1704:
2844:
The Joint Summer
Research Conferences in the Mathematical Sciences were held at the University of Massachusetts from June 7 to July 4, 1990. These were sponsored by the AMS, SIAM, and the Institute for Mathematical Statistics (IMS). Topics in 1990 were: Probability models and statistical analysis for
87:
The general solution to this full-information expected rank problem is unknown. The major difficulty is that the problem is fully history-dependent, that is, the optimal rule depends at every stage on all preceding values, and not only on simpler sufficient statistics of these. Only bounds are known
1717:
would be solved. But the major reason is to understand how to cope with full history dependence in a (deceptively easy-looking) problem. On the Ester's Book
International Conference in Israel (2006) Robbins' problem was accordingly named one of the four most important problems in the field of
1180:
100: < 2.329. It is known that there is some room to improve the lower bound by further computations for a truncated version of the problem. It is still not known how to improve on the upper bound which stems from the subclass of memoryless threshold rules.
1553:
978:
2845:
ranking data, Inverse scattering on the line, Deformation theory of algebras and quantization with applications to physics, Strategies for sequential search and selection in real time, Schottky problems, and Logic, fields, and subanalytic sets.
1019:
854:
83:'s sequentially and must stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation, and what is its corresponding value?
739:
772:
has finite second moment, then after subtracting the mean and dividing by the standard deviation, we get a distribution with mean zero and variance one. Consequently it suffices to study the case of
879:
328:
is the relative rank of the ith observation and n is the total number of items. This rule has added flexibility. A curtailed version thereof can be used to select an item with a given probability
103:
It was proposed the continuous time version of the problem where the observations follow a
Poisson arrival process of homogeneous rate 1. Under some assumptions, the corresponding value function
571:
667:
1548:
1494:
220:
424:
299:
1298:
492:
1699:{\displaystyle \Delta k_{n}=\left\lceil {\alpha {\sqrt {n}}\,\,-1/2\,\,+\,\,{\frac {\left({-2\zeta (-1/2)}\right){\sqrt {\alpha }}}{\sqrt {\pi }}}{n^{-1/4}}}\right\rceil }
1452:
1008:
2452:
1246:
1391:
1423:
1359:
874:
392:
1200:
372:
326:
439:
159:
130:
1321:
790:
770:
615:
591:
512:
346:
243:
179:
225:
A simple suboptimal rule, which performs almost as well as the optimal rule, was proposed by
Krieger & Samuel-Cahn. The rule stops with the smallest
798:
1175:{\displaystyle \beta _{n}=\alpha {\sqrt {n}}-1/2+{\frac {(-2\zeta (-1/2)){\sqrt {\alpha }}}{\sqrt {\pi }}}n^{-1/4}+O\left(n^{-7/24}\right)}
672:
593:
is the stopping time? The probability of eventually stopping must be 1 (that is, you are not allowed to keep sampling and never stop).
1924:. Athens Conference on Applied Probability and Time Series Analysis. Vol. 114. New York, NY: Springer New York. pp. 1–17.
1323:
tosses, with k heads and (n-k) tails, should one continue or should one stop? Since 1D random walk is recurrent, starting at any
1939:
132:
is bounded and
Lipschitz continuous, and the differential equation for this value function is derived. The limiting value of
517:
2174:
2153:
Elton, John H. (2023-06-06). "Exact
Solution to the Chow-Robbins Game for almost all n, using the Catalan Triangle".
2861:
2066:
1965:
71:
973:{\displaystyle \alpha =\left(1-\alpha ^{2}\right)\int _{0}^{\infty }e^{\lambda \alpha -\lambda ^{2}/2}d\lambda }
2019:
1871:
1821:
620:
1499:
1016:
When the game is a fair coin toss game, with heads being +1 and tails being -1, then there is a sharper result
1783:
1460:
16:
This article is about
Robbins' problem of optimal stopping. For the Robbins problem on Boolean algebras, see
2421:
184:
2062:"The secretary problem of minimizing the expected rank: a simple suboptimal approach with generalization"
397:
248:
1251:
445:
1774:
1431:
987:
1713:
One of the motivations to study
Robbins' problem is that with its solution all classical (four)
1224:
374:. The rule can be used to select two or more items. The problem of selecting a fixed percentage
2373:
1742:
1364:
1396:
617:
with finite second moment, there exists an optimal strategy, defined by a sequence of numbers
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8:
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1992:
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775:
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331:
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24:
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2222:
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2057:
1984:
1956:
1935:
1890:
1749:. Scientists working in the field of optimal stopping have since called this problem
1714:
39:
2417:
2390:
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2433:
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2295:
2253:
2214:
2120:
2075:
2028:
1974:
1925:
1880:
1830:
1792:
1719:
30:
1961:"The secretary problem: Minimizing the expected rank with i.i.d. random variables"
2104:
2010:
1858:
1812:
1770:
1734:
67:
35:
17:
2187:
Dvoretzky, Aryeh. "Existence and properties of certain optimal stopping rules."
