3009:
3025:
2650:
2217:
3004:{\displaystyle \mathbb {P} (q(X)\geq q^{*}|p_{\theta }\leq p_{0}^{*},\theta )={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{\mathbb {P} (p_{\theta }\leq p_{0}^{*}|\theta )}}={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{p_{0}^{*}}}={\frac {\mathbb {P} (q(X)\geq q^{*}|\theta )}{\mathbb {P} (q(X)>q^{*}|0)}}.}
387:
1922:
1188:
3640:
theory) versions of the confidence principle are incompatible with the likelihood principle, and therefore no frequentist method can be regarded as a truly complete solution to the problems raised by considering conditional properties of confidence intervals.
3805:
1714:
from the background-only hypothesis, and thus declaring that such small values are excluded (in favor of the background-only hypothesis) seems inappropriate. To overcome this difficulty Zech suggested conditioning the probability that
83:
Upper limits derived with the CLs method always contain the zero value of the parameter and hence the coverage probability at this point is always 100%. The definition of CLs does not follow from any precise theoretical framework of
251:
2212:{\displaystyle \mathbb {P} (n\leq n^{*}|n_{b}\leq n^{*},s+b)={\frac {\mathbb {P} (n\leq n^{*},n_{b}\leq n^{*}|s+b)}{\mathbb {P} (n_{b}\leq n^{*}|s+b)}}={\frac {\mathbb {P} (n\leq n^{*}|s+b)}{\mathbb {P} (n\leq n^{*}|b)}}.}
2612:
3180:: if the outcome of an experiment is to be only reported in a form of a "accept"/"reject" decision, then the overall procedure is equivalent to an experiment that has only two possible outcomes, with probabilities
1042:
482:
1916:. That is, not the over-all error probability should be reported but the conditional probability given the knowledge one has on the number of background events in the sample. This conditional probability is
3105:, and thus such considerations may only be used to suggest plausibility, but not theoretical completeness from the foundational point of view. (The same however can be said on any frequentist method if the
2401:
1529:
2283:
1253:
The original motivation for CLs was based on a conditional probability calculation suggested by physicist G. Zech for an event counting experiment. Suppose an experiment consists of measuring
741:
1358:
566:
159:
80:
necessarily produce empty intervals with some fixed probability when the parameter value is zero, and this property is considered undesirable by most physicists and statisticians.
3445:, making the basis of such an interpretation questionable. Birnbaum later described this as having "at most heuristic, but not substantial, value for evidential interpretation".
3694:
3443:
3399:
3150:
838:
3861:
3682:
3401:
and hence should determine the evidential interpretation of this result. (Since, for a test of two simple hypotheses, the likelihood ratio is a compact representation of the
1786:
1223:
605:
513:
241:
198:
3599:
1014:
3328:
915:
863:
2454:
1746:
946:
890:
689:
638:
3230:
3534:
3256:
1646:
1555:
972:
3282:
771:
3944:
3905:
3885:
3632:
Such definition of confidence can naturally seem to be satisfied by the definition of CLs. It remains true that both this and the more common (as associated with the
3198:
3174:
2494:
2474:
1672:
1614:
1034:
662:
3828:
3626:
3561:
3508:
3481:
3359:
2642:
1894:
1867:
1840:
1813:
1445:
798:
2320:
1581:
2517:
2427:
4381:
Robert D. Cousins (2011). "Negatively Biased
Relevant Subsets Induced by the Most-Powerful One-Sided Upper Confidence Limits for a Bounded Physical Parameter".
1914:
1712:
1692:
1465:
1418:
1398:
1378:
1315:
1295:
1271:
382:{\displaystyle {\frac {\mathbb {P} (\theta _{up}(X)<\theta |\theta )}{\mathbb {P} (\theta _{up}(X)<\theta |0)}}\leq \alpha '{\text{ for all }}\theta .}
3405:). On the other hand, if the likelihood principle is to be followed consistently, then the likelihood ratio of the original outcome should be used and not
1233:
Upper limits based on the CLs method were used in numerous publications of experimental results obtained at particle accelerator experiments such as
2525:
4167:
1183:{\displaystyle {\frac {\mathbb {P} (q_{\theta }(X)\geq q_{\theta }^{*}|\theta )}{\mathbb {P} (q_{\theta }(X)\geq q_{\theta }^{*}|0)}}=\alpha '.}
3035:
4218:"The Neyman-Pearson Theory as Decision Theory, and as Inference Theory; with a Criticism of the Lindley-Savage Argument for Bayesian Theory"
423:
3085:
of statistical inference, although they express a more generalized notion of conditionality which do not require the existence of an
2332:
72:, however it differs from standard confidence intervals in that the stated confidence level of the interval is not equal to its
4422:
1470:
3043:
2229:
1842:
is small the procedure is more likely to produce an error (i.e., an interval that does not cover the true value) than when
3831:
414:
3068:
1694:. Such a result is difficult to interpret because the experiment cannot essentially distinguish very small values of
4258:
G. Cowan; K. Cranmer; E. Gross; O. Vitells (2011). "Asymptotic formulae for likelihood-based tests of new physics".
694:
3452:, which, unlike the more common version, refers to error probabilities of both kinds. This is stated as follows:
4417:
1320:
525:
126:
3800:{\displaystyle \theta _{up}={\hat {\theta }}+\sigma \Phi ^{-1}(1-\alpha '\Phi ({\hat {\theta }}/\sigma )),}
3685:
77:
4016:
1400:
is the parameter to be estimated by the experiment. The standard procedure for setting an upper limit on
3408:
3364:
3115:
803:
3837:
3448:
A more direct approach leading to a similar conclusion can be found in
Birnbaum's formulation of the
1751:
1196:
641:
571:
207:
3566:
3050:
983:
3287:
3106:
3090:
3082:
117:
3039:
2432:
2223:
1718:
924:
868:
667:
616:
3656:
3203:
2222:
which correspond to the above definition of CLs. The first equality just uses the definition of
487:
172:
3513:
3235:
1619:
1534:
951:
3261:
895:
843:
750:
44:
used for parameters that can take only non-negative values. Although CLs are said to refer to
4402:
3998:
3929:
3890:
3870:
3183:
3159:
2479:
2459:
1651:
1586:
1019:
647:
85:
4277:
4196:
4136:
3972:
3813:
3649:
If certain regularity conditions are met, then a general likelihood function will become a
3604:
3539:
3486:
3459:
3337:
3177:
3102:
3094:
2620:
1872:
1845:
1818:
1791:
1423:
1274:
776:
418:
73:
69:
2476:
is unknown) random variable, whose distribution is uniform between 0 and 1 independent of
2296:
1815:
is the (unmeasurable) number of background events. The reasoning behind this is that when
8:
3456:"A concept of statistical evidence is not plausible unless it finds 'strong evidence for
3402:
3331:
3086:
2285:
and the number of background events is by definition independent of the signal strength.
