20:
773:
928:
214:
1235:
1666:
1796:
attempt to only achieve the confidence level on each individual point, which can allow for a tighter bound. The DKW bounds runs parallel to, and is equally above and below, the empirical CDF. The equally spaced confidence interval around the empirical CDF allows for different rates of violations
1985:
598:
1441:
429:
23:
The above chart shows an example application of the DKW inequality in constructing confidence bounds (in purple) around an empirical distribution function (in light blue). In this random draw, the true CDF (orange) is entirely contained within the DKW
2077:
784:
70:
1797:
across the support of the distribution. In particular, it is more common for a CDF to be outside of the CDF bound estimated using the DKW inequality near the median of the distribution than near the endpoints of the distribution.
1009:
1484:
1861:
238:
and McCarty. In 2021, Michael Naaman proved the multivariate version of the DKW inequality and generalized
Massart's tightness result to the multivariate case, which results in a sharp constant of twice the dimension
768:{\displaystyle \Pr {\Bigl (}\sup _{x\in \mathbb {R} }{\bigl (}F_{n}(x)-F(x){\bigr )}>\varepsilon {\Bigr )}\leq e^{-2n\varepsilon ^{2}}\qquad {\text{for every }}\varepsilon \geq {\sqrt {{\tfrac {1}{2n}}\ln 2}},}
1290:
315:
1695:
1721:
1853:
1996:
923:{\displaystyle \Pr {\Bigl (}\sup _{x\in \mathbb {R} }|F_{n}(x)-F(x)|>\varepsilon {\Bigr )}\leq 2e^{-2n\varepsilon ^{2}}\qquad {\text{for every }}\varepsilon >0.}
545:
209:{\displaystyle \Pr {\Bigl (}\sup _{x\in \mathbb {R} }|F_{n}(x)-F(x)|>\varepsilon {\Bigr )}\leq Ce^{-2n\varepsilon ^{2}}\qquad {\text{for every }}\varepsilon >0.}
1748:
1827:
461:
1768:
1230:{\displaystyle \sup _{x\in \mathbb {R} }|F_{n}(x)-F(x)|\;{\stackrel {d}{=}}\;\sup _{x\in \mathbb {R} }|G_{n}(F(x))-F(x)|\leq \sup _{0\leq t\leq 1}|G_{n}(t)-t|,}
569:
509:
489:
1661:{\displaystyle \Pr {\Bigl (}{\sqrt {n}}\sup _{t\in [0,\infty )}|(1-G(t))(F_{n}(t)-F(t))|>\varepsilon {\Bigr )}\leq 2.5e^{-2\varepsilon ^{2}+C\varepsilon }}
1793:
1779:
1980:{\displaystyle F_{n}(x)-\varepsilon \leq F(x)\leq F_{n}(x)+\varepsilon \;{\text{ where }}\varepsilon ={\sqrt {\frac {\ln {\frac {2}{\alpha }}}{2n}}}}
2417:
285:
2422:
2358:
2327:
2145:
1436:{\displaystyle \Pr {\Bigl (}\sup _{t\in \mathbb {R} ^{k}}|F_{n}(t)-F(t)|>\varepsilon {\Bigr )}\leq (n+1)ke^{-2n\varepsilon ^{2}}}
954:
1990:
which is also a special case of the asymptotic procedure for the multivariate case, whereby one uses the following critical value
292:
49:
1785:
306:
45:
2373:
Bitouze, D.; Laurent, B.; Massart, P. (1999), "A Dvoretzky–Kiefer–Wolfowitz type inequality for the Kaplan–Meier estimator",
934:
1789:
946:
2132:
2098:; moreover, the multivariate test described by Naaman can be generalized to account for heterogeneity and dependence.