1930:
434:
Another optimal stopping problem bearing
Robbins' name is the Chow–Robbins game:
981:
43:
2341:
2300:
2283:
2242:"Optimally stopping the sample mean of a Wiener process with an unknown drift"
1920:(1996). "Half-Prophets and Robbins' Problem of Minimizing the expected rank".
1496:, and it found an almost always optimal decision rule, of stopping as soon as
2855:
2476:
2399:
2350:
2309:
2265:
2226:
2134:
2125:
2108:
2080:
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2033:
2014:
1988:
1894:
1835:
1816:
1010:, the discrete time problem becomes the same as the continuous time problem.
980:
which can be proved by solving the same problem with continuous time, with a
849:{\displaystyle \lim _{n}\beta _{n}/{\sqrt {n}}\approx \alpha =0.8399236757}
1361:, the probability of eventually having more heads than tails is 1. So, if
2468:
2089:
2042:
1844:
1221:
is small, the asymptotic bound does not apply, and finding the value of
2358:
2324:
2218:
1996:
1902:
1885:
1866:
1797:
1778:
2203:"Existence of optimal stopping rules for linear and quadratic rewards"
514:, how to decide when to stop, in order to maximize the sample average
161:
presents the solution of
Robbins’ problem. It is shown that for large
2202:
1979:
1960:
2159:
734:{\displaystyle {\frac {1}{n}}(X_{1}+\cdots X_{n})\geq \beta _{n}}
2207:
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
2374:"Regarding stopping rules for Brownian motion and random walks"
1303:
For the fair coin toss, a strategy is a binary decision: after
1739:
International Conference on Search and Selection in Real Time
222:. This estimation coincides with the bounds mentioned above.
1768:
2284:"Explicit Solutions to Some Problems of Optimal Stopping"
1425:, it is tricky to decide whether to stop or continue.
1248:
is much more difficult. Even the simplest case, where
2422:"Rigorous Computer Analysis of the Chow–Robbins Game"
1747:
I should like to see this problem solved before I die
1556:
1502:
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1367:
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500:
448:
400:
380:
354:
334:
307:
251:
231:
187:
167:
138:
109:
2049:
1867:"Minimizing the expected rank with full information"
744:
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1698:
1542:
1488:
1446:
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1385:
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1315:
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1240:
1194:
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1002:
972:
868:
848:
784:
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661:
609:
585:
566:{\displaystyle {\frac {1}{n}}(X_{1}+\cdots X_{n})}
565:
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418:
386:
366:
340:
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237:
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173:
153:
124:
2412:
2853:
2451:Christensen, Sören; Fischer, Simon (June 2022).
2200:
1954:
1909:
1851:
1745:, 1990. He concluded his address with the words
803:
1948:
1209:
2189:Proc. Fifth Berkeley Symp. Math. Statist. Prob
96:goes to infinity, namely 1.908 <
2378:Bulletin of the American Mathematical Society
1915:
1857:
1737:presented the above described problem at the
2240:Simons, Gordon; Yao, Yi-Ching (1989-08-01).
2246:Stochastic Processes and Their Applications
2148:
2146:
2144:
2109:"On optimal stopping rules for $ S_{n}/n$ "
2102:
1300:are fair coin tosses, is not fully solved.
2003:
1779:"Optimal Selection Based on Relative Rank"
1393:, one should always continue. However, if
2389:
2340:
2299:
2239:
2233:
2201:Teicher, H.; Wolfowitz, J. (1966-12-01).
