841:
75:
is to provide the basis for testing whether a set of observations can reasonably be modelled as arising from a specified distribution. Specifically, the probability integral transform is applied to construct an equivalent set of values, and a test is then made of whether a uniform distribution is
1531:
606:
95:
for a set of random variables is simplified or reduced in apparent complexity by applying the probability integral transform to each of the components and then working with a joint distribution for which the marginal variables have uniform distributions.
1342:
836:{\displaystyle {\begin{aligned}F_{Y}(y)&=\operatorname {P} (Y\leq y)\\&=\operatorname {P} (F_{X}(X)\leq y)\\&=\operatorname {P} (X\leq F_{X}^{-1}(y))\\&=F_{X}(F_{X}^{-1}(y))\\&=y\end{aligned}}}
43:. This holds exactly provided that the distribution being used is the true distribution of the random variables; if the distribution is one fitted to the data, the result will hold approximately in large samples.
1053:
611:
99:
A third use is based on applying the inverse of the probability integral transform to convert random variables from a uniform distribution to have a selected distribution: this is known as
1566:
1215:
384:
1334:
946:
1720:
981:
236:
888:
536:
1773:
1631:
340:
1820:
1133:
1091:
598:
454:
167:
1160:
908:
492:
266:
46:
The result is sometimes modified or extended so that the result of the transformation is a standard distribution other than the uniform distribution, such as the
1593:
1654:
1295:
1235:
556:
412:
318:
187:
129:
91:
which are a means of both defining and working with distributions for statistically dependent multivariate data. Here the problem of defining or manipulating a
1267:
298:
1526:{\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}{\rm {e}}^{-t^{2}/2}\,{\rm {d}}t={\frac {1}{2}}{\Big },\quad x\in \mathbb {R} ,\,}
1931:
59:
40:
986:
136:
92:
1539:
1165:
242:
348:
1300:
81:
1834:
913:
100:
1666:
951:
195:
1657:
849:
497:
47:
1731:
1598:
323:
1860:
132:
36:
1784:
1096:
1058:
561:
417:
142:
1138:
893:
459:
251:
8:
343:
88:
1575:
1893:
1874:"The Probability Integral Transformation When Parameters are Estimated from the Sample"
1639:
1280:
1220:
541:
397:
303:
172:
114:
20:
1240:
271:
1885:
32:
1778:
has a uniform distribution. Moreover, by symmetry of the uniform distribution,
1569:
1925:
72:
54:
77:
1897:
1873:
1725:
and the immediate result of the probability integral transform is that
890:
does not exist, then it can be replaced in this proof by the function
1889:
31:) relates to the result that data values that are modeled as being
87:
A second use for the transformation is in the theory related to
76:
appropriate for the constructed dataset. Examples of this are
71:
One use for the probability integral transform in statistical
1297:
be a random variable with a standard normal distribution
1048:{\displaystyle \chi (y)\equiv \inf\{x:F_{X}(x)\geq y\}}
1787:
1734:
1669:
1642:
1601:
1578:
1542:
1345:
1303:
1283:
1243:
1223:
1168:
1141:
1099:
1061:
989:
954:
916:
896:
852:
609:
564:
544:
500:
462:
420:
400:
351:
326:
306:
274:
254:
198:
175:
145:
117:
1814:
1767:
1714:
1648:
1625:
1587:
1560:
1525:
1328:
1289:
1261:
1229:
1209:
1154:
1127:
1085:
1047:
975:
940:
902:
882:
835:
592:
550:
530:
486:
448:
406:
378:
334:
312:
292:
260:
230:
181:
161:
123:
1499:
1491:
1472:
1452:
1923:
1005:
39:can be converted to random variables having a
1910:
1871:
1042:
1008:
1237:has a uniform distribution on the interval
1911:Casella, George; Berger, Roger L. (2002).
1857:The Oxford Dictionary of Statistical Terms
1522:
1515:
1496:
1457:
1426:
328:
224:
60:Statistical Methods for Research Workers
1851:
1849:
1561:{\displaystyle \operatorname {erf} (),}
1277:For a first, illustrative example, let
1210:{\displaystyle \mathrm {Uniform} (0,1)}
538:exists (i.e., if there exists a unique
1924:
137:cumulative distribution function (CDF)
1915:(2nd ed.). Theorem 2.1.10, p.54.
1872:David, F. N.; Johnson, N. L. (1948).
394:Given any random continuous variable
1846:
379:{\displaystyle \mu \circ F_{X}^{-1}}
1932:Theory of probability distributions
1329:{\displaystyle {\mathcal {N}}(0,1)}
13:
1608:
1429:
1397:
1384:
1346:
1306:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
970:
935:
718:
671:
640:
14:
1943:
1825:also has a uniform distribution.
