2084:
2094:
675:
Using the complex parameter also let easily prove the invariance of f-divergences (e.g., Kullback-Leibler divergence, chi-squared divergence, etc.) with respect to real linear fractional transformations (group action of SL(2,R)), and show that all f-divergences between univariate Cauchy densities are
337:
448:
604:
215:
Although the parameter is notionally expressed using a complex number, the density is still a density over the real line. In particular the density can be written using the real-valued parameters
625:. ...the induced transformation on the parameter space has the same fractional linear form as the transformation on the sample space only if the parameter space is taken to be the complex plane.
229:
99:
482:
364:
742:
672:." In addition, McCullagh shows that the complex-valued parameterisation allows a simple relationship to be made between the Cauchy and the "circular Cauchy distribution".
871:
501:
1354:
1262:
2049:
1915:
1127:
886:
735:
1810:
1574:
1248:
1569:
1513:
1173:
811:
1319:
1855:
1589:
1442:
1117:
861:
2097:
1314:
2087:
1759:
1735:
728:
689:
1956:
1584:
2118:
1833:
1794:
1766:
1740:
1658:
1007:
755:
1944:
1910:
1776:
1771:
1616:
1424:
1122:
876:
1694:
1607:
1579:
1488:
1437:
1411:
1309:
1092:
1057:
1708:
1625:
1462:
1209:
1087:
1062:
926:
921:
916:
1386:
896:
891:
185:, uses the two parameters of the non-standardised distribution to form a single complex-valued parameter, specifically, the
2024:
1890:
1598:
1447:
1379:
1364:
1257:
1231:
1163:
1002:
833:
818:
1541:
1920:
1860:
1850:
1467:
1168:
1027:
29:
1269:
1012:
941:
1905:
1900:
1845:
1781:
1546:
1324:
1221:
806:
332:{\displaystyle f(x)={1 \over \pi \left\vert \sigma \right\vert \left(1+{\frac {(x-\mu )^{2}}{\sigma ^{2}}}\right)}\,,}
1725:
1533:
2039:
1815:
1634:
1416:
1369:
1238:
1214:
1194:
1037:
911:
791:
2044:
987:
38:
1828:
1789:
1663:
1500:
1344:
1289:
1187:
1151:
1022:
1730:
1518:
1284:
1243:
1158:
1112:
1052:
1017:
906:
801:
751:
25:
843:
443:{\displaystyle f(x)={\left\vert \Im {\theta }\right\vert \over \pi \left\vert x-\theta \right\vert ^{2}}\,,}
2029:
1971:
1642:
1429:
1339:
1294:
1279:
1199:
1097:
1047:
1042:
823:
456:
1895:
1883:
1872:
1754:
1650:
1457:
901:
881:
786:
2019:
1976:
1820:
1495:
1349:
1329:
1226:
796:
182:
120:
713:
2069:
2064:
2059:
2054:
1991:
1961:
1840:
1483:
1374:
1274:
977:
936:
931:
828:
2003:
1528:
1508:
1478:
1452:
1406:
1334:
1146:
1082:
21:
2034:
1523:
1304:
1299:
1204:
1141:
1136:
992:
982:
866:
17:
660:
McCullagh also wrote, "The distribution of the first exit point from the upper half-plane of a
346: = 0. An alternative form for the density can be written using the complex parameter
1932:
1359:
1102:
1032:
997:
946:
1107:
781:
205:
685:
488:
174:
1180:
661:
599:{\displaystyle Y^{*}={aY+b \over cY+d}\sim C\left({a\theta +b \over c\theta +d}\right)}
186:
699:
2112:
1803:
1551:
838:
495:
To this question I can give no better answer than to present the curious result that
720:
108:
694:
178:
115:
0, and first and third quartiles respectively −1 and +1. Generally, a
487:
To the question "Why introduce complex numbers when only real-valued
112:
708:
208:. It also extends the usual range of scale parameter to include
724:
641:
defined above has a Cauchy distribution with parameter (
668:
is the Cauchy density on the real line with parameter
223:, which can each take positive or negative values, as
119:
is any probability distribution belonging to the same
504:
459:
367:
232:
151:
and whose first and third quartiles are respectively
41:
342:
where the distribution is regarded as degenerate if
2012:
1970:
1871:
1707:
1685:
1676:
1560:
1395:
1071:
968:
959:
852:
772:
763:
598:
476:
442:
331:
93:
633:has a Cauchy distribution with complex parameter
709:"On f-divergences between Cauchy distributions"
736:
8:
94:{\displaystyle f(x)={1 \over \pi (1+x^{2})}}
1682:
965:
769:
743:
729:
721:
147:has a Cauchy distribution whose median is
698:, volume 79 (1992), pages 247–259.