1560:
61:
49:
41:
4281:
4140:
4121:
3976:
2499:
2409:
405:
The inequality is used in the definition to account for cases where the distribution of
4382:
4369:
4351:
4330:
4267:
4237:
4184:
4102:
4063:
3916:
3912:
3908:
3864:
2293:
Zech's conditional argument can be formally extended to the general case. Suppose that
1899:
1697:
1677:
1450:
1403:
1383:
1363:
1300:
1280:
1256:
417:
then this should be replaced by an equality. Note that the definition implies that the
4342:
Fraser, D. A. S.; Reid N.; Wong, A. C. M. (2004). "Inference for bounded parameters".
3984:
4289:
4148:
3923:
3919:
3650:
744:
4373:
4241:
4106:
3915:
matrix or by using the "Asimov" data set. This result happens to be equivalent to a
4361:
4320:
4285:
4229:
4176:
4144:
4094:
4053:
3980:
45:
17:
409:
is discrete and an equality can not be achieved precisely. If the distribution of
4192:
519:
201:
3653:
in the large sample limit. In such case the CLs upper limit at confidence level
4365:
4213:
4162:
3098:
3093:
however, already in its original more restricted version, formally implies the
2323:
978:
97:
4411:
2607:{\displaystyle p_{\theta }\leq \mathbb {P} (q(X)>q^{*}|0)\equiv p_{0}^{*}}
113:
68:
method in the sense that the properties of the limit are defined by means of
48:, "The method's name is ... misleading, as the CLs exclusion region is not a
4325:
4308:
4058:
4041:
4003:
3963:
Read, A. L. (2002). "Presentation of search results: The CL(s) technique".
3637:
3633:
76:. The reason for this deviation is that standard upper limits based on a
65:
4356:
4309:"[Setting Confidence Intervals for Bounded Parameters]: Comment"
4334:
4233:
4188:
4098:
4067:
3081:
The arguments given above can be viewed as following the spirit of the
25:
1273:
events coming from signal and background processes, both described by
4341:
977:
The calculation of the upper limit is usually done by constructing a
121:
37:
4257:
4180:
4217:
4122:"Upper limits in experiments with background or measurement errors"
4082:
1238:
4387:
4272:
477:{\displaystyle \mathbb {P} (\theta _{up}(X)\geq \theta |\theta )}
3112:
Birnbaum himself suggested in his 1962 paper that the CLs ratio
2396:{\displaystyle p_{\theta }=\mathbb {P} (q(X)>q^{*}|\theta )}
2288:
4403:
The
Particle Data Group (PDG) review of statistical methods
1648:
is excluded at 95% confidence level. But this implies that
57:
3999:
Particle
Physics at the Tercentenary of Mikhail Lomonosov
1524:{\displaystyle \mathbb {P} (n\leq n^{*}|s+b)\leq \alpha }
1242:
1234:
53:
3911:; the latter might be estimated from the inverse of the
3644:
52:." It was first introduced by physicists working at the
4165:(1962). "On the foundations of statistical inference".
2326:
from which the confidence interval is derived, and let
3176:
alone. This followed from a simple application of the
3014:
2278:{\displaystyle n\leq n^{*}\Rightarrow n_{b}\leq n^{*}}
2226:, and the second equality comes from the fact that if
518:
An equivalent definition can be made by considering a
4083:"Allan Birnbaum's Conception of Statistical Evidence"
4042:"Setting Confidence Intervals for Bounded Parameters"
3932:
3893:
3873:
3840:
3816:
3697:
3659:
3607:
3569:
3542:
3516:
3489:
3462:
3411:
3367:
3340:
3290:
3264:
3238:
3206:
3186:
3162:
3118:
2653:
2623:
2528:
2502:
2482:
2462:
2435:
2412:
2335:
2299:
2232:
1925:
1902:
1875:
1848:
1821:
1794:
1754:
1721:
1700:
1680:
1654:
1622:
1589:
1563:
1537:
1473:
1453:
1426:
1406:
1386:
1366:
1323:
1303:
1283:
1259:
1199:
1045:
1022:
986:
954:
927:
898:
871:
865:. This can be interpreted intuitively as saying that
846:
806:
779:
753:
697:
670:
650:
619:
574:
528:
490:
426:
254:
210:
175:
129:
3965:Journal of Physics G: Nuclear and Particle Physics
3938:
3899:
3879:
3855:
3822:
3799:
3676:
3620:
3593:
3555:
3528:
3502:
3475:
3437:
3393:
3353:
3322:
3276:
3250:
3224:
3192:
3168:
3144:
3003:
2636:
2606:
2511:
2488:
2468:
2448:
2421:
2395:
2314:
2277:
2211:
1908:
1888:
1861:
1834:
1807:
1780:
1740:
1706:
1686:
1666:
1640:
1608:
1575:
1549:
1523:
1459:
1439:
1412:
1392:
1372:
1352:
1309:
1289:
1265:
1245:, most notable in the searches for new particles.
1217:
1182:
1028:
1008:
966:
940:
917:less likely to observe such an extreme outcome as
909:
884:
857:
832:
792:
765:
735:
683:
656:
632:
599:
560:
507:
476:
381:
235:
192:
153:
92:. It has however close resemblance to concepts of
4380:
4409:
2429:is the outcome observed by the experiment. Then
4306:
4168:Journal of the American Statistical Association
4080:
4039:
3152:should be used as a measure of the strength of
1228:
1557:coverage. Consider, for example, a case where
736:{\displaystyle \theta _{up}(X)<\theta _{0}}
3156:provided by significance tests, rather than
2496:. If the test is unbiased then the outcome
1674:is excluded, namely all possible values of
1225:is the observed outcome of the experiment.