2181:
2158:
2124:
2079:
2032:
2009:
1978:
1929:
1884:
1834:
1805:
1796:
1610:
1609:
1605:
1604:
1589:
1588:
669:. The strategy is to keep sampling until
662:{\displaystyle \beta _{1},\beta _{2},...}
38:, is sometimes referred to as the fourth
2141:
1543:{\displaystyle k-(n-k)\geq \Delta k_{n}}
66:be independent, identically distributed
2015:"What is known about Robbins' Problem?"
1817:"What is known about Robbins' Problem?"
1489:{\displaystyle n\leq 9.06\times 10^{7}}
2854:
2371:
2325:"Optimal Stopping in a Markov Process"
2322:
1762:
215:{\displaystyle 1\leq w(t)\leq 2.33183}
2829:"Solution did not converge"
2329:The Annals of Mathematical Statistics
2288:The Annals of Mathematical Statistics
2278:
2152:
2096:
1811:
2438:10.4169/amer.math.monthly.120.10.893
1457:Elton found exact solutions for all
429:
13:
1557:
1527:
1013:This was proved independently by.
997:
925:
14:
2873:
2426:The American Mathematical Monthly
1922:Lecture Notes in Statistics (LNS)
792:with mean zero and variance one.
419:{\displaystyle 0<\alpha <1}
294:{\displaystyle R_{i}<ic/(n+i)}
42:or the problem of minimizing the
1753:. Robbins himself died in 2001.
1428:found an exact solution for all
745:Optimal strategy for very large
2838:
2493:
2444:
2406:
2391:10.1090/S0002-9904-1969-12140-3
2365:
2316:
2272:
2194:
2113:Illinois Journal of Mathematics
2067:Advances in Applied Probability
1966:Advances in Applied Probability
1293:{\displaystyle X_{1},X_{2},...}
876:is the solution to the equation
487:{\displaystyle X_{1},X_{2},...}
2457:Journal of Applied Probability
2020:Journal of Applied Probability
1872:Journal of Applied Probability
1822:Journal of Applied Probability
1645:
1628:
1521:
1509:
1348:
1336:
1098:
1095:
1078:
1066:
994:
715:
686:
560:
531:
438:Given an infinite sequence of
301:for a given constant c, where
288:
276:
203:
197:
148:
142:
119:
113:
1:
2173:: CS1 maint: date and year (
1784:Israel Journal of Mathematics
1756:
1708:
2486:
2258:10.1016/0304-4149(89)90084-7
1447:{\displaystyle n\leq 489241}
1003:{\displaystyle n\to \infty }
46:rank with full information.
7:
1931:10.1007/978-1-4612-0749-8_1
1210:Optimal strategy for small
10:
2878:
2323:Taylor, Howard M. (1968).
1769:Chow, Y.S.; Moriguti, S.;
1729:
1241:{\displaystyle \beta _{n}}
15:
2372:Walker, Leroy H. (1969).
1386:{\displaystyle k\leq n-k}
426:, of n, is also treated.
2499:
2013:; Swan, Yvik C. (2009).
1418:{\displaystyle k>n-k}
2862:Stochastic optimization
2342:10.1214/aoms/1177698259
2301:10.1214/aoms/1177697604
1354:{\displaystyle k,(n-k)}
869:{\displaystyle \alpha }
387:{\displaystyle \alpha }
88:for the limiting value
2126:10.1215/ijm/1256068146
2081:10.1239/aap/1261669585
2034:10.1239/jap/1238592113
1836:10.1239/jap/1110381374
1771:Robbins, Herbert Ellis
1700:
1544:
1490:
1448:
1419:
1387:
1355:
1317:
1294:
1242:
1196:
1195:{\displaystyle \zeta }
1176:
1004:
974:
870:
850:
786:
766:
735:
663:
611:
595:
587:
567:
508:
488:
420:
388:
368:
367:{\displaystyle P<1}
342:
322:
295:
239:
216:
175:
155:
126:
85:
2453:"On the Sn/n problem"
1701:
1545:
1491:
1449:
1420:
1388:
1356:
1318:
1295:
1243:
1204:Riemann zeta function
1197:
1177:
1005:
975:
871:
851:
787:
767:
736:
664:
612:
597:For any distribution
588:
568:
509:
489:
436:
421:
389:
369:
343:
323:
321:{\displaystyle R_{i}}
296:
240:
217:
176:
156:
127:
48:
2748:# Print the solution
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185:
165:
154:{\displaystyle w(t)}
136:
125:{\displaystyle w(t)}
107:
74:on . We observe the
29:Robbins' problem of
2793:with a residual of
2469:10.1017/jpr.2021.73
1918:Ferguson, S. Thomas
1863:Ferguson, S. Thomas
1775:Samuels, Stephen M.