57:
1475:
2427:
1784:
The
Dvoretzky–Kiefer–Wolfowitz inequality is one method for generating CDF-based confidence bounds and producing a
1792:. The purpose of this confidence interval is to contain the entire CDF at the specified confidence level, while
2107:
424:{\displaystyle F_{n}(x)={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{\{X_{i}\leq x\}},\qquad x\in \mathbb {R} .}
2141:"Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator"
2187:
1674:
2386:
2225:"A distribution-free upper confidence bound for Pr{Y<X}, based on independent samples of X and Y"
938:
1700:
2300:
2072:{\displaystyle {\frac {d(\alpha ,k)}{\sqrt {n}}}={\sqrt {\frac {\ln {\frac {2k}{\alpha }}}{2n}}}}
1832:
2354:
2323:
514:
592: > 0 anywhere on the real line. More precisely, there is the one-sided estimate
2394:
2290:
2254:
2236:
2196:
2154:
2250:
2210:
2168:
1726:
2258:
2246:
2206:
2164:
2136:
2128:
1803:
575:
437:
288:
61:
53:
2390:
2346:
1753:
554:
494:
474:
235:
227:
2398:
2411:
2342:
2304:
2224:
2140:
2279:"On the tight constant in the multivariate Dvoretzky–Kiefer–Wolfowitz inequality"
2182:
29:
2295:
2278:
2241:
2159:
1462: + 1) term can be replaced with a 2 for any sufficiently large
33:
2201:
1478:
which is a right-censored data analog of the empirical distribution function
2318:
Kosorok, M.R. (2008), "Chapter 11: Additional
Empirical Process Results",
574:
The
Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the
2382:
19:
2183:"The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality"
945:
tends to infinity. It also estimates the tail probability of the
2320:
Introduction to
Empirical Processes and Semiparametric Inference
1474:
The
Dvoretzky–Kiefer–Wolfowitz inequality is obtained for the
1003:
where these are independent and
Uniform(0,1), and noting that
735:
1999:
1864:
1835:
1806:
1756:
1729:
1703:
1677:
1487:
1293:
1012:
787:
601:
557:
517:
497:
477:
440:
318:
73:
949:. The inequalities above follow from the case where
44:) provides a bound on the worst case distance of an
2351:
Empirical
Processes with Applications to Statistics
2094: + 1) for a test that holds for all
2110:– a summary of bounds on sets of random variables.
2071:
1979:
1847:
1821:
1762:
1742:
1715:
1689:
1660:
1435:
1229:
922:
767:
563:
539:
503:
483:
455:
423:
243:of the space in which the observations are found:
223:in front of the exponent on the right-hand side.
208:
1615:
1493:
1381:
1299:
868:
793:
686:
607:
219:with an unspecified multiplicative constant
154:
79:
1506:
1488:
1305:
1294:
1168:
1094:
1014:
799:
788:
613:
602:
85:
74:
234: = 2, confirming a conjecture due to
230:proved the inequality with the sharp constant
673:
632:
38:Dvoretzky–Kiefer–Wolfowitz–Massart inequality
8:
2082:for the multivariate test; one may replace 2
398:
379:
46:empirically determined distribution function
1780:CDF-based nonparametric confidence interval
1933:
1092:
1075:
551:of random variables that are smaller than
2294:
2240:
2200:
2158:
2042:
2032:
2000:
1998:
1955:
1945:
1934:
1912:
1869:
1863:
1834:
1805:
1800:The interval that contains the true CDF,
1755:
1734:
1728:
1702:
1676:
1641:
1630:
1614:
1613:
1602:
1569:
1533:
1509:
1498:
1492:
1491:
1486:
1425:
1411:
1380:
1379:
1368:
1338:
1329:
1321:
1317:
1316:
1308:
1298:
1297:
1292:
1284:is the multivariate empirical cdf, then
1219:
1198:
1189:
1171:
1159:
1120:
1111:
1105:
1104:
1097:
1084:
1079:
1077:
1076:
1070:
1040:
1031:
1025:
1024:
1017:
1011:
906:
897:
883:
867:
866:
855:
825:
816:
810:
809:
802:
792:
791:
786:
734:
732:
721:
712:
698:
685:
684:
672:
671:
641:
631:
630:
624:
623:
616:
606:
605:
600:
556:
522:
516:
496:
476:
439:
414:
413:
386:
378:
373:
366:
355:
341:
323:
317:
192:
183:
169:
153:
152:
141:
111:
102:
96:
95:
88:
78:
77:
72:
2223:Birnbaum, Z. W.; McCarty, R. C. (1958).
1770:is the censoring distribution function.