3563:is true, and with much larger probability
2456:can be regarded as an unmeasurable (since
2289:Generalization of the conditional argument
4386:
4355:
4324:
4271:
4057:
3069:Learn how and when to remove this message
2952:
2906:
2838:
2782:
2736:
2655:
2617:from which, similarly to conditioning on
2543:
2350:
2169:
2126:
2069:
2000:
1927:
1616:events are observed, then one finds that
1475:
1353:{\displaystyle n\sim {\text{Poiss}}(s+b)}
1108:
1050:
561:{\displaystyle H_{0}:\theta =\theta _{0}}
428:
308:
259:
4253:
4251:
4212:
4161:
3049:Relevant discussion may be found on the
948:is true than it is when the alternative
88:and is therefore described sometimes as
4119:
4014:
3832:standard normal cumulative distribution
4410:
154:{\displaystyle \theta \in [0,\infty )}
4248:
4017:"Statistical methods in CMS searches"
3645:Calculation in the large sample limit
4074:
3962:
3334:associated with the outcome "reject
3018:
245:
4129:Nucl. Instrum. Methods Phys. Res. A
4113:
4008:
3956:
3015:Relation to foundational principles
200:, is a statistic (i.e., observable
13:
4300:
3817:
3762:
3733:
3438:{\displaystyle \alpha /(1-\beta )}
3394:{\displaystyle \alpha /(1-\beta )}
3145:{\displaystyle \alpha /(1-\beta )}
2644:in the previous case, one obtains
1869:is large, and the distribution of
833:{\displaystyle \alpha /(1-\beta )}
145:
14:
4434:
4396:
3856:{\displaystyle {\hat {\theta }}}
3023:
1447:consists of excluding values of
60:and has since been used by many
1781:{\displaystyle n_{b}\leq n^{*}}
1218:{\displaystyle q_{\theta }^{*}}
773:). The criterion for rejecting
600:{\displaystyle H_{1}:\theta =0}
236:{\displaystyle \theta _{up}(X)}
4290:10.1140/epjc/s10052-011-1554-0
4206:
4155:
4033:
3991:
3847:
3791:
3788:
3774:
3765:
3745:
3720:
3594:{\displaystyle \ (1-\beta )\ }
3585:
3573:
3523:
3517:
3432:
3420:
3388:
3376:
3317:
3304:
3271:
3265:
3219:
3207:
3139:
3127:
2992:
2985:
2968:
2962:
2956:
2946:
2939:
2922:
2916:
2910:
2878:
2871:
2854:
2848:
2842:
2825:
2818:
2786:
2776:
2769:
2752:
2746:
2740:
2726:
2688:
2671:
2665:
2659:
2583:
2576:
2559:
2553:
2547:
2390:
2383:
2366:
2360:
2354:
2309:
2303:
2249:
2200:
2193:
2173:
2163:
2150:
2130:
2113:
2100:
2073:
2063:
2050:
2004:
1990:
1951:
1931:
1512:
1499:
1479:
1420:given an experimental outcome
1347:
1335:
1160:
1153:
1131:
1125:
1112:
1102:
1095:
1073:
1067:
1054:
1009:{\displaystyle q_{\theta }(X)}
1003:
997:
827:
815:
717:
711:
471:
464:
454:
448:
432:
351:
344:
334:
328:
312:
302:
295:
285:
279:
263:
230:
224:
165:upper limit for the parameter
148:
136:
1:
4423:Experimental particle physics
3949:
3323:{\displaystyle H_{1},(H_{2})}
3097:, a result famously shown by
800:thus requires that the ratio
743:) and the denominator to the
103:
96:proposed by the statistician
4149:10.1016/0168-9002(89)90795-X
3686:uniformly most powerful test
1531:, which guarantees at least
1229:Usage in high energy physics
7:
3985:10.1088/0954-3899/28/10/313
3109:is regarded as necessary).
2449:{\displaystyle p_{\theta }}
1741:{\displaystyle n\leq n^{*}}
1380:is assumed to be known and
941:{\displaystyle \theta _{0}}
885:{\displaystyle \theta _{0}}
684:{\displaystyle \theta _{0}}
633:{\displaystyle \theta _{0}}
609:
395:
10:
4439:
4366:10.1103/PhysRevD.69.033002
3677:{\displaystyle 1-\alpha '}
3225:{\displaystyle (1-\beta )}
892:is excluded because it is
508:{\displaystyle 1-\alpha '}
193:{\displaystyle 1-\alpha '}
3529:{\displaystyle (\alpha )}
3510:' with small probability
3251:{\displaystyle 1-\alpha }
1896:itself is independent of
1641:{\displaystyle s+b\geq 3}
1550:{\displaystyle 1-\alpha }
1248:
1016:and finding the value of
967:{\displaystyle \theta =0}
607:. Then the numerator in (
120:with a real non-negative
4307:Leon Jay Gleser (2002).
4081:Ronald N. Giere (1977).
4040:Mark Mandelkern (2002).
3277:{\displaystyle (\beta )}
3107:conditionality principle
3101:. CLs does not obey the
3091:conditionality principle
3083:conditionality principle
1748:on the observation that
910:{\displaystyle \alpha '}
858:{\displaystyle \alpha '}
766:{\displaystyle 1-\beta }
642:type-I error probability
568:against the alternative
243:which has the property:
169:, with confidence level
118:probability distribution
3939:{\displaystyle \theta }
3900:{\displaystyle \sigma }
3880:{\displaystyle \theta }
3193:{\displaystyle \alpha }
3169:{\displaystyle \alpha }
2489:{\displaystyle \theta }
2469:{\displaystyle \theta }