690:"Conditional inference and Cauchy models"
560:
518:
509:
503:
463:
458:
436:
427:
393:
383:
366:
325:
309:
298:
279:
248:
231:
79:
57:
40:
629:In other words, if the random variable
127:has a standard Cauchy distribution and
477:{\displaystyle \Im {\theta }=\sigma }
7:
2093:
707:Frank Nielsen and Kazuki Okamura,
460:
390:
14:
491:are involved?", McCullagh wrote:
2092:
2083:
2082:
377:
371:
295:
282:
242:
236:
85:
66:
51:
45:
1:
30:probability density function
637:, then the random variable
171:McCullagh's parametrization
2135:
1916:Wrapped asymmetric Laplace
887:Extended negative binomial
702:from McCullagh's homepage.
2078:
1575:Generalized extreme value
1355:Relativistic Breit–Wigner
752:Probability distributions
2119:Continuous distributions
135: > 0, then
26:probability distribution
1570:Generalized chi-squared
1514:Normal-inverse Gaussian
131:is any real number and
123:as this one. Thus, if
1882:Univariate (circular)
1443:Generalized hyperbolic
872:Conway–Maxwell–Poisson
862:Beta negative binomial
627:
600:
478:
444:
333:
95:
1927:Bivariate (spherical)
1425:Kaniadakis κ-Gaussian
609:for all real numbers
601:
493:
479:
445:
334:
183:University of Chicago
121:location-scale family
96:
1992:Dirac delta function
1939:Bivariate (toroidal)
1896:Univariate von Mises
1767:Multivariate Laplace
1659:Shifted log-logistic
1008:Continuous Bernoulli
502:
457:
365:
230:
39:
2040:Natural exponential
1945:Bivariate von Mises
1911:Wrapped exponential
1777:Multivariate stable
1772:Multivariate normal
1093:Benktander 2nd kind
1088:Benktander 1st kind
877:Discrete phase-type
212: < 0.
155: −
117:Cauchy distribution
22:Cauchy distribution
1695:Rectified Gaussian
1580:Generalized Pareto
1438:Generalized normal
1310:Matrix-exponential
596:
474:
440:
329:
91:
18:probability theory
2106:
2105:
1703:
1702:
1672:
1671:
1563:whose type varies
1509:Normal (Gaussian)
1463:Hyperbolic secant
1412:Exponential power
1315:Maxwell–Boltzmann
1063:Wigner semicircle
955:
954:
927:Parabolic fractal
917:Negative binomial
662:Brownian particle
590:
548:
434:
323:
315:
89:
20:, the "standard"
2126:
2096:
2095:
2086:
2085:
2025:Compound Poisson
2000:
1988:
1957:von Mises–Fisher
1953:
1941:
1929:
1891:Circular uniform
1887:
1807:
1751:
1722:
1683:
1585:Marchenko–Pastur
1448:Geometric stable
1365:Truncated normal
1258:Inverse Gaussian
1164:Hyperexponential
1003:Beta rectangular
971:bounded interval
966:
834:Discrete uniform
819:Poisson binomial
770:
745:
738:
731:
722:
714:arXiv 2101.12459
605:
603:
602:
597:
595:
591:
589:
575:
561:
549:
547:
533:
519:
514:
513:
489:random variables
483:
481:
480:
475:
467:
449:
447:
446:
441:
435:
433:
432:
431:
426:
422:
402:
398:
397:
384:
338:
336:
335:
330:
324:
322:
321:
317:
316:
314:
313:
304:
303:
302:
280:
267:
249:
173:, introduced by
100:
98:
97:
92:
90:
88:
84:
83:
58:
2134:
2133:
2129:
2128:
2127:
2125:
2124:
2123:
2109:
2108:
2107:
2102:
2074:
2050:Maximum entropy
2008:
1996:
1984:
1974:
1966:
1949:
1937:
1925:
1880:
1867:
1804:Matrix-valued:
1801:
1747:
1718:
1710:
1699:
1687:
1678:
1668:
1562:
1556:
1473:
1399:
1397:
1391:
1320:Maxwell–Jüttner
1169:Hypoexponential
1075:
1073:
1072:supported on a
1067:
1028:Noncentral beta
988:Balding–Nichols
970:
969:supported on a
961:
951:
854:
848:
844:Zipf–Mandelbrot
774:
765:
759:
749:
686:Peter McCullagh
682:
576:
562:
556:
534:
520:
505:
500:
499:
455:
454:
412:
408:
407:
403:
389:
385:
363:
362:
305:
294:
281:
272:
268:
257:
253:
228:
227:
177:, professor of
175:Peter McCullagh
75:
62:
37:
36:
12:
11:
5:
2132:
2130:
2122:
2121:
2111:
2110:
2104:
2103:
2101:
2100:
2090:
2079:
2076:
2075:
2073:
2072:
2067:
2062:
2057:
2052:
2047:
2045:Location–scale
2042:
2037:
2032:
2027:
2022:
2016:
2014:
2010:
2009:
2007:
2006:
2001:
1994:
1989:
1981:
1979:
1968:
1967:
1965:
1964:
1959:
1954:
1947:
1942:
1935:
1930:
1923:
1918:
1913:
1908:
1906:Wrapped Cauchy
1903:
1901:Wrapped normal
1898:
1893:
1888:
1877:
1875:
1869:
1868:
1866:
1865:
1864:
1863:
1858:
1856:Normal-inverse
1853:
1848:
1838:
1837:
1836:
1826:
1818:
1813:
1808:
1799:
1798:
1797:
1787:
1779:
1774:
1769:
1764:
1763:
1762:
1752:
1745:
1744:
1743:
1738:
1728:
1723:
1715:
1713:
1705:
1704:
1701:
1700:
1698:
1697:
1691:
1689:
1680:
1674:
1673:
1670:
1669:
1667:
1666:
1661:
1656:
1648:
1640:
1632:
1623:
1614:
1605:
1596:
1587:
1582:
1577:
1572:
1566:
1564:
1558:
1557:
1555:
1554:
1549:
1547:Variance-gamma
1544:
1539:
1531:
1526:
1521:
1516:
1511:
1506:
1498:
1493:
1492:
1491:
1481:
1476:
1471:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1427:
1422:
1414:
1409:
1403:
1401:
1393:
1392:
1390:
1389:
1387:Wilks's lambda
1384:
1383:
1382:
1372:
1367:
1362:
1357:
1352:
1347:
1342:
1337:
1332:
1327:
1325:Mittag-Leffler
1322:
1317:
1312:
1307:
1302:
1297:
1292:
1287:
1282:
1277:
1272:
1267:
1266:
1265:
1255:
1246:
1241:
1236:
1235:
1234:
1224:
1222:gamma/Gompertz
1219:
1218:
1217:
1212:
1202:
1197:
1192:
1191:
1190:
1178:
1177:
1176:
1171:
1166:
1156:
1155:
1154:
1144:
1139:
1134:
1133:
1132:
1131:
1130:
1120:
1110:
1105:
1100:
1095:
1090:
1085:
1079:
1077:
1074:semi-infinite
1069:
1068:
1066:
1065:
1060:
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
1000:
995:
990:
985:
980:
974:
972:
963:
957:
956:
953:
952:
950:
949:
944:
939:
934:
929:
924:
919:
914:
909:
904:
899:
894:
889:
884:
879:
874:
869:
864:
858:
856:
853:with infinite
850:
849:
847:
846:
841:
836:
831:
826:
821:
816:
815:
814:
807:Hypergeometric
804:
799:
794:
789:
784:
778:
776:
767:
761:
760:
750:
748:
747:
740:
733:
725:
719:
718:
704:
703:
681:
678:
607:
606:
594:
588:
585:
582:
579:
574:
571:
568:
565:
559:
555:
552:
546:
543:
540:
537:
532:
529:
526:
523:
517:
512:
508:
473:
470:
466:
462:
451:
450:
439:
430:
425:
421:
418:
415:
411:
406:
401:
396:
392:
388:
382:
379:
376:
373:
370:
340:
339:
328:
320:
312:
308:
301:
297:
293:
290:
287:
284:
278:
275:
271:
266:
263:
260:
256:
252:
247:
244:
241:
238:
235:
206:imaginary unit
187:complex number
102:
101:
87:
82:
78:
74:
71:
68:
65:
61:
56:
53:
50:
47:
44:
13:
10:
9:
6:
4:
3:
2:
2131:
2120:
2117:
2116:
2114:
2099:
2091:
2089:
2081:
2080:
2077:
2071:
2068:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2017:
2015:
2011:
2005:
2002:
1999:
1995:
1993:
1990:
1987:
1983:
1982:
1980:
1978:
1973:
1969:
1963:
1960:
1958:
1955:
1952:
1948:
1946:
1943:
1940:
1936:
1934:
1931:
1928:
1924:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1886:
1885:
1879:
1878:
1876:
1874:
1870:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1843:
1842:
1839:
1835:
1832:
1831:
1830:
1827:
1825:
1824:
1819:
1817:
1816:Matrix normal
1814:
1812:
1809:
1806:
1805:
1800:
1796:
1793:
1792:
1791:
1788:
1786:
1785:
1782:Multivariate
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1761:
1758:
1757:
1756:
1753:
1750:
1746:
1742:
1739:
1737:
1734:
1733:
1732:
1729:
1727:
1724:
1721:
1717:
1716:
1714:
1712:
1709:Multivariate
1706:
1696:
1693:
1692:
1690:
1684:
1681:
1675:
1665:
1662:
1660:
1657:
1655:
1653:
1649:
1647:
1645:
1641:
1639:
1637:
1633:
1631:
1629:
1624:
1622:
1620:
1615:
1613:
1611:
1606:
1604:
1602:
1597:
1595:
1593:
1588:
1586:
1583:
1581:
1578:
1576:
1573:
1571:
1568:
1567:
1565:
1561:with support
1559:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1537:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1504:
1499:
1497:
1494:
1490:
1487:
1486:
1485:
1482:
1480:
1477:
1475:
1474:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1433:
1428:
1426:
1423:
1421:
1420:
1415:
1413:
1410:
1408:
1405:
1404:
1402:
1398:on the whole
1394:
1388:
1385:
1381:
1378:
1377:
1376:
1373:
1371:
1370:type-2 Gumbel
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1306:
1303:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1271:
1268:
1264:
1261:
1260:
1259:
1256:
1254:
1252:
1247:
1245:
1242:
1240:
1239:Half-logistic
1237:
1233:
1230:
1229:
1228:
1225:
1223:
1220:
1216:
1213:
1211:
1208:
1207:
1206:
1203:
1201:
1198:
1196:
1195:Folded normal
1193:
1189:
1186:
1185:
1184:
1183:
1179:
1175:
1172:
1170:
1167:
1165:
1162:
1161:
1160:
1157:
1153:
1150:
1149:
1148:
1145:
1143:
1140:
1138:
1135:
1129:
1126:
1125:
1124:
1121:
1119:
1116:
1115:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1091:
1089:
1086:
1084:
1081:
1080:
1078:
1070:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1038:Raised cosine
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
975:
973:
967:
964:
958:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
912:Mixed Poisson
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
863:
860:
859:
857:
851:
845:
842:
840:
837:
835:
832:
830:
827:
825:
822:
820:
817:
813:
810:
809:
808:
805:
803:
800:
798:
795:
793:
792:Beta-binomial
790:
788:
785:
783:
780:
779:
777:
771:
768:
762:
757:
753:
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1951:Multivariate
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1926:
1921:Wrapped Lévy
1881:
1829:Matrix gamma
1822:
1802:
1790:Normal-gamma
1783:
1749:Continuous:
1748:
1719:
1664:Tukey lambda
1651:
1643:
1638:-exponential
1635:
1627:
1618:
1609:
1600:
1594:-exponential
1591:
1535:
1502:
1469:
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1418:
1345:Poly-Weibull
1290:Log-logistic
1250:
1249:Hotelling's
1181:
1023:Logit-normal
897:Gauss–Kuzmin
892:Flory–Schulz
773:with finite
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2035:Exponential
1884:directional
1873:Directional
1760:Generalized
1731:Multinomial
1686:continuous-
1626:Kaniadakis
1617:Kaniadakis
1608:Kaniadakis
1599:Kaniadakis
1590:Kaniadakis
1542:Tracy–Widom
1519:Skew normal
1501:Noncentral
1285:Log-Laplace
1263:Generalized
1244:Half-normal
1210:Generalized
1174:Logarithmic
1159:Exponential
1113:Chi-squared
1053:U-quadratic
1018:Kumaraswamy
960:Continuous
907:Logarithmic
802:Categorical
676:symmetric.
2030:Elliptical
1986:Degenerate
1972:Degenerate
1720:Discrete:
1679:univariate
1534:Student's
1489:Asymmetric
1468:Johnson's
1396:supported
1340:Phase-type
1295:Log-normal
1280:Log-Cauchy
1270:Kolmogorov
1188:Noncentral
1118:Noncentral
1098:Beta prime
1048:Triangular
1043:Reciprocal
1013:Irwin–Hall
962:univariate
942:Yule–Simon
824:Rademacher
766:univariate
695:Biometrika
680:References
179:statistics
1755:Dirichlet
1736:Dirichlet
1646:-Gaussian
1621:-Logistic
1458:Holtsmark
1430:Gaussian
1417:Fisher's
1400:real line
902:Geometric
882:Delaporte
787:Bernoulli
764:Discrete
581:θ
567:θ
551:∼
511:∗
472:σ
465:θ
461:ℑ
420:θ
417:−
405:π
395:θ
391:ℑ
307:σ
292:μ
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255:π
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2113:Category
2088:Category
2020:Circular
2013:Families
1998:Singular
1977:singular
1741:Negative
1688:discrete
1654:-Weibull
1612:-Weibull
1496:Logistic
1380:Discrete
1350:Rayleigh
1330:Nakagami
1253:-squared
1227:Gompertz
1076:interval
812:Negative
797:Binomial
200:, where
2098:Commons
2070:Wrapped
2065:Tweedie
2060:Pearson
2055:Mixture
1962:Bingham
1861:Complex
1851:Inverse
1841:Wishart
1834:Inverse
1821:Matrix
1795:Inverse
1711:(joint)
1630:-Erlang
1484:Laplace
1375:Weibull
1232:Shifted
1215:Inverse
1200:Fréchet
1123:Inverse
1058:Uniform
978:Arcsine
937:Skellam
932:Poisson
855:support
829:Soliton
782:Benford
775:support
717:(2021).
204:is the
181:at the
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2004:Cantor
1846:Normal
1677:Mixed
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1529:Stable
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1453:Gumbel
1407:Cauchy
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1147:Erlang
1128:Scaled
1083:Benini
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453:where
113:median
28:whose
1726:Ewens
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1524:Slash
1305:Lomax
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1205:Gamma
1152:Hyper
1142:Davis
1137:Dagum
993:Bates
983:ARGUS
867:Borel
1975:and
1933:Kent
1360:Rice
1275:Lévy
1103:Burr
1033:PERT
998:Beta
947:Zeta
839:Zipf
756:list
621:and
219:and
159:and
109:real
104:for
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1108:Chi
700:PDF
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