1724:sequential analysis
929:
2219:10.1007/BF00535366
2107:(September 1965).
2058:Samuel-Cahn, Ester
2056:Krieger, Abba M.;
1957:Samuel-Cahn, Ester
1886:10.1007/bf02759948
1798:10.1007/bf02759948
1715:secretary problems
1696:
1540:
1486:
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1415:
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1000:
984:. At the limit of
970:
915:
866:
846:
811:
782:
762:
731:
659:
607:
583:
563:
504:
494:with distribution
484:
416:
384:
364:
338:
318:
291:
235:
212:
171:
151:
122:
25:probability theory
2775:"Solved α =
1941:978-0-387-94788-4
1916:Bruss, F.Thomas;
1666:
1665:
1658:
1586:
1316:{\displaystyle n}
1114:
1113:
1106:
1044:
832:
802:
785:{\displaystyle F}
765:{\displaystyle F}
684:
610:{\displaystyle F}
586:{\displaystyle n}
529:
507:{\displaystyle F}
442:random variables
430:Chow–Robbins game
341:{\displaystyle P}
238:{\displaystyle i}
174:{\displaystyle t}
40:secretary problem
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2335:(4): 1333–1344.
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2105:Robbins, Herbert
2100:
2094:
2093:
2083:
2074:(4): 1041–1058.
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2047:
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2011:Bruss, F. Thomas
2007:
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1720:optimal stopping
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68:random variables
31:optimal stopping
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2517:scipy.integrate
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2494:
2489:
2484:
2449:
2445:
2418:Wästlund, Johan
2414:Häggström, Olle
2411:
2407:
2370:
2366:
2321:
2317:
2294:(3): 993–1010.
2277:
2273:
2238:
2234:
2199:
2195:
2191:. Vol. 1. 1967.
2186:
2182:
2166:
2165:
2151:
2142:
2101:
2097:
2054:
2050:
2008:
2004:
1980:10.2307/1428183
1953:
1949:
1942:
1914:
1910:
1859:Bruss, F.Thomas
1856:
1852:
1810:
1806:
1767:
1763:
1759:
1735:Herbert Robbins
1732:
1711:
1680:
1673:
1669:
1668:
1653:
1637:
1618:
1614:
1613:
1611:
1596:
1581:
1577:
1573:
1564:
1560:
1555:
1552:
1551:
1534:
1530:
1501:
1498:
1497:
1480:
1476:
1462:
1459:
1458:
1433:
1430:
1429:
1398:
1395:
1394:
1366:
1363:
1362:
1328:
1325:
1324:
1308:
1305:
1304:
1272:
1268:
1259:
1255:
1253:
1250:
1249:
1232:
1228:
1226:
1223:
1222:
1215:
1187:
1184:
1183:
1158:
1151:
1147:
1143:
1127:
1120:
1116:
1101:
1087:
1065:
1063:
1052:
1039:
1027:
1023:
1021:
1018:
1017:
989:
986:
985:
954:
948:
944:
934:
930:
924:
919:
904:
900:
893:
889:
881:
878:
877:
861:
858:
857:
827:
822:
816:
812:
806:
800:
797:
796:
777:
774:
773:
757:
754:
753:
750:
725:
721:
709:
705:
693:
689:
676:
674:
671:
670:
641:
637:
628:
624:
622:
619:
618:
602:
599:
598:
578:
575:
574:
554:
550:
538:
534:
521:
519:
516:
515:
499:
496:
495:
466:
462:
453:
449:
447:
444:
443:
432:
399:
396:
395:
379:
376:
375:
353:
350:
349:
333:
330:
329:
312:
308:
306:
303:
302:
271:
256:
252:
250:
247:
246:
230:
227:
226:
186:
183:
182:
166:
163:
162:
137:
134:
133:
108:
105:
104:
82:
65:
56:
36:Herbert Robbins
21:
18:Robbins algebra
12:
11:
5:
2875:
2865:
2864:
2848:
2847:
2837:
2529:scipy.optimize
2500:
2491:
2490:
2488:
2485:
2483:
2482:
2463:(2): 571–583.
2443:
2405:
2364:
2315:
2271:
2252:(2): 347–354.