2224:Conditional probability
1667:{\displaystyle s\geq 0}
1609:{\displaystyle n^{*}=0}
1029:{\displaystyle \theta }
657:{\displaystyle \alpha }
522:of the null hypothesis
40:, a particular form of
3940:
3901:
3881:
3857:
3824:
3801:
3678:
3630:
3622:
3595:
3557:
3530:
3504:
3477:
3439:
3395:
3355:
3324:
3278:
3252:
3226:
3194:
3170:
3146:
3005:
2638:
2608:
2513:
2490:
2470:
2450:
2423:
2397:
2316:
2279:
2213:
1910:
1890:
1863:
1836:
1809:
1782:
1742:
1708:
1688:
1668:
1642:
1610:
1577:
1551:
1525:
1461:
1441:
1414:
1394:
1374:
1354:
1311:
1291:
1277:with respective rates
1267:
1219:
1184:
1030:
1010:
968:
942:
911:
886:
859:
834:
794:
767:
737:
685:
658:
634:
601:
562:
509:
484:is always larger than
478:
383:
237:
194:
155:
4418:Statistical intervals
4326:10.1214/ss/1030550859
4059:10.1214/ss/1030550859
3941:
3902:
3882:
3858:
3825:
3823:{\displaystyle \Phi }
3802:
3679:
3623:
3621:{\displaystyle H_{2}}
3596:
3558:
3556:{\displaystyle H_{1}}
3531:
3505:
3503:{\displaystyle H_{1}}
3478:
3476:{\displaystyle H_{2}}
3454:
3440:
3396:
3356:
3354:{\displaystyle H_{1}}
3325:
3279:
3253:
3227:
3195:
3171:
3147:
3036:synthesis of material
3006:
2639:
2637:{\displaystyle n_{b}}
2609:
2514:
2491:
2471:
2451:
2424:
2398:
2317:
2280:
2214:
1911:
1891:
1889:{\displaystyle n_{b}}
1864:
1862:{\displaystyle n_{b}}
1837:
1835:{\displaystyle n_{b}}
1810:
1808:{\displaystyle n_{b}}
1783:
1743:
1709:
1689:
1669:
1643:
1611:
1578:
1552:
1526:
1462:
1442:
1440:{\displaystyle n^{*}}
1415:
1395:
1375:
1355:
1312:
1292:
1275:Poisson distributions
1268:
1220:
1185:
1031:
1011:
969:
943:
912:
887:
860:
840:will be smaller than
835:
795:
793:{\displaystyle H_{0}}
768:
738:
686:
664:) of the test (i.e.,
659:
635:
613:), when evaluated at
602:
563:
510:
479:
384:
238:
195:
156:
86:statistical inference
64:experiments. It is a
3930:
3891:
3871:
3838:
3814:
3695:
3657:
3605:
3567:
3540:
3514:
3487:
3460:
3450:Confidence principle
3409:
3365:
3338:
3288:
3262:
3236:
3204:
3184:
3178:likelihood principle
3160:
3154:statistical evidence
3116:
3103:likelihood principle
3095:likelihood principle
2651:
2621:
2526:
2500:
2480:
2460:
2433:
2410:
2333:
2315:{\displaystyle q(X)}
2297:
2230:
1923:
1900:
1873:
1846:
1819:
1792:
1752:
1719:
1698:
1678:
1652:
1620:
1587:
1561:
1535:
1471:
1451:
1424:
1404:
1384:
1364:
1321:
1301:
1281:
1257:
1197:
1043:
1020:
984:
952:
925:
896:
869:
844:
804:
777:
751:
695:
668:
648:
640:, correspond to the
617:
572:
526:
488:
424:
419:coverage probability
252:
208:
173:
127:
94:statistical evidence
74:coverage probability
4313:Statistical Science
4282:2011EPJC...71.1554C
4141:1989NIMPA.277..608Z
4046:Statistical Science
3977:2002JPhG...28.2693R
3403:likelihood function
3087:ancillary statistic
2896:
2816:
2719:
2603:
1576:{\displaystyle b=3}
1214:
1151:
1093:
370: for all
70:error probabilities
62:high energy physics
50:confidence interval
42:interval estimation
28:method for setting
4234:10.1007/bf00485690
4202:(With discussion.)
4099:10.1007/bf00485688
3936:
3913:Fisher information
3909:standard deviation
3897:
3877:
3865:maximum likelihood
3853:
3820:
3797:
3684:(derived from the
3674:
3618:
3591:
3553:
3526:
3500:
3473:
3435:
3391:
3351:
3320:
3274:
3248:
3222:
3190:
3166:
3142:
3046:to the main topic.
3040:verifiably mention
3034:possibly contains
3001:
2882:
2802:
2705:
2634:
2604:
2589:
2512:{\displaystyle q*}
2509:
2486:
2466:
2446:
2422:{\displaystyle q*}
2419:
2393:
2312:
2275:
2209:
1906:
1886:
1859:
1832:
1805:
1778:
1738:
1704:
1684:
1664:
1638:
1606:
1573:
1547:
1521:
1457:
1437:
1410:
1390:
1370:
1350:
1307:
1287:
1263:
1215:
1200:
1180:
1137:
1079:
1026:
1006:
964:
938:
907:
882:
855:
830:
790:
763:
733:
681:
654:
630:
597:
558:
505:
474:
379:
233:
190:
151:
78:most powerful test
3971:(10): 2693–2704.
3920:credible interval
3850:
3777:
3723:
3651:Gaussian function
3590:
3572:
3079:
3078:
3071:
2996:
2897:
2829:
2204:
2117:
1909:{\displaystyle s}
1707:{\displaystyle s}
1687:{\displaystyle s}
1460:{\displaystyle s}
1413:{\displaystyle s}
1393:{\displaystyle s}
1373:{\displaystyle b}
1333:
1310:{\displaystyle b}
1290:{\displaystyle s}
1266:{\displaystyle n}
1164:
691:is rejected when
403:
402:
371:
355:
46:Confidence Levels
4430:
4392:
4390:
4377:
4359:
4338:
4328:
4294:
4293:
4275:
4255:
4246:
4245:
4210:
4204:
4200:
4175:(298): 269–326.
4159:
4153:
4152:
4135:(2–3): 608–610.
4126:
4120:G. Zech (1989).