778:which also implies a two-sided estimate
18:
2120:
286:independent and identically distributed
2375:Annales de l'Institut Henri Poincaré B
7:
2272:
2270:
2268:
64:, who in 1956 proved the inequality
1750:is the Kaplan–Meier estimator, and
2283:Statistics and Probability Letters
1790:Kolmogorov–Smirnov confidence band
1710:
1525:
14:
2229:Annals of Mathematical Statistics
2146:Annals of Mathematical Statistics
1690:{\displaystyle \varepsilon >0}
982:is the empirical distribution of
1788:, which is sometimes called the
374:
293:cumulative distribution function
905:
720:
406:
307:empirical distribution function
191:
48:from its associated population
2418:Asymptotic theory (statistics)
2018:
2006:
1924:
1918:
1902:
1896:
1881:
1875:
1816:
1810:
1603:
1599:
1596:
1590:
1581:
1575:
1562:
1559:
1556:
1550:
1538:
1534:
1528:
1516:
1401:
1389:
1369:
1365:
1359:
1350:
1344:
1330:
1220:
1210:
1204:
1190:
1160:
1156:
1150:
1141:
1138:
1132:
1126:
1112:
1071:
1067:
1061:
1052:
1046:
1032:
964:has the same distributions as
856:
852:
846:
837:
831:
817:
668:
662:
653:
647:
588:by more than a given constant
534:
528:
450:
444:
335:
329:
142:
138:
132:
123:
117:
103:
1:
2399:10.1016/S0246-0203(99)00112-0
1240:with equality if and only if
1716:{\displaystyle C<\infty }
947:Kolmogorov–Smirnov statistic
2444:
1777:
1252:In the multivariate case,
2322:, Springer, p. 210,
2296:10.1016/j.spl.2021.109088
1848:{\displaystyle 1-\alpha }
1277:-dimensional vectors. If
1273:is an i.i.d. sequence of
935:Glivenko–Cantelli theorem
2423:Statistical inequalities
2277:Naaman, Michael (2021).
2108:Concentration inequality
1458: > 0. The (
540:{\displaystyle F_{n}(x)}
2242:10.1214/aoms/1177706631
2160:10.1214/aoms/1177728174
259:Given a natural number
2202:10.1214/aop/1176990746
2073:
1981:
1855:is often specified as
1849:
1823:
1794:alternative approaches
1764:
1744:
1717:
1697:and for some constant
1691:
1662:
1476:Kaplan–Meier estimator
1470:Kaplan–Meier estimator
1437:
1231:
953:corresponds to be the
924:
769:
565:
541:
505:
485:
457:
425:
371:
305:denote the associated
210:
25:
16:Statistical inequality
2188:Annals of Probability
2074:
1982:
1850:
1824:
1765:
1745:
1743:{\displaystyle F_{n}}
1718:
1692:
1663:
1438:
1232:
933:This strengthens the
925:
770:
566:
542:
506:
486:
458:
426:
351:
211:
50:distribution function
22:
2181:Massart, P. (1990),
1997:
1862:
1833:
1822:{\displaystyle F(x)}
1804:
1754:
1727:
1701:
1675:
1485:
1291:
1010:
955:uniform distribution
785:
599:
555:
515:
495:
475:
456:{\displaystyle F(x)}
438:
316:
71:
52:. It is named after
2391:1999AIHPB..35..735B
1829:, with probability
939:rate of convergence
937:by quantifying the
2069:
1977:
1845:
1819:
1774:Building CDF bands
1760:
1740:
1713:
1687:
1658:
1532:
1433:
1328:
1227:
1188:
1110:
1030:
920:
815:
765:
749:
629:
561:
537:
501:
481:
453:
421:
255:The DKW inequality
206:
101:
26:
2428:Empirical process
2067:
2066:
2055:
2027:
2026:
1975:
1974:
1963:
1937:
1936: where
1763:{\displaystyle G}
1505:
1503:
1304:
1248:Multivariate case
1167:
1093:
1089:
1013:
909:
798:
760:
748:
724:
612:
564:{\displaystyle x}
504:{\displaystyle x}
484:{\displaystyle X}
349:
195:
84:
28:In the theory of
2435:
2402:
2401:
2370:
2364:
2363:
2339:
2333:
2332:
2315:
2309:
2308:
2298:
2274:
2263:
2262:
2244:
2220:
2214:
2213:
2204:
2195:(3): 1269–1283,
2178:
2172:
2171:
2162:
2125:
2078:
2076:
2075:
2070:
2068:
2065:
2057:
2056:
2051:
2043:
2034:
2033:
2028:
2022:
2021:
2001:
1986:
1984:
1983:
1978:
1976:
1973:
1965:
1964:
1956:
1947:
1946:
1938:
1935:
1917:
1916:
1874:
1873:
1854:
1852:
1851:
1846:
1828:
1826:
1825:
1820:
1769:
1767:
1766:
1761:
1749:
1747:
1746:
1741:
1739:
1738:
1722:
1720:
1719:
1714:
1696:
1694:
1693:
1688:
1667:
1665:
1664:
1659:
1657:
1656:
1646:
1645:
1619:
1618:
1606:
1574:
1573:
1537:
1531:
1504:
1499:
1497:
1496:
1442:
1440:
1439:
1434:
1432:
1431:
1430:
1429:
1385:
1384:
1372:
1343:
1342:
1333:
1327:
1326:
1325:
1320:
1303:
1302:
1236:
1234:
1233:
1228:
1223:
1203:
1202:
1193:
1187:
1163:
1125:
1124:
1115:
1109:
1108:
1091:
1090:
1088:
1083:
1078:
1074:
1045:
1044:
1035:
1029:
1028:
929:
927:
926:
921:
910:
907:
904:
903:
902:
901:
872:
871:
859:
830:
829:
820:
814:
813:
797:
796:
774:
772:
771:
766:
761:
750:
747:
736:
733:
725:
722:
719:
718:
717:
716:
690:
689:
677:
676:
646:
645:
636:
635:
628:
627:
611:
610:
570:
568:
567:
562:
546:
544:
543:
538:
527:
526:
510:
508:
507:
502:
491:is smaller than
490:
488:
487:
482:
471:random variable
462:
460:
459:
454:
430:
428:
427:
422:
417:
402:
401:
391:
390:
377:
370:
365:
350:
342:
328:
327:
289:random variables
215:
213:
212:
207:
196:
193:
190:
189:
188:
187:
158:
157:
145:
116:
115:
106:
100:
99:
83:
82:
2443:
2442:
2438:
2437:
2436:
2434:
2433:
2432:
2408:
2407:
2406:
2405:
2372:
2371:
2367:
2361:
2341:
2340:
2336:
2330:
2317:
2316:
2312:
2276:
2275:
2266:
2222:
2221:
2217:
2180:
2179:
2175:
2127:
2126:
2122:
2117:
2104:
2058:
2044:
2035:
2002:
1995:
1994:
1966:
1948:
1908:
1865:
1860:
1859:
1831:
1830:
1802:
1801:
1786:confidence band
1782:
1776:
1752:
1751:
1730:
1725:
1724:
1699:
1698:
1673:
1672:
1637:
1626:
1565:
1483:
1482:
1472:
1421:
1407:
1334:
1315:
1289:
1288:
1282:
1271:
1265:
1258:
1250:
1244:is continuous.
1194:
1116:
1036:
1008:
1007:
1001:
995:
988:
980:
969:
962:
908:for every
893:
879:
821:
783:
782:
740:
723:for every
708:
694:
637:
597:
596:
582:
576:random function
553:
552:
518:
513:
512:
493:
492:
473:
472:
436:
435:
382:
372:
319:
314:
313:
303:
284:be real-valued
282:
276:
269:
257:
194:for every
179:
165:
107:
69:
68:
62:Jacob Wolfowitz
54:Aryeh Dvoretzky
17:
12:
11:
5:
2441:
2439:
2431:
2430:
2425:
2420:
2410:
2409:
2404:
2403:
2365:
2359:
2334:
2328:
2310:
2264:
2215:
2173:
2153:(3): 642–669,
2119:
2118:
2116:
2113:
2112:
2111:
2103:
2100:
2080:
2079:
2064:
2061:
2054:
2050:
2047:
2041:
2038:
2031:
2025:
2020:
2017:
2014:
2011:
2008:
2005:
1988:
1987:
1972:
1969:
1962:
1959:
1954:
1951:
1944:
1941:
1932:
1929:
1926:
1923:
1920:
1915:
1911:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1872:
1868:
1844:
1841:
1838:
1818:
1815:
1812:
1809:
1775:
1772:
1759:
1737:
1733:
1712:
1709:
1706:
1686:
1683:
1680:
1669:
1668:
1655:
1652:
1649:
1644:
1640:
1636:
1633:
1629:
1625:
1622:
1617:
1612:
1609:
1605:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1572:
1568:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1536:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1508:
1502:
1495:
1490:
1471:
1468:
1444:
1443:
1428:
1424:
1420:
1417:
1414:
1410:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1383:
1378:
1375:
1371:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1341:
1337:
1332:
1324:
1319:
1314:
1311:
1307:
1301:
1296:
1280:
1269:
1263:
1256:
1249:
1246:
1238:
1237:
1226:
1222:
1218:
1215:
1212:
1209:
1206:
1201:
1197:
1192:
1186:
1183:
1180:
1177:
1174:
1170:
1166:
1162:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1123:
1119:
1114:
1107:
1103:
1100:
1096:
1087:
1082:
1073:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1043:
1039:
1034:
1027:
1023:
1020:
1016:
999:
993:
986:
978:
967:
960:
931:
930:
919:
916:
913:
900:
896:
892:
889:
886:
882:
878:
875:
870:
865:
862:
858:
854:
851:
848:
845:
842:
839:
836:
833:
828:
824:
819:
812:
808:
805:
801:
795:
790:
776:
775:
764:
759:
756:
753:
746:
743:
739:
731:
728:
715:
711:
707:
704:
701:
697:
693:
688:
683:
680:
675:
670:
667:
664:
661:
658:
655:
652:
649:
644:
640:
634:
626:
622:
619:
615:
609:
604:
580:
560:
536:
533:
530:
525:
521:
500:
480:
452:
449:
446:
443:
432:
431:
420:
416:
412:
409:
405:
400:
397:
394:
389:
385:
381:
376:
369:
364:
361:
358:
354:
348:
345:
340:
337:
334:
331:
326:
322:
301:
280:
274:
267:
256:
253:
247: = 2
228:Pascal Massart
226:In 1990,
217:
216:
205:
202:
199:
186:
182:
178:
175:
172:
168:
164:
161:
156:
151:
148:
144:
140:
137:
134:
131:
128:
125:
122:
119:
114:
110:
105:
98:
94:
91:
87:
81:
76:
42:DKW inequality
15:
13:
10:
9:
6:
4:
3:
2:
2440:
2429:
2426:
2424:
2421:
2419:
2416:
2415:
2413:
2400:
2396:
2392:
2388:
2384:
2380:
2376:
2369:
2366:
2362:
2360:0-471-86725-X
2356:
2352:
2348:
2347:Wellner, J.A.
2344:
2343:Shorack, G.R.
2338:
2335:
2331:
2329:9780387749778
2325:
2321:
2314:
2311:
2306:
2302:
2297:
2292:
2288:
2284:
2280:
2273:
2271:
2269:
2265:
2260:
2256:
2252:
2248:
2243:
2238:
2234:
2230:
2226:
2219:
2216:
2212:
2208:
2203:
2198:
2194:
2190:
2189:
2184:
2177:
2174:
2170:
2166:
2161:
2156:
2152:
2148:
2147:
2142:
2138:
2137:Wolfowitz, J.
2134:
2130:
2129:Dvoretzky, A.
