36:
amongst its elements. In particular, when elements of the random vector must add up to certain sum, then an element in the vector is neutral with respect to the others if the distribution of the vector created by expressing the remaining elements as proportions of their total is independent of the
924:
1284:
582:
1375:
728:
1112:
431:
1025:
129:
284:
224:
1104:
720:
666:
1055:
954:
612:
419:
388:
341:
314:
160:
70:
1395:
361:
1492:
1497:
1304:
919:{\displaystyle X_{1,2}^{*}=\left({\frac {X_{3}}{1-X_{1}-X_{2}}},{\frac {X_{4}}{1-X_{1}-X_{2}}},\ldots ,{\frac {X_{k}}{1-X_{1}-X_{2}}}\right).}
1279:{\displaystyle X_{1,\ldots ,j}^{*}=\left({\frac {X_{j+1}}{1-X_{1}-\cdots -X_{j}}},\ldots ,{\frac {X_{k}}{1-X_{1}-\cdots -X_{j}}}\right).}
1426:
Connor, R. J.; Mosimann, J. E. (1969). "Concepts of
Independence for Proportions with a Generalization of the Dirichlet Distribution".
1406:
316:
are interpreted as lengths whose sum is unity. In a variety of contexts, it is often desirable to eliminate a proportion, say
1397:
is completely neutral. In 1980, James and
Mosimann showed that the Dirichlet distribution is characterised by neutrality.
577:{\displaystyle X_{1}^{*}=\left({\frac {X_{2}}{1-X_{1}}},{\frac {X_{3}}{1-X_{1}}},\ldots ,{\frac {X_{k}}{1-X_{1}}}\right).}
959:
422:
75:
33:
343:, and consider the distribution of the remaining intervals within the remaining length. The first element of
235:
171:
1060:
21:
671:
617:
1466:
1435:
1033:
932:
590:
397:
366:
319:
292:
138:
48:
29:
1380:
346:
1486:
17:
1471:
1454:
1455:"A new characterization of the Dirichlet distribution through neutrality"
1370:{\displaystyle X=(X_{1},\ldots ,X_{K})\sim \operatorname {Dir} (\alpha )}
1439:
135:
proportions of all the other elements are independent of
1383:
1307:
1115:
1063:
1036:
962:
935:
731:
674:
620:
593:
434:
400:
369:
349:
322:
295:
238:
174:
141:
78:
51:
668:is independent of the remaining interval: that is,
165:Formally, consider the vector of random variables
1389:
1369:
1278:
1098:
1049:
1019:
948:
918:
714:
660:
606:
576:
413:
382:
355:
335:
308:
278:
218:
154:
123:
64:
1428:Journal of the American Statistical Association
1294:A vector for which each element is neutral is
1377:is drawn from a Dirichlet distribution, then
1020:{\displaystyle Y=(X_{2},X_{3},\ldots ,X_{k})}
8:
32:is one that exhibits a particular type of
1470:
1453:James, Ian R.; Mosimann, James E (1980).
1382:
1340:
1321:
1306:
1259:
1240:
1223:
1217:
1199:
1180:
1157:
1151:
1137:
1120:
1114:
1084:
1068:
1062:
1041:
1035:
1008:
989:
976:
961:
940:
934:
899:
886:
869:
863:
845:
832:
815:
809:
797:
784:
767:
761:
747:
736:
730:
703:
685:
679:
673:
649:
631:
625:
619:
598:
592:
557:
540:
534:
516:
499:
493:
481:
464:
458:
444:
439:
433:
405:
399:
374:
368:
348:
327:
321:
300:
294:
264:
254:
243:
237:
207:
188:
173:
146:
140:
124:{\displaystyle X_{1},X_{2},\ldots ,X_{k}}
115:
96:
83:
77:
56:
50:
1418:
279:{\displaystyle \sum _{i=1}^{k}X_{i}=1.}
219:{\displaystyle X=(X_{1},\ldots ,X_{k})}
20:, and specifically in the study of the
7:
1099:{\displaystyle X_{1},\ldots X_{j-1}}
1493:Theory of probability distributions
1407:Generalized Dirichlet distribution
14:
1498:Independence (probability theory)
956:, viewed as the first element of
715:{\displaystyle X_{2}/(1-X_{1})}
661:{\displaystyle X_{2}/(1-X_{1})}
1364:
1358:
1346:
1314:
1014:
969:
709:
690:
655:
636:
213:
181:
1:
37:element that was omitted.