4117:
4111:
4110:
4078:
4072:
4071:
4061:
4037:
4031:
4030:
4028:
4027:
4022:. indico.cern.ch
4021:
4012:
4006:
3995:
3989:
3988:
3960:
3945:
3943:
3942:
3937:
3906:
3904:
3903:
3898:
3886:
3884:
3883:
3878:
3862:
3860:
3859:
3854:
3852:
3851:
3843:
3829:
3827:
3826:
3821:
3806:
3804:
3803:
3798:
3784:
3779:
3778:
3770:
3761:
3744:
3743:
3725:
3724:
3716:
3710:
3709:
3683:
3681:
3680:
3675:
3673:
3627:
3625:
3624:
3619:
3617:
3616:
3600:
3598:
3597:
3592:
3588:
3570:
3562:
3560:
3559:
3554:
3552:
3551:
3535:
3533:
3532:
3527:
3509:
3507:
3506:
3501:
3499:
3498:
3482:
3480:
3479:
3474:
3472:
3471:
3444:
3442:
3441:
3436:
3419:
3400:
3398:
3397:
3392:
3375:
3360:
3358:
3357:
3352:
3350:
3349:
3332:likelihood ratio
3329:
3327:
3326:
3321:
3316:
3315:
3300:
3299:
3283:
3281:
3280:
3275:
3257:
3255:
3254:
3249:
3231:
3229:
3228:
3223:
3199:
3197:
3196:
3191:
3175:
3173:
3172:
3167:
3151:
3149:
3148:
3143:
3126:
3074:
3067:
3063:
3060:
3054:
3027:
3026:
3019:
3010:
3008:
3007:
3002:
2997:
2995:
2988:
2983:
2982:
2955:
2949:
2942:
2937:
2936:
2909:
2903:
2898:
2895:
2890:
2881:
2874:
2869:
2868:
2841:
2835:
2830:
2828:
2821:
2815:
2810:
2798:
2797:
2785:
2779:
2772:
2767:
2766:
2739:
2733:
2718:
2713:
2701:
2700:
2691:
2686:
2685:
2658:
2643:
2641:
2640:
2635:
2633:
2632:
2613:
2611:
2610:
2605:
2602:
2597:
2579:
2574:
2573:
2546:
2538:
2537:
2518:
2516:
2515:
2510:
2495:
2493:
2492:
2487:
2475:
2473:
2472:
2467:
2455:
2453:
2452:
2447:
2445:
2444:
2428:
2426:
2425:
2420:
2402:
2400:
2399:
2394:
2386:
2381:
2380:
2353:
2345:
2344:
2321:
2319:
2318:
2313:
2284:
2282:
2281:
2276:
2274:
2273:
2261:
2260:
2248:
2247:
2218:
2216:
2215:
2210:
2205:
2203:
2196:
2191:
2190:
2172:
2166:
2153:
2148:
2147:
2129:
2123:
2118:
2116:
2103:
2098:
2097:
2085:
2084:
2072:
2066:
2053:
2048:
2047:
2035:
2034:
2022:
2021:
2003:
1997:
1977:
1976:
1964:
1963:
1954:
1949:
1948:
1930:
1915:
1913:
1912:
1907:
1895:
1893:
1892:
1887:
1885:
1884:
1868:
1866:
1865:
1860:
1858:
1857:
1841:
1839:
1838:
1833:
1831:
1830:
1814:
1812:
1811:
1806:
1804:
1803:
1787:
1785:
1784:
1779:
1777:
1776:
1764:
1763:
1747:
1745:
1744:
1739:
1737:
1736:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1673:
1671:
1670:
1665:
1647:
1645:
1644:
1639:
1615:
1613:
1612:
1607:
1599:
1598:
1582:
1580:
1579:
1574:
1556:
1554:
1553:
1548:
1530:
1528:
1527:
1522:
1502:
1497:
1496:
1478:
1466:
1464:
1463:
1458:
1446:
1444:
1443:
1438:
1436:
1435:
1419:
1417:
1416:
1411:
1399:
1397:
1396:
1391:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1334:
1331:
1316:
1314:
1313:
1308:
1296:
1294:
1293:
1288:
1272:
1270:
1269:
1264:
1224:
1222:
1221:
1216:
1213:
1208:
1189:
1187:
1186:
1181:
1176:
1165:
1163:
1156:
1150:
1145:
1124:
1123:
1111:
1105:
1098:
1092:
1087:
1066:
1065:
1053:
1047:
1035:
1033:
1032:
1027:
1015:
1013:
1012:
1007:
996:
995:
973:
971:
970:
965:
947:
945:
944:
939:
937:
936:
916:
914:
913:
908:
906:
891:
889:
888:
883:
881:
880:
864:
862:
861:
856:
854:
839:
837:
836:
831:
814:
799:
797:
796:
791:
789:
788:
772:
770:
769:
764:
742:
740:
739:
734:
732:
731:
710:
709:
690:
688:
687:
682:
680:
679:
663:
661:
660:
655:
639:
637:
636:
631:
629:
628:
606:
604:
603:
598:
584:
583:
567:
565:
564:
559:
557:
556:
538:
537:
514:
512:
511:
506:
504:
483:
481:
480:
475:
467:
447:
446:
431:
397:
388:
386:
385:
380:
372:
369:
367:
356:
354:
347:
327:
326:
311:
305:
298:
278:
277:
262:
256:
246:
242:
240:
239:
234:
223:
222:
199:
197:
196:
191:
189:
160:
158:
157:
152:
34:exclusion limits
18:particle physics
4438:
4437:
4433:
4432:
4431:
4429:
4428:
4427:
4408:
4407:
4399:
4357:physics/0303111
4303:
4301:Further reading
4298:
4297:
4260:Eur. Phys. J. C
4256:
4249:
4214:Birnbaum, Allan
4211:
4207:
4181:10.2307/2281640
4163:Birnbaum, Allan
4160:
4156:
4124:
4118:
4114:
4079:
4075:
4038:
4034:
4025:
4023:
4019:
4013:
4009:
3996:
3992:
3961:
3957:
3952:
3931:
3928:
3927:
3892:
3889:
3888:
3872:
3869:
3868:
3842:
3841:
3839:
3836:
3835:
3815:
3812:
3811:
3780:
3769:
3768:
3754:
3736:
3732:
3715:
3714:
3702:
3698:
3696:
3693:
3692:
3666:
3658:
3655:
3654:
3647:
3612:
3608:
3606:
3603:
3602:
3568:
3565:
3564:
3547:
3543:
3541:
3538:
3537:
3515:
3512:
3511:
3494:
3490:
3488:
3485:
3484:
3467:
3463:
3461:
3458:
3457:
3415:
3410:
3407:
3406:
3371:
3366:
3363:
3362:
3361:" is therefore
3345:
3341:
3339:
3336:
3335:
3311:
3307:
3295:
3291:
3289:
3286:
3285:
3263:
3260:
3259:
3237:
3234:
3233:
3205:
3202:
3201:
3185:
3182:
3181:
3161:
3158:
3157:
3122:
3117:
3114:
3113:
3075:
3064:
3058:
3055:
3048:
3038:which does not
3028:
3024:
3017:
2984:
2978:
2974:
2951:
2950:
2938:
2932:
2928:
2905:
2904:
2902:
2891:
2886:
2870:
2864:
2860:
2837:
2836:
2834:
2817:
2811:
2806:
2793:
2789:
2781:
2780:
2768:
2762:
2758:
2735:
2734:
2732:
2714:
2709:
2696:
2692:
2687:
2681:
2677:
2654:
2652:
2649:
2648:
2628:
2624:
2622:
2619:
2618:
2598:
2593:
2575:
2569:
2565:
2542:
2533:
2529:
2527:
2524:
2523:
2501:
2498:
2497:
2481:
2478:
2477:
2461:
2458:
2457:
2440:
2436:
2434:
2431:
2430:
2411:
2408:
2407:
2382:
2376:
2372:
2349:
2340:
2336:
2334:
2331:
2330:
2298:
2295:
2294:
2291:
2269:
2265:
2256:
2252:
2243:
2239:
2231:
2228:
2227:
2192:
2186:
2182:
2168:
2167:
2149:
2143:
2139:
2125:
2124:
2122:
2099:
2093:
2089:
2080:
2076:
2068:
2067:
2049:
2043:
2039:
2030:
2026:
2017:
2013:
1999:
1998:
1996:
1972:
1968:
1959:
1955:
1950:
1944:
1940:
1926:
1924:
1921:
1920:
1901:
1898:
1897:
1880:
1876:
1874:
1871:
1870:
1853:
1849:
1847:
1844:
1843:
1826:
1822:
1820:
1817:
1816:
1799:
1795:
1793:
1790:
1789:
1772:
1768:
1759:
1755:
1753:
1750:
1749:
1732:
1728:
1720:
1717:
1716:
1699:
1696:
1695:
1679:
1676:
1675:
1653:
1650:
1649:
1621:
1618:
1617:
1594:
1590:
1588:
1585:
1584:
1562:
1559:
1558:
1536:
1533:
1532:
1498:
1492:
1488:
1474:
1472:
1469:
1468:
1452:
1449:
1448:
1431:
1427:
1425:
1422:
1421:
1405:
1402:
1401:
1385:
1382:
1381:
1365:
1362:
1361:
1330:
1322:
1319:
1318:
1302:
1299:
1298:
1282:
1279:
1278:
1258:
1255:
1254:
1251:
1231:
1209:
1204:
1198:
1195:
1194:
1169:
1152:
1146:
1141:
1119:
1115:
1107:
1106:
1094:
1088:
1083:
1061:
1057:
1049:
1048:
1046:
1044:
1041:
1040:
1021:
1018:
1017:
991:
987:
985:
982:
981:
953:
950:
949:
932:
928:
926:
923:
922:
899:
897:
894:
893:
876:
872:
870:
867:
866:
847:
845:
842:
841:
810:
805:
802:
801:
784:
780:
778:
775:
774:
752:
749:
748:
727:
723:
702:
698:
696:
693:
692:
675:
671:
669:
666:
665:
649:
646:
645:
624:
620:
618:
615:
614:
579:
575:
573:
570:
569:
552:
548:
533:
529:
527:
524:
523:
520:hypothesis test
497:
489:
486:
485:
463:
439:
435:
427:
425:
422:
421:
368:
360:
343:
319:
315:
307:
306:
294:
270:
266:
258:
257:
255:
253:
250:
249:
215:
211:
209:
206:
205:
202:random variable
182:
174:
171:
170:
128:
125:
124:
106:
12:
11:
5:
4436:
4426:
4425:
4420:
4406:
4405:
4398:
4397:External links
4395:
4394:
4393:
4378:
4339:
4319:(2): 161–163.
4302:
4299:
4296:
4295:
4247:
4205:
4154:
4112:
4073:
4052:(2): 149–159.
4032:
4007:
3990:
3954:
3953:
3951:
3948:
3935:
3896:
3876:
3849:
3846:
3819:
3808:
3807:
3796:
3793:
3790:
3787:
3783:
3776:
3773:
3767:
3764:
3760:
3757:
3753:
3750:
3747:
3742:
3739:
3735:
3731:
3728:
3722:
3719:
3713:
3708:
3705:
3701:
3688:) is given by
3672:
3669:
3665:
3662:
3646:
3643:
3615:
3611:
3587:
3584:
3581:
3578:
3575:
3550:
3546:
3525:
3522:
3519:
3497:
3493:
3470:
3466:
3434:
3431:
3428:
3425:
3422:
3418:
3414:
3390:
3387:
3384:
3381:
3378:
3374:
3370:
3348:
3344:
3319:
3314:
3310:
3306:
3303:
3298:
3294:
3273:
3270:
3267:
3247:
3244:
3241:
3221:
3218:
3215:
3212:
3209:
3189:
3165:
3141:
3138:
3135:
3132:
3129:
3125:
3121:
3077:
3076:
3031:
3029:
3022:
3016:
3013:
3012:
3011:
3000:
2994:
2991:
2987:
2981:
2977:
2973:
2970:
2967:
2964:
2961:
2958:
2954:
2948:
2945:
2941:
2935:
2931:
2927:
2924:
2921:
2918:
2915:
2912:
2908:
2901:
2894:
2889:
2885:
2880:
2877:
2873:
2867:
2863:
2859:
2856:
2853:
2850:
2847:
2844:
2840:
2833:
2827:
2824:
2820:
2814:
2809:
2805:
2801:
2796:
2792:
2788:
2784:
2778:
2775:
2771:
2765:
2761:
2757:
2754:
2751:
2748:
2745:
2742:
2738:
2731:
2728:
2725:
2722:
2717:
2712:
2708:
2704:
2699:
2695:
2690:
2684:
2680:
2676:
2673:
2670:
2667:
2664:
2661:
2657:
2631:
2627:
2615:
2614:
2601:
2596:
2592:
2588:
2585:
2582:
2578:
2572:
2568:
2564:
2561:
2558:
2555:
2552:
2549:
2545:
2541:
2536:
2532:
2508:
2505:
2485:
2465:
2443:
2439:
2418:
2415:
2404:
2403:
2392:
2389:
2385:
2379:
2375:
2371:
2368:
2365:
2362:
2359:
2356:
2352:
2348:
2343:
2339:
2324:test statistic
2311:
2308:
2305:
2302:
2290:
2287:
2272:
2268:
2264:
2259:
2255:
2251:
2246:
2242:
2238:
2235:
2220:
2219:
2208:
2202:
2199:
2195:
2189:
2185:
2181:
2178:
2175:
2171:
2165:
2162:
2159:
2156:
2152:
2146:
2142:
2138:
2135:
2132:
2128:
2121:
2115:
2112:
2109:
2106:
2102:
2096:
2092:
2088:
2083:
2079:
2075:
2071:
2065:
2062:
2059:
2056:
2052:
2046:
2042:
2038:
2033:
2029:
2025:
2020:
2016:
2012:
2009:
2006:
2002:
1995:
1992:
1989:
1986:
1983:
1980:
1975:
1971:
1967:
1962:
1958:
1953:
1947:
1943:
1939:
1936:
1933:
1929:
1905:
1883:
1879:
1856:
1852:
1829:
1825:
1802:
1798:
1775:
1771:
1767:
1762:
1758:
1735:
1731:
1727:
1724:
1703:
1683:
1663:
1660:
1657:
1637:
1634:
1631:
1628:
1625:
1605:
1602:
1597:
1593:
1572:
1569:
1566:
1546:
1543:
1540:
1520:
1517:
1514:
1511:
1508:
1505:
1501:
1495:
1491:
1487:
1484:
1481:
1477:
1456:
1434:
1430:
1409:
1389:
1369:
1349:
1346:
1343:
1340:
1337:
1329:
1326:
1306:
1286:
1262:
1250:
1247:
1230:
1227:
1212:
1207:
1203:
1191:
1190:
1179:
1175:
1172:
1168:
1162:
1159:
1155:
1149:
1144:
1140:
1136:
1133:
1130:
1127:
1122:
1118:
1114:
1110:
1104:
1101:
1097:
1091:
1086:
1082:
1078:
1075:
1072:
1069:
1064:
1060:
1056:
1052:
1025:
1005:
1002:
999:
994:
990:
979:test statistic
963:
960:
957:
935:
931:
905:
902:
879:
875:
853:
850:
829:
826:
823:
820:
817:
813:
809:
787:
783:
762:
759:
756:
730:
726:
722:
719:
716:
713:
708:
705:
701:
678:
674:
653:
627:
623:
596:
593:
590:
587:
582:
578:
555:
551:
547:
544:
541:
536:
532:
503:
500:
496:
493:
473:
470:
466:
462:
459:
456:
453:
450:
445:
442:
438:
434:
430:
401:
400:
391:
389:
378:
375:
366:
363:
359:
353:
350:
346:
342:
339:
336:
333:
330:
325:
322:
318:
314:
310:
304:
301:
297:
293:
290:
287:
284:
281:
276:
273:
269:
265:
261:
232:
229:
226:
221:
218:
214:
188:
185:
181:
178:
150:
147:
144:
141:
138:
135:
132:
105:
102:
98:Allan Birnbaum
56:experiment at
9:
6:
4:
3:
2:
4435:
4424:
4421:
4419:
4416:
4415:
4413:
4404:
4401:
4400:
4389:
4384:
4379:
4375:
4371:
4367:
4363:
4358:
4353:
4350:(3): 033002.
4349:
4345:
4340:
4336:
4332:
4327:
4322:
4318:
4314:
4310:
4305:
4304:
4291:
4287:
4283:
4279:
4274:
4269:
4265:
4261:
4254:
4252:
4243:
4239:
4235:
4231:
4227:
4223:
4219:
4215:
4209:
4203:
4198:
4194:
4190:
4186:
4182:
4178:
4174:
4170:
4169:
4164:
4158:
4150:
4146:
4142:
4138:
4134:
4130:
4123:
4116:
4108:
4104:
4100:
4096:
4092:
4088:
4084:
4077:
4069:
4065:
4060:
4055:
4051:
4047:
4043:
4036:
4018:
4015:Amnon Harel.
4011:
4005:
4001:
4000:
3994:
3986:
3982:
3978:
3974:
3970:
3966:
3959:
3955:
3947:
3933:
3925:
3922:if a uniform
3921:
3918:
3914:
3910:
3894:
3874:
3867:estimator of
3866:
3844:
3833:
3794:
3785:
3781:
3771:
3758:
3755:
3751:
3748:
3740:
3737:
3729:
3726:
3717:
3711:
3706:
3703:
3699:
3691:
3690:
3689:
3687:
3670:
3667:
3663:
3660:
3652:
3642:
3639:
3635:
3629:
3613:
3609:
3582:
3579:
3576:
3548:
3544:
3520:
3495:
3491:
3468:
3464:
3453:
3451:
3446:
3429:
3426:
3423:
3416:
3412:
3404:
3385:
3382:
3379:
3372:
3368:
3346:
3342:
3333:
3312:
3308:
3301:
3296:
3292:
3268:
3245:
3242:
3239:
3216:
3213:
3210:
3187:
3179:
3163:
3155:
3136:
3133:
3130:
3123:
3119:
3110:
3108:
3104:
3100:
3096:
3092:
3088:
3084:
3073:
3070:
3062:
3052:
3047:
3045:
3041:
3037:
3032:This section
3030:
3021:
3020:
2998:
2989:
2979:
2975:
2971:
2965:
2959:
2943:
2933:
2929:
2925:
2919:
2913:
2899:
2892:
2887:
2883:
2875:
2865:
2861:
2857:
2851:
2845:
2831:
2822:
2812:
2807:
2803:
2799:
2794:
2790:
2773:
2763:
2759:
2755:
2749:
2743:
2729:
2723:
2720:
2715:
2710:
2706:
2702:
2697:
2693:
2682:
2678:
2674:
2668:
2662:
2647:
2646:
2645:
2629:
2625:
2599:
2594:
2590:
2586:
2580:
2570:
2566:
2562:
2556:
2550:
2539:
2534:
2530:
2522:
2521:
2520:
2506:
2503:
2483:
2463:
2441:
2437:
2416:
2413:
2387:
2377:
2373:
2369:
2363:
2357:
2346:
2341:
2337:
2329:
2328:
2327:
2325:
2306:
2300:
2286:
2270:
2266:
2262:
2257:
2253:
2244:
2240:
2236:
2233:
2225:
2206:
2197:
2187:
2183:
2179:
2176:
2160:
2157:
2154:
2144:
2140:
2136:
2133:
2119:
2110:
2107:
2104:
2094:
2090:
2086:
2081:
2077:
2060:
2057:
2054:
2044:
2040:
2036:
2031:
2027:
2023:
2018:
2014:
2010:
2007:
1993:
1987:
1984:
1981:
1978:
1973:
1969:
1965:
1960:
1956:
1945:
1941:
1937:
1934:
1919:
1918:
1917:
1903:
1881:
1877:
1854:
1850:
1827:
1823:
1800:
1796:
1773:
1769:
1765:
1760:
1756:
1733:
1729:
1725:
1722:
1701:
1681:
1661:
1658:
1655:
1635:
1632:
1629:
1626:
1623:
1603:
1600:
1595:
1591:
1570:
1567:
1564:
1544:
1541:
1538:
1518:
1515:
1509:
1506:
1503:
1493:
1489:
1485:
1482:
1454:
1432:
1428:
1407:
1387:
1367:
1344:
1341:
1338:
1327:
1324:
1304:
1284:
1276:
1260:
1246:
1244:
1240:
1236:
1226:
1210:
1205:
1201:
1177:
1173:
1170:
1166:
1157:
1147:
1142:
1138:
1134:
1128:
1120:
1116:
1099:
1089:
1084:
1080:
1076:
1070:
1062:
1058:
1039:
1038:
1037:
1023:
1000:
992:
988:
980:
975:
961:
958:
955:
933:
929:
920:
903:
900:
877:
873:
851:
848:
824:
821:
818:
811:
807:
785:
781:
760:
757:
754:
746:
728:
724:
720:
714:
706:
703:
699:
676:
672:
651:
643:
625:
621:
612:
611:
594:
591:
588:
585:
580:
576:
553:
549:
545:
542:
539:
534:
530:
521:
516:
501:
498:
494:
491:
468:
460:
457:
451:
443:
440:
436:
420:
416:
412:
408:
399:
392:
390:
376:
373:
364:
361:
357:
348:
340:
337:
331:
323:
320:
316:
299:
291:
288:
282:
274:
271:
267:
248:
247:
244:
227:
219:
216:
212:
203:
186:
183:
179:
176:
168:
164:
142:
139:
133:
130:
123:
119:
115:
114:random sample
111:
101:
99:
95:
91:
87:
81:
79:
75:
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
32:(also called
31:
27:
24:represents a
23:
19:
4347:
4344:Phys. Rev. D
4343:
4316:
4312:
4263:
4259:
4228:(1): 19–49.