2232:
2213:(4): 361–368.
2193:
2180:
2140:
2119:(3): 444–454.
2095:
2048:
2002:
1973:(3): 828–852.
1955:Assaf, David;
1947:
1940:
1908:
1879:(3): 616–626.
1850:
1829:(1): 108–120.
1804:
1760:
1758:
1755:
1731:
1728:
1710:
1707:
1694:
1687:
1683:
1679:
1676:
1672:
1664:
1657:
1651:
1647:
1644:
1640:
1636:
1633:
1630:
1627:
1624:
1621:
1617:
1608:
1603:
1599:
1595:
1592:
1585:
1580:
1576:
1572:
1567:
1563:
1559:
1537:
1533:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1483:
1479:
1475:
1472:
1469:
1466:
1443:
1440:
1437:
1414:
1411:
1408:
1405:
1402:
1382:
1379:
1376:
1373:
1370:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1312:
1289:
1286:
1283:
1280:
1275:
1271:
1267:
1262:
1258:
1235:
1231:
1214:
1208:
1191:
1170:
1165:
1161:
1157:
1154:
1150:
1146:
1142:
1139:
1134:
1130:
1126:
1123:
1119:
1112:
1105:
1100:
1097:
1094:
1090:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1062:
1059:
1055:
1051:
1048:
1043:
1038:
1035:
1030:
1026:
999:
996:
993:
982:Wiener process
969:
966:
961:
957:
951:
947:
943:
940:
937:
933:
927:
922:
918:
913:
907:
903:
899:
896:
892:
888:
885:
865:
845:
842:
839:
836:
831:
825:
819:
815:
809:
805:
781:
761:
749:
743:
728:
724:
720:
717:
712:
708:
704:
701:
696:
692:
688:
683:
680:
658:
655:
652:
649:
644:
640:
636:
631:
627:
606:
582:
562:
557:
553:
549:
546:
541:
537:
533:
528:
525:
503:
483:
480:
477:
474:
469:
465:
461:
456:
452:
431:
428:
415:
412:
409:
406:
403:
383:
363:
360:
357:
337:
315:
311:
290:
287:
284:
281:
278:
274:
270:
267:
264:
259:
255:
234:
211:
208:
205:
202:
199:
196:
193:
190:
170:
150:
147:
144:
141:
121:
118:
115:
112:
78:
61:
54:
34:, named after
9:
6:
4:
3:
2:
2874:
2863:
2860:
2859:
2857:
2841:
2496:
2492:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2447:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2409:
2401:
2397:
2392:
2387:
2383:
2379:
2375:
2368:
2360:
2356:
2352:
2348:
2343:
2338:
2334:
2330:
2326:
2319:
2311:
2307:
2302:
2297:
2293:
2289:
2285:
2282:(June 1969).
2281:
2275:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2236:
2228:
2224:
2220:
2216:
2212:
2208:
2204:
2197:
2190:
2184:
2176:
2170:
2161:
2156:
2149:
2147:
2145:
2136:
2132:
2127:
2122:
2118:
2114:
2110:
2106:
2103:Chow, Y. S.;
2099:
2091:
2087:
2082:
2077:
2073:
2069:
2068:
2063:
2059:
2052:
2044:
2040:
2035:
2030:
2026:
2022:
2021:
2016:
2012:
2006:
1998:
1994:
1990:
1986:
1981:
1976:
1972:
1968:
1967:
1962:
1958:
1951:
1943:
1937:
1932:
1927:
1923:
1919:
1912:
1904:
1900:
1896:
1892:
1887:
1882:
1878:
1874:
1873:
1868:
1864:
1860:
1854:
1846:
1842:
1837:
1832:
1828:
1824:
1823:
1818:
1814:
1808:
1799:
1794:
1790:
1786:
1785:
1780:
1776:
1772:
1765:
1761:
1754:
1752:
1748:
1744:
1740:
1736:
1727:
1725:
1721:
1716:
1706:
1692:
1685:
1681:
1677:
1674:
1670:
1662:
1655:
1649:
1642:
1638:
1634:
1631:
1625:
1622:
1619:
1615:
1606:
1601:
1597:
1593:
1590:
1583:
1578:
1574:
1570:
1565:
1561:
1535:
1531:
1524:
1518:
1515:
1512:
1506:
1503:
1481:
1477:
1473:
1470:
1467:
1464:
1455:
1441:
1438:
1435:
1426:
1412:
1409:
1406:
1403:
1400:
1380:
1377:
1374:
1371:
1368:
1345:
1342:
1339:
1333:
1330:
1310:
1301:
1287:
1284:
1281:
1278:
1273:
1269:
1265:
1260:
1256:
1233:
1229:
1220:
1213:
1207:
1205:
1189:
1168:
1163:
1159:
1155:
1152:
1148:
1144:
1140:
1137:
1132:
1128:
1124:
1121:
1117:
1110:
1103:
1092:
1088:
1084:
1081:
1075:
1072:
1069:
1060:
1057:
1053:
1049:
1046:
1041:
1036:
1033:
1028:
1024:
1014:
1011:
991:
983:
967:
964:
959:
955:
949:
945:
941:
938:
935:
931:
920:
916:
911:
905:
901:
897:
894:
890:
886:
883:
863:
843:
840:
837:
834:
829:
823:
817:
813:
807:
793:
779:
759:
748:
742:
726:
722:
718:
710:
706:
702:
699:
694:
690:
681:
678:
656:
653:
650:
647:
642:
638:
634:
629:
625:
604:
594:
580:
555:
551:
547:
544:
539:
535:
526:
523:
501:
481:
478:
475:
472:
467:
463:
459:
454:
450:
441:
435:
427:
413:
410:
407:
404:
401:
381:
361:
358:
355:
335:
313:
309:
285:
282:
279:
272:
268:
265:
262:
257:
253:
232:
223:
209:
206:
200:
194:
191:
188:
168:
145:
139:
116:
110:
101:
99:
95:
91:
84:
81:
77:
73:
69:
64:
60:
53:
47:
45:
41:
37:
33:
32:
26:
19:
2840:
2495:
2460:
2456:
2446:
2429:
2425:
2408:
2384:(1): 46–50.
2381:
2377:
2367:
2332:
2328:
2318:
2291:
2287:
2280:Shepp, L. A.
2274:
2249:
2245:
2235:
2210:
2206:
2196:
2188:
2183:
2116:
2112:
2098:
2071:
2065:
2051:
2024:
2018:
2005:
1970:
1964:
1950:
1921:
1911:
1876:
1870:
1853:
1826:
1820:
1807:
1791:(2): 81–90.
1788:
1782:
1764:
1750:
1746:
1738:
1733:
1712:
1456:
1427:
1302:
1218:
1216:
1211:
1015:
1012:
844:0.8399236757
794:
751:
746:
596:
437:
433:
224:
102:
97:
93:
89:
86:
79:
75:
62:
58:
51:
49:
28:
22:
2432:(10): 893.
2027:(1): 1–18.
795:With this,
2169:cite arXiv
2160:2205.13499
1757:References
1709:Importance
245:such that
2487:Footnotes
2477:0021-9002
2400:0002-9904
2351:0003-4851
2310:0003-4851
2266:0304-4149
2227:1432-2064
2135:0019-2082
1989:0001-8678
1895:0021-9002
1675:−
1663:π
1656:α
1632:−
1626:ζ
1620:−
1591:−
1579:α
1558:Δ
1528:Δ
1525:≥
1516:−
1507:−
1474:×
1468:≤
1439:≤
1410:−
1378:−
1372:≤
1343:−
1230:β
1190:ζ
1153:−
1122:−
1111:π
1104:α
1082:−
1076:ζ
1070:−
1047:−
1037:α
1025:β
998:∞
995:→
968:λ
946:λ
942:−
939:α
936:λ
926:∞
917:∫
902:α
898:−
884:α
864:α
838:α
835:≈
814:β
723:β
719:≥
703:⋯
639:β
626:β
548:⋯
408:α
382:α
207:≤
192:≤
2856:Category
2799:solution
2781:solution
2754:solution
2724:equation
2712:solution
2679:integral
2619:integral
2607:equation
2420:(2013).
2090:27793918
2060:(2009).
2043:30040773
1959:(1996).
1865:(1993).
1845:30040773
1815:(2005).
1777:(1964).