1514:
423:statistically independent
1459:The Annals of Statistics
34:statistical independence
1472:10.1214/aos/1176344900
1391:
1371:
1280:
1100:
1051:
1021:
950:
920:
722:being independent of
716:
662:
608:
578:
415:
384:
357:
337:
310:
280:
259:
220:
156:
125:
66:
22:Dirichlet distribution
1392:
1372:
1281:
1101:
1052:
1050:{\displaystyle X_{j}}
1030:In general, variable
1022:
951:
949:{\displaystyle X_{2}}
921:
717:
663:
609:
607:{\displaystyle X_{2}}
579:
416:
414:{\displaystyle X_{1}}
385:
383:{\displaystyle X_{1}}
358:
338:
336:{\displaystyle X_{1}}
311:
309:{\displaystyle X_{i}}
281:
239:
221:
157:
155:{\displaystyle X_{i}}
126:
67:
65:{\displaystyle X_{i}}
1381:
1305:
1113:
1061:
1034:
960:
933:
729:
672:
618:
591:
432:
398:
367:
347:
320:
293:
236:
172:
139:
76:
49:
1290:Complete neutrality
1142:
752:
449:
72:of a random vector
1387:
1367:
1296:completely neutral
1276:
1116:
1106:is independent of
1096:
1047:
1017:
946:
916:
732:
712:
658:
604:
574:
435:
411:
380:
353:
333:
306:
276:
216:
152:
131:is neutral if the
121:
62:
1390:{\displaystyle X}
1266:
1206:
906:
852:
804:
564:
523:
488:
356:{\displaystyle X}
45:A single element
1505:
1477:
1476:
1474:
1450:
1444:
1443:
1434:(325): 194–206.
1423:
1396:
1394:
1393:
1388:
1376:
1374:
1373:
1368:
1345:
1344:
1326:
1325:
1285:
1283:
1282:
1277:
1272:
1268:
1267:
1265:
1264:
1263:
1245:
1244:
1228:
1227:
1218:
1207:
1205:
1204:
1203:
1185:
1184:
1168:
1167:
1152:
1141:
1136:
1105:
1103:
1102:
1097:
1095:
1094:
1073:
1072:
1056:
1054:
1053:
1048:
1046:
1045:
1026:
1024:
1023:
1018:
1013:
1012:
994:
993:
981:
980:
955:
953:
952:
947:
945:
944:
925:
923:
922:
917:
912:
908:
907:
905:
904:
903:
891:
890:
874:
873:
864:
853:
851:
850:
849:
837:
836:
820:
819:
810:
805:
803:
802:
801:
789:
788:
772:
771:
762:
751:
746:
721:
719:
718:
713:
708:
707:
689:
684:
683:
667:
665:
664:
659:
654:
653:
635:
630:
629:
613:
611:
610:
605:
603:
602:
583:
581:
580:
575:
570:
566:
565:
563:
562:
561:
545:
544:
535:
524:
522:
521:
520:
504:
503:
494:
489:
487:
486:
485:
469:
468:
459:
448:
443:
420:
418:
417:
412:
410:
409:
389:
387:
386:
381:
379:
378:
362:
360:
359:
354:
342:
340:
339:
334:
332:
331:
315:
313:
312:
307:
305:
304:
285:
283:
282:
277:
269:
268:
258:
253:
225:
223:
222:
217:
212:
211:
193:
192:
161:
159:
158:
153:
151:
150:
130:
128:
127:
122:
120:
119:
101:
100:
88:
87:
71:
69:
68:
63:
61:
60:
30:random variables
1513:
1512:
1508:
1507:
1506:
1504:
1503:
1502:
1483:
1482:
1481:
1480:
1452:
1451:
1447:
1440:10.2307/2283728
1425:
1424:
1420:
1415:
1403:
1379:
1378:
1336:
1317:
1303:
1302:
1292:
1255:
1236:
1229:
1219:
1195:
1176:
1169:
1153:
1150:
1146:
1111:
1110:
1080:
1064:
1059:
1058:
1037:
1032:
1031:
1004:
985:
972:
958:
957:
936:
931:
930:
895:
882:
875:
865:
841:
828:
821:
811:
793:
780:
773:
763:
760:
756:
727:
726:
699:
675:
670:
669:
645:
621:
616:
615:
594:
589:
588:
553:
546:
536:
512:
505:
495:
477:
470:
460:
457:
453:
430:
429:
425:of the vector
401:
396:
395:
370:
365:
364:
345:
344:
323:
318:
317:
296:
291:
290:
260:
234:
233:
203:
184:
170:
169:
142:
137:
136:
111:
92:
79:
74:
73:
52:
47:
46:
43:
12:
11:
5:
1511:
1509:
1501:
1500:
1495:
1485:
1484:
1479:
1478:
1465:(1): 183–189.