2124:
2121:
2114:
2109:
2106:
2105:
2101:
2099:
2097:
2093:
2089:
2085:
2062:
2059:
2052:
2048:
2045:
2039:
2036:
2029:
2023:
2015:
2012:
2009:
2003:
1993:
1992:
1991:
1970:
1967:
1960:
1957:
1952:
1949:
1942:
1939:
1930:
1927:
1921:
1913:
1909:
1905:
1899:
1893:
1890:
1887:
1884:
1878:
1870:
1866:
1858:
1857:
1856:
1842:
1839:
1836:
1813:
1807:
1798:
1795:
1791:
1787:
1781:
1773:
1771:
1757:
1735:
1731:
1707:
1704:
1684:
1681:
1678:
1653:
1650:
1647:
1642:
1638:
1634:
1631:
1627:
1623:
1620:
1610:
1607:
1593:
1587:
1584:
1578:
1570:
1566:
1553:
1547:
1544:
1541:
1522:
1519:
1513:
1510:
1500:
1481:
1480:
1479:
1477:
1469:
1467:
1465:
1461:
1457:
1453:
1449:
1426:
1422:
1418:
1415:
1412:
1408:
1404:
1398:
1395:
1392:
1386:
1376:
1373:
1362:
1356:
1353:
1347:
1339:
1335:
1322:
1312:
1309:
1287:
1286:
1285:
1283:
1276:
1272:
1262:
1255:
1247:
1245:
1243:
1224:
1216:
1213:
1207:
1199:
1195:
1184:
1181:
1178:
1175:
1172:
1164:
1153:
1147:
1144:
1135:
1129:
1121:
1117:
1101:
1098:
1085:
1080:
1064:
1058:
1055:
1049:
1041:
1037:
1021:
1018:
1006:
1005:
1004:
1002:
992:
985:
981:
974:
970:
963:
956:
952:
948:
944:
940:
936:
917:
914:
911:
898:
894:
890:
887:
884:
880:
876:
873:
863:
860:
849:
843:
840:
834:
826:
822:
806:
803:
781:
780:
779:
762:
757:
754:
751:
744:
741:
737:
729:
726:
713:
709:
705:
702:
699:
695:
691:
681:
678:
665:
659:
656:
650:
642:
638:
620:
617:
595:
594:
593:
591:
587:
584:differs from
583:
577:
572:
558:
550:
531:
523:
519:
498:
478:
470:
466:
447:
441:
418:
410:
407:
403:
395:
392:
387:
383:
367:
362:
359:
356:
352:
346:
343:
338:
332:
324:
320:
312:
311:
310:
308:
304:
297:
294:
290:
287:
283:
273:
266:
262:
254:
252:
250:
246:
242:
237:
233:
229:
224:
222:
203:
200:
197:
184:
180:
176:
173:
170:
166:
162:
159:
149:
146:
135:
129:
126:
120:
112:
108:
92:
89:
67:
66:
65:
63:
59:
55:
51:
47:
43:
39:
35:
31:
21:
2378:
2374:
2368:
2350:
2337:
2319:
2313:
2286:
2282:
2232:
2228:
2218:
2192:
2186:
2176:
2150:
2144:
2123:
2095:
2091:
2087:
2083:
2081:
1989:
1799:
1783:
1670:
1473:
1463:
1459:
1455:
1451:
1447:
1445:
1278:
1274:
1267:
1260:
1253:
1251:
1241:
1239:
997:
990:
983:
976:
972:
965:
958:
950:
942:
932:
777:
589:
585:
578:
573:
548:
468:
464:
433:
299:
295:
278:
271:
264:
260:
258:
248:
244:
240:
231:
225:
220:
218:
41:
37:
27:
2385:: 735–763,
2235:: 558–562.
1671:for every
465:probability
309:defined by
58:Jack Kiefer
30:probability
2412:Categories
2259:0087.34002
2133:Kiefer, J.
2115:References
1778:See also:
1446:for every
298:(·). Let
34:statistics
2353:, Wiley,
2305:233844405
2053:α
2040:
2010:α
1961:α
1953:
1940:ε
1931:ε
1906:≤
1891:≤
1888:ε
1885:−
1843:α
1840:−
1711:∞
1679:ε
1654:ε
1639:ε
1632:−
1621:≤
1611:ε
1585:−
1545:−
1526:∞
1514:∈
1423:ε
1413:−
1387:≤
1377:ε
1354:−
1313:∈
1214:−
1182:≤
1176:≤
1165:≤
1145:−
1102:∈
1056:−
1022:∈
912:ε
895:ε
885:−
874:≤
864:ε
841:−
807:∈
755:
730:≥
727:ε
710:ε
700:−
692:≤
682:ε
657:−
621:∈
411:∈
393:≤
353:∑
198:ε
181:ε
171:−
160:≤
150:ε
127:−
93:∈
2383:Elsevier
2349:(1986),
2139:(1956),
2102:See also
1723:, where
975:) where
957:on as
549:fraction
236:Birnbaum
2387:Bibcode
2289:: 1–8.
2251:0093874
2211:1062069
2169:0083864
547:is the
467:that a
463:is the
24:bounds.
2357:
2326:
2303:
2257:
2249:
2209:
2167:
511:, and
469:single
263:, let
60:, and
36:, the
2381:(6),
2301:S2CID
2086:with
1266:, …,
996:, …,
291:with
277:, …,
2355:ISBN
2324:ISBN
1708:<
1682:>
1608:>
1374:>
915:>
861:>
679:>
201:>
147:>
32:and
2395:doi
2291:doi
2287:173
2255:Zbl
2237:doi
2197:doi
2155:doi
1624:2.5
1507:sup
1306:sup
1169:sup
1095:sup
1015:sup
941:as
800:sup
614:sup
434:so
251:.