1660:with unit mean, then its CDF is
941:{\displaystyle \chi (0)=-\infty }
1715:{\displaystyle F(x)=1-\exp(-x),}
976:{\displaystyle \chi (1)=\infty }
57:in his 1932 edition of the book
53:The transform was introduced by
1572:. Then the new random variable
1507:
111:Suppose that a random variable
66:
1904:
1865:
1809:
1800:
1762:
1753:
1706:
1697:
1679:
1673:
1617:
1611:
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1311:
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1116:
1110:
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999:
993:
964:
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926:
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813:
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724:
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658:
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624:
581:
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525:
519:
481:
469:
443:
437:
287:
275:
231:{\displaystyle Y:=F_{X}(X)\,,}
221:
215:
93:joint probability distribution
25:probability integral transform
1:
1840:
883:{\displaystyle F_{X}^{-1}(y)}
531:{\displaystyle F_{X}^{-1}(y)}
243:standard uniform distribution
41:standard uniform distribution
1768:{\displaystyle Y=1-\exp(-X)}
1626:{\displaystyle Y:=\Phi (X),}
1093:, with the same result that
335:{\displaystyle \mathbb {R} }
106:
16:Probability theory operation
7:
1828:
1272:
29:universality of the uniform
10:
1948:
1835:Inverse transform sampling
1815:{\displaystyle Z=\exp(-X)}
1633:is uniformly distributed.
1128:{\displaystyle F_{Y}(y)=y}
1086:{\displaystyle y\in (0,1)}
593:{\displaystyle F_{X}(x)=y}
449:{\displaystyle Y=F_{X}(X)}
268:is the uniform measure on
101:inverse transform sampling
1217:random variable, so that
169:Then the random variable
1658:exponential distribution
389:
82:Kolmogorov–Smirnov tests
48:exponential distribution
1861:Oxford University Press
133:continuous distribution
37:continuous distribution
1816:
1769:
1716:
1650:
1636:As second example, if
1627:
1589:
1562:
1527:
1330:
1291:
1263:
1231:
1211:
1156:
1129:
1087:
1049:
977:
942:
904:
884:
837:
594:
552:
532:
488:
450:
408:
380:
336:
314:
300:, the distribution of
294:
262:
232:
183:
163:
162:{\displaystyle F_{X}.}
125:
1913:Statistical Inference
1817:
1770:
1717:
1651:
1628:
1590:
1563:
1528:
1331:
1292:
1264:
1232:
1212:
1162:is just the CDF of a
1157:
1155:{\displaystyle F_{Y}}
1130:
1088:
1050:
978:
943:
905:
903:{\displaystyle \chi }
885:
838:
595:
553:
533:
489:
487:{\displaystyle y\in }
451:
409:
381:
337:
315:
295:
263:
233:
184:
164:
126:
1785:
1732:
1667:
1640:
1599:
1576:
1540:
1343:
1301:
1281:
1241:
1221:
1166:
1139:
1097:
1059:
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952:
914:
894:
850:
607:
562:
542:
498:
460:
418:
398:
349:
324:
304:
272:
261:{\displaystyle \mu }
252:
196:
173:
143:
115:
1393:
870:
803:
750:
518:
375:
344:pushforward measure
1812:
1765:
1712:
1646:
1623:
1588:{\displaystyle Y,}
1585:
1558:
1523:
1376:
1336:. Then its CDF is
1326:
1287:
1259:
1227:
1207:
1152:
1125:
1083:
1045:
973:
938:
910:, where we define
900:
880:
853:
833:
831:
786:
733:
590:
548:
528:
501:
484:
446:
404:
376:
358:
332:
310:
290:
258:
228:
179:
159:
121:
21:probability theory
1855:Dodge, Y. (2006)
1649:{\displaystyle X}
1487:
1486:
1448:
1374:
1373:
1290:{\displaystyle X}
1230:{\displaystyle Y}
551:{\displaystyle x}
407:{\displaystyle X}
313:{\displaystyle X}
248:Equivalently, if
182:{\displaystyle Y}
124:{\displaystyle X}
1939:
1917:
1916:
1908:
1902:
1901:
1869:
1863:
1853:
1821:
1819:
1818:
1813:
1774:
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1771:
1766:
1721:
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1655:
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1532:
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1524:
1518:
1503:
1502:
1495:
1494:
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1456:
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1449:
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1415:
1414:
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1392:
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1335:
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1332:
1327:
1310:
1309:
1296:
1294:
1293:
1288:
1268:
1266:
1265:
1262:{\displaystyle }
1260:
1236:
1234:
1233:
1228:
1216:
1214:
1213:
1208:
1191:
1161:
1159:
1158:
1153:
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1126:
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1108:
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1089:
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1054:
1052:
1051:
1046:
1026:
1025:
982:
980:
979:
974:
947:
945:
944:
939:
909:
907:
906:
901:
889:
887:
886:
881:
869:
861:
842:
840:
839:
834:
832:
819:
802:
794:
782:
781:
766:
749:
741:
711:
689:
688:
664:
623:
622:
599:
597:
596:
591:
574:
573:
557:
555:
554:
549:
537:
535:
534:
529:
517:
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493:
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490:
485:
455:
453:
452:
447:
436:
435:
413:
411:
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405:
385:
383:
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377:
374:
366:
341:
339:
338:
333:
331:
319:
317:
316:
311:
299:
297:
296:
293:{\displaystyle }
291:
267:
265:
264:
259:
237:
235:
234:
229:
214:
213:
188:
186:
185:
180:
168:
166:
165:
160:
155:
154:
130:
128:
127:
122:
33:random variables
1947:
1946:
1942:
1941:
1940:
1938:
1937:
1936:
1922:
1921:
1920:
1909:
1905:
1890:10.2307/2332638
1870:
1866:
1854:
1847:
1843:
1831:
1786:
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1733:
1730:
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1668:
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1641:
1638:
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1600:
1597:
1596:
1577:
1574:
1573:
1541:
1538:
1537:
1514:
1498:
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1489:
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1471:
1470:
1451:
1450:
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1428:
1427:
1416:
1410:
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1402:
1396:
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1388:
1380:
1361:
1344:
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1340:
1305:
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1302:
1299:
1298:
1282:
1279:
1278:
1275:
1242:
1239:
1238:
1222:
1219:
1218:
1169:
1167:
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1163:
1146:
1142:
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1136:
1104:
1100:
1098:
1095:
1094:
1060:
1057:
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1021:
1017:
988:
985:
984:
953:
950:
949:
915:
912:
911:
895:
892:
891:
862:
857:
851:
848:
847:
830:
829:
817:
816:
795:
790:
777:
773:
764:
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742:
737:
709:
708:
684:
680:
662:
661:
633:
618:
614:
610:
608:
605:
604:
569:
565:
563:
560:
559:
543:
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510:
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431:
427:
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367:
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350:
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327:
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321:
305:
302:
301:
273:
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269:
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209:
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174:
171:
170:
150:
146:
144:
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140:
116:
113:
112:
109:
69:
35:from any given
27:(also known as
17:
12:
11:
5:
1945:
1935:
1934:
1919:
1918:
1903:
1864:
1844:
1842:
1839:
1838:
1837:
1830:
1827:
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1822:
1811:
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1802:
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1675:
1672:
1645:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1584:
1581:
1570:error function
1557:
1554:
1551:
1548:
1545:
1534:
1533:
1521:
1517:
1513:
1510:
1506:
1501:
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1379:
1372:
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1325:
1322:
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1308:
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1249:
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1226:
1206:
1203:
1200:
1197:
1194:
1190:
1187:
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1149:
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1118:
1115:
1112:
1107:
1103:
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1070:
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1035:
1032:
1029:
1024:
1020:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
972:
969:
966:
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960:
957:
937:
934:
931:
928:
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922:
919:
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843:
828:
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822:
820:
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815:
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789:
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178:
158:
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149:
135:for which the
120:
108:
105:
68:
65:
15:
9:
6:
4:
3:
2:
1944:
1933:
1930:
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1927:
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73:data analysis
64:
62:
61:
56:
55:Ronald Fisher
51:
49:
44:
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67:Applications
58:
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189:defined as
1878:Biometrika
1841:References
558:such that
1804:−
1798:
1757:−
1751:
1745:−
1701:−
1695:
1689:−
1609:Φ
1547:
1512:∈
1468:
1404:−
1385:∞
1382:−
1378:∫
1371:π
1347:Φ
1066:∈
1037:≥
1003:≡
991:χ
971:∞
956:χ
936:∞
933:−
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898:χ
864:−
797:−
744:−
731:≤
722:
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675:
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644:
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512:−
467:∈
414:, define
369:−
356:∘
353:μ
256:μ
107:Statement
78:P–P plots
1926:Category
1829:See also
1273:Examples
1135:. Thus,
456:. Given
1898:2332638
1656:has an
1568:is the
342:is the
89:copulas
1896:
1536:where
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241:has a
131:has a
23:, the
1894:JSTOR
494:, if
390:Proof
1055:for
80:and
1886:doi
1795:exp
1748:exp
1692:exp
1544:erf
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1006:inf
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320:on
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276:[
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216:(
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207:F
200:Y
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157:.
152:X
148:F
119:X
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