2006). [[Prior_Art#Molon.2C_2006.2C_The_Trouble_With_3-Shot_Groups|'''The Trouble With 3-Shot Groups''' &ndash; ''detailed in Prior Art'']].
 
  
 
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].
 
* Rifleslinger (2014). [http://artoftherifleblog.com/on-zeroing/2014/02/on-zeroing.html '''On Zeroing'''].
 
* 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. [[Media:Some Problems Connected With Rayleigh Distributions - Siddiqui 1961.pdf|'''(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.'') [[Media:Statistical Inference for Rayleigh Distributions - Siddiqui, 1964.pdf|'''(cached PDF)''']]
 

Latest revision as of 21:52, 18 June 2015

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.

  • 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

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.

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).

Derivation

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.