1445:
1417:
1416:
1414:
1411:
1410:
1409:
1402:
1399:
1386:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1343:
1339:
1335:
1332:
1329:
1324:
1320:
1316:
1313:
1310:
1291:
1288:
1287:
1286:
1275:
1271:
1262:
1258:
1254:
1251:
1248:
1243:
1239:
1235:
1232:
1226:
1222:
1216:
1213:
1210:
1202:
1198:
1194:
1191:
1188:
1183:
1179:
1175:
1172:
1166:
1163:
1160:
1156:
1149:
1145:
1140:
1135:
1132:
1129:
1126:
1123:
1119:
1093:
1090:
1087:
1083:
1079:
1076:
1071:
1067:
1057:is neutral if
1044:
1040:
1027:, is neutral.
1016:
1011:
1007:
1003:
1000:
997:
992:
988:
984:
979:
975:
971:
968:
965:
943:
939:
927:
926:
915:
911:
902:
898:
894:
889:
885:
881:
878:
872:
868:
862:
859:
856:
848:
844:
840:
835:
831:
827:
824:
818:
814:
808:
800:
796:
792:
787:
783:
779:
776:
770:
766:
759:
755:
750:
745:
742:
739:
735:
711:
706:
702:
698:
695:
692:
688:
682:
678:
657:
652:
648:
644:
641:
638:
634:
628:
624:
614:is neutral if
601:
597:
585:
584:
573:
569:
560:
556:
552:
549:
543:
539:
533:
530:
527:
519:
515:
511:
508:
502:
498:
492:
484:
480:
476:
473:
467:
463:
456:
452:
447:
442:
438:
408:
404:
390:is defined as
377:
373:
352:
330:
326:
303:
299:
287:
286:
275:
272:
267:
263:
257:
252:
249:
246:
242:
227:
226:
215:
210:
206:
202:
199:
196:
191:
187:
183:
180:
177:
149:
145:
118:
114:
110:
107:
104:
99:
95:
91:
86:
82:
59:
55:
42:
39:
26:neutral vector
13:
10:
9:
6:
4:
3:
2:
1510:
1499:
1496:
1494:
1491:
1490:
1488:
1473:
1468:
1464:
1460:
1456:
1449:
1446:
1441:
1437:
1433:
1429:
1422:
1419:
1412:
1408:
1405:
1404:
1400:
1398:
1384:
1361:
1355:
1352:
1349:
1341:
1337:
1333:
1330:
1327:
1322:
1318:
1311:
1308:
1299:
1297:
1289:
1273:
1269:
1260:
1256:
1252:
1249:
1246:
1241:
1237:
1233:
1230:
1224:
1220:
1214:
1211:
1208:
1200:
1196:
1192:
1189:
1186:
1181:
1177:
1173:
1170:
1164:
1161:
1158:
1154:
1147:
1143:
1138:
1133:
1130:
1127:
1124:
1121:
1117:
1109:
1108:
1107:
1091:
1088:
1085:
1081:
1077:
1074:
1069:
1065:
1042:
1038:
1028:
1009:
1005:
1001:
998:
995:
990:
986:
982:
977:
973:
966:
963:
941:
937:
913:
909:
900:
896:
892:
887:
883:
879:
876:
870:
866:
860:
857:
854:
846:
842:
838:
833:
829:
825:
822:
816:
812:
806:
798:
794:
790:
785:
781:
777:
774:
768:
764:
757:
753:
748:
743:
740:
737:
733:
725:
724:
723:
704:
700:
696:
693:
686:
680:
676:
650:
646:
642:
639:
632:
626:
622:
599:
595:
571:
567:
558:
554:
550:
547:
541:
537:
531:
528:
525:
517:
513:
509:
506:
500:
496:
490:
482:
478:
474:
471:
465:
461:
454:
450:
445:
440:
436:
428:
427:
426:
424:
406:
402:
393:
375:
371:
350:
328:
324:
301:
297:
273:
270:
265:
261:
255:
250:
247:
244:
240:
232:
231:
230:
208:
204:
200:
197:
194:
189:
185:
178:
175:
168:
167:
166:
163:
147:
143:
134:
116:
112:
108:
105:
102:
97:
93:
89:
84:
80:
57:
53:
40:
38:
35:
31:
27:
23:
19:
1462:
1458:
1448:
1431:
1427:
1421:
1300:
1295:
1293:
1029:
928:
586:
391:
288:
228:
164:
132:
44:
25:
15:
289:The values
1487:Categories
1413:References
41:Definition
18:statistics
1362:α
1356:
1350:∼
1331:…
1253:−
1250:⋯
1247:−
1234:−
1212:…
1193:−
1190:⋯
1187:−
1174:−
1139:∗
1128:…
1089:−
1078:…
999:…
893:−
880:−
858:…
839:−
826:−
791:−
778:−
749:∗
697:−
643:−
587:Variable
551:−
529:…
510:−
475:−
446:∗
241:∑
198:…
106:…
1401:See also
133:relative
392:neutral
363:, viz
229:where
929:Thus
24:, a
1467:doi
1436:doi
1353:Dir
1301:If
421:is
394:if
28:of
16:In
1489::
1461:.
1457:.
1432:64
1430:.
1298:.
274:1.
162:.
1475:.
1469::
1463:8
1442:.
1438::
1385:X
1365:)
1359:(
1347:)
1342:K
1338:X
1334:,
1328:,
1323:1
1319:X
1315:(
1312:=
1309:X
1274:.
1270:)
1261:j
1257:X
1242:1
1238:X
1231:1
1225:k
1221:X
1215:,
1209:,
1201:j
1197:X
1182:1
1178:X
1171:1
1165:1
1162:+
1159:j
1155:X
1148:(
1144:=
1134:j
1131:,
1125:,
1122:1
1118:X
1092:1
1086:j
1082:X
1075:,
1070:1
1066:X
1043:j
1039:X
1015:)
1010:k
1006:X
1002:,
996:,
991:3
987:X
983:,
978:2
974:X
970:(
967:=
964:Y
942:2
938:X
914:.
910:)
901:2
897:X
888:1
884:X
877:1
871:k
867:X
861:,
855:,
847:2
843:X
834:1
830:X
823:1
817:4
813:X
807:,
799:2
795:X
786:1
782:X
775:1
769:3
765:X
758:(
754:=
744:2
741:,
738:1
734:X
710:)
705:1
701:X
694:1
691:(
687:/
681:2
677:X
656:)
651:1
647:X
640:1
637:(
633:/
627:2
623:X
600:2
596:X
572:.
568:)
559:1
555:X
548:1
542:k
538:X
532:,
526:,
518:1
514:X
507:1
501:3
497:X
491:,
483:1
479:X
472:1
466:2
462:X
455:(
451:=
441:1
437:X
407:1
403:X
376:1
372:X
351:X
329:1
325:X
302:i
298:X
271:=
266:i
262:X
256:k
251:1
248:=
245:i
214:)
209:k
205:X
201:,
195:,
190:1
186:X
182:(
179:=
176:X
148:i
144:X
117:k
113:X
109:,
103:,
98:2
94:X
90:,
85:1
81:X
58:i
54:X
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