86:sup
2414::
2393:,
2379:35
2377:,
2345:;
2299:.
2285:.
2281:.
2267:^
2253:.
2247:MR
2245:.
2233:29
2231:.
2227:.
2207:MR
2205:,
2193:18
2191:,
2185:,
2165:MR
2163:,
2151:27
2149:,
2143:,
2135:;
2131:;
2037:ln
1950:ln
1489:Pr
1466:.
1454:,
1450:,
1295:Pr
1259:,
989:,
918:0.
789:Pr
752:ln
603:Pr
571:.
270:,
204:0.
75:Pr
56:,
2397::
2389::
2307:.
2293::
2261:.
2239::
2199::
2157::
2096:n
2092:n
2090:(
2088:k
2084:k
2063:n
2060:2
2049:k
2046:2
2030:=
2024:n
2019:)
2016:k
2013:,
2007:(
2004:d
1971:n
1968:2
1958:2
1943:=
1928:+
1925:)
1922:x
1919:(
1914:n
1910:F
1903:)
1900:x
1897:(
1894:F
1882:)
1879:x
1876:(
1871:n
1867:F
1837:1
1817:)
1814:x
1811:(
1808:F
1758:G
1736:n
1732:F
1705:C
1685:0
1651:C
1648:+
1643:2
1635:2
1628:e
1616:)
1604:|
1600:)
1597:)
1594:t
1591:(
1588:F
1582:)
1579:t
1576:(
1571:n
1567:F
1563:(
1560:)
1557:)
1554:t
1551:(
1548:G
1542:1
1539:(
1535:|
1529:)
1523:,
1520:0
1517:[
1511:t
1501:n
1494:(
1464:n
1460:n
1456:k
1452:n
1448:ε
1427:2
1419:n
1416:2
1409:e
1405:k
1402:)
1399:1
1396:+
1393:n
1390:(
1382:)
1370:|
1366:)
1363:t
1360:(
1357:F
1351:)
1348:t
1345:(
1340:n
1336:F
1331:|
1323:k
1318:R
1310:t
1300:(
1281:n
1279:F
1275:k
1270:n
1268:X
1264:2
1261:X
1257:1
1254:X
1242:F
1225:,
1221:|
1217:t
1211:)
1208:t
1205:(
1200:n
1196:G
1191:|
1185:1
1179:t
1173:0
1161:|
1157:)
1154:x
1151:(
1148:F
1142:)
1139:)
1136:x
1133:(
1130:F
1127:(
1122:n
1118:G
1113:|
1106:R
1099:x
1086:d
1081:=
1072:|
1068:)
1065:x
1062:(
1059:F
1053:)
1050:x
1047:(
1042:n
1038:F
1033:|
1026:R
1019:x
1000:n
998:U
994:2
991:U
987:1
984:U
979:n
977:G
973:F
971:(
968:n
966:G
961:n
959:F
951:F
943:n
899:2
891:n
888:2
881:e
877:2
869:)
857:|
853:)
850:x
847:(
844:F
838:)
835:x
832:(
827:n
823:F
818:|
811:R
804:x
794:(
763:,
758:2
745:n
742:2
738:1
714:2
706:n
703:2
696:e
687:)
674:)
669:)
666:x
663:(
660:F
654:)
651:x
648:(
643:n
639:F
633:(
625:R
618:x
608:(
590:ε
586:F
581:n
579:F
559:x
535:)
532:x
529:(
524:n
520:F
499:x
479:X
451:)
448:x
445:(
442:F
419:.
415:R
408:x
404:,
399:}
396:x
388:i
384:X
380:{
375:1
368:n
363:1
360:=
357:i
347:n
344:1
339:=
336:)
333:x
330:(
325:n
321:F
302:n
300:F
296:F
281:n
279:X
275:2
272:X
268:1
265:X
261:n
249:k
245:C
241:k
232:C
221:C
185:2
177:n
174:2
167:e
163:C
155:)
143:|
139:)
136:x
133:(
130:F
124:)
121:x
118:(
113:n
109:F
104:|
97:R
90:x
80:(
40:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