1693:⌉
1575:⌈
856:, where
57:, ... ,
44:expected
2760:success
2730:0.83992
2586:lambda_
2574:lambda_
2547:lambda_
2359:2239702
1997:1428183
1903:3214770
1743:Amherst
1730:History
1202:is the
210:2.33183
72:uniform
2811:"
2676:return
2559:return
2532:import
2520:import
2502:import
2475:
2398:
2357:
2349:
2308:
2264:
2225:
2133:
2088:
2041:
1995:
1987:
1938:
1901:
1893:
1843:
1442:489241
1182:where
573:where
2823:print
2766:print
2742:1e-15
2709:alpha
2694:alpha
2670:alpha
2625:error
2613:alpha
2580:alpha
2553:alpha
2505:numpy
2355:JSTOR
2155:arXiv
2086:JSTOR
2039:JSTOR
1993:JSTOR
1899:JSTOR
1841:JSTOR
1550:where
1217:When
2817:else
2718:root
2661:args
2631:quad
2535:root
2526:from
2523:quad
2514:from
2473:ISSN
2396:ISSN
2347:ISSN
2306:ISSN
2262:ISSN
2223:ISSN
2175:link
2131:ISSN
1985:ISSN
1936:ISBN
1891:ISSN
1722:and
1471:9.06
1404:>
411:<
405:<
359:<
263:<
50:Let
2805:fun
2736:tol
2655:inf
2604:def
2568:exp
2538:def
2465:doi
2434:doi
2430:120
2386:doi
2337:doi
2296:doi
2254:doi
2215:doi
2121:doi
2076:doi
2029:doi
1975:doi
1926:doi
1881:doi
1831:doi
1793:doi
1741:in
804:lim
752:If
440:IID
92:as
23:In
2858::
2751:if
2697:**
2673:))
2649:np
2616:):
2589:**
2562:np
2556:):
2511:np
2508:as
2471:.
2461:59
2459:.
2455:.
2428:.
2424:.
2416:;
2394:.
2382:75
2380:.
2376:.
2353:.
2345:.
2333:39
2331:.
2327:.
2304:.
2292:40
2290:.
2286:.
2260:.
2250:32
2248:.
2244:.
2221:.
2209:.
2205:.
2171:}}
2167:{{
2143:^
2129:.
2115:.
2111:.
2084:.
2072:41
2070:.
2064:.
2037:.
2025:46
2023:.
2017:.
1991:.
1983:.
1971:28
1969:.
1963:.
1934:.
1897:.
1889:.
1877:30
1875:.
1869:.
1861:;
1839:.
1827:42
1825:.
1819:.
1787:.
1781:.
1773:;
1726:.
1478:10
1454:.
1206:.
1164:24
741:.
394:,
348:,
181:,
70:,
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2826:(
2820::
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2802:.
2796:{
2790:}
2787:x
2784:.
2778:{
2772:f
2769:(
2763::
2757:.
2745:)
2739:=
2733:,
2727:,
2721:(
2715:=
2706:-
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2700:2
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2685:(
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2667:(
2664:=
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2652:.
2646:,
2643:0
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2610:(
2601:)
2598:2
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2577:*
2571:(
2565:.
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2544:(
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2436::
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2388::
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2298::
2268:.
2256::
2229:.
2217::
2211:5
2177:)
2163:.
2157::
2137:.
2123::
2117:9
2092:.
2078::
2045:.
2031::
1999:.
1977::
1944:.
1928::
1905:.
1883::
1847:.
1833::
1801:.
1795::
1789:2
1686:4
1682:/
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1629:(
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1616:(
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1234:n
1219:n
1212:n
1169:)
1160:/
1156:7
1149:n
1145:(
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1138:+
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1093:2
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1079:(
1073:2
1067:(
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1034:=
1029:n
992:n
965:d
960:2
956:/
950:2
932:e
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906:2
895:1
891:(
887:=
841:=
830:n
824:/
818:n
808:n
780:F
760:F
747:n
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716:)
711:n
707:X
700:+
695:1
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687:(
682:n
679:1
657:.
654:.
651:.
648:,
643:2
635:,
630:1
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581:n
561:)
556:n
552:X
545:+
540:1
536:X
532:(
527:n
524:1
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482:.
479:.
476:.
473:,
468:2
464:X
460:,
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414:1
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362:1
356:P
336:P
314:i
310:R
289:)
286:i
283:+
280:n
277:(
273:/
269:c
266:i
258:i
254:R
233:i
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201:t
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195:w
189:1
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146:t
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140:w
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117:t
114:(
111:w
98:v
94:n
90:v
80:k
76:X
63:n
59:X
55:1
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20:.
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