4225:
4221:
4208:
4201:
4172:
4166:
4157:
4132:
4128:
4115:
4090:
4086:
4076:
4049:
4045:
4035:
4024:. Retrieved
4010:
4004:Google Books
4002:, p. 13, at
3997:
3993:
3968:
3964:
3958:
3809:
3648:
3631:
3455:
3449:
3447:
3153:
3111:
3080:
3065:
3056:
3033:
2616:
2405:
2292:
2221:
1252:
1232:
1192:
976:
918:
608:
517:
410:
406:
404:
393:
166:
162:
109:
107:
93:
89:
82:
33:
30:upper limits
29:
21:
15:
4266:(2): 1554.
4093:(1): 5–13.
3483:as against
66:frequentist
36:) on model
26:statistical
4412:Categories
4026:2015-04-10
3950:References
3628:is true."
3059:April 2016
1467:for which
1036:for which
415:continuous
104:Definition
38:parameters
4388:1109.2023
4273:1007.1727
3946:is used.
3934:θ
3895:σ
3875:θ
3848:^
3845:θ
3818:Φ
3786:σ
3775:^
3772:θ
3763:Φ
3756:α
3752:−
3738:−
3734:Φ
3730:σ
3721:^
3718:θ
3700:θ
3668:α
3664:−
3583:β
3580:−
3521:α
3430:β
3427:−
3413:α
3386:β
3383:−
3369:α
3269:β
3246:α
3243:−
3217:β
3214:−
3188:α
3164:α
3137:β
3134:−
3120:α
3051:talk page
2980:∗
2944:θ
2934:∗
2926:≥
2893:∗
2876:θ
2866:∗
2858:≥
2823:θ
2813:∗
2800:≤
2795:θ
2774:θ
2764:∗
2756:≥
2724:θ
2716:∗
2703:≤
2698:θ
2683:∗
2675:≥
2600:∗
2587:≡
2571:∗
2540:≤
2535:θ
2507:∗
2484:θ
2464:θ
2442:θ
2417:∗
2388:θ
2378:∗
2342:θ
2271:∗
2263:≤
2250:⇒
2245:∗
2237:≤
2188:∗
2180:≤
2145:∗
2137:≤
2095:∗
2087:≤
2045:∗
2037:≤
2019:∗
2011:≤
1974:∗
1966:≤
1946:∗
1938:≤
1774:∗
1766:≤
1734:∗
1726:≤
1659:≥
1633:≥
1596:∗
1545:α
1542:−
1519:α
1516:≤
1494:∗
1486:≤
1433:∗
1328:∼
1317:, namely
1211:∗
1206:θ
1171:α
1148:∗
1143:θ
1135:≥
1121:θ
1100:θ
1090:∗
1085:θ
1077:≥
1063:θ
1024:θ
993:θ
974:is true.
956:θ
930:θ
901:α
874:θ
849:α
825:β
822:−
808:α
761:β
758:−
725:θ
700:θ
673:θ
652:α
622:θ
589:θ
550:θ
543:θ
499:α
495:−
469:θ
461:θ
458:≥
437:θ
374:θ
362:α
358:≤
341:θ
317:θ
300:θ
292:θ
268:θ
213:θ
184:α
180:−
146:∞
134:∈
131:θ
122:parameter
4374:18947032
4242:35027844
4222:Synthese
4216:(1977).
4107:46973213
4087:Synthese
3917:Bayesian
3759:′
3671:′
3099:Birnbaum
2519:implies
1788:, where
1241:and the
1239:Tevatron
1174:′
904:′
852:′
502:′
365:′
187:′
4335:3182818
4278:Bibcode
4197:0138176
4189:2281640
4137:Bibcode
4068:3182816
3973:Bibcode
3907:is its
3863:is the
3830:is the
3638:Pearson
116:from a
4372:
4333:
4240:
4195:
4187:
4105:
4066:
3810:where
3634:Neyman
3589:
3571:
3330:. The
3284:under
3089:. The
3044:relate
2406:where
1249:Origin
1237:, the
1193:where
90:ad hoc
4383:arXiv
4370:S2CID
4352:arXiv
4331:JSTOR
4268:arXiv
4238:S2CID
4185:JSTOR
4125:(PDF)
4103:S2CID
4064:JSTOR
4020:(PDF)
3924:prior
3601:when
3536:when
2322:is a
1332:Poiss
921:when
745:power
112:be a
3926:for
3887:and
3232:and
2972:>
2563:>
2370:>
1583:and
1297:and
721:<
338:<
289:<
161:. A
108:Let
58:CERN
4362:doi
4321:doi
4286:doi
4230:doi
4177:doi
4145:doi
4133:277
4095:doi
4054:doi
3981:doi
3042:or
1243:LHC
1235:LEP
413:is
163:CLs
54:LEP
22:CLs
16:In
4414::
4368:.
4360:.
4348:69
4346:.
4329:.
4317:17
4315:.
4311:.
4284:.
4276:.
4264:71
4262:.
4250:^
4236:.
4226:36
4224:.
4220:.
4193:MR
4191:.
4183:.
4173:57
4171:.
4143:.
4131:.
4127:.
4101:.
4091:36
4089:.
4085:.
4062:.
4050:17
4048:.
4044:.
3979:.
3969:28
3967:.
3834:,
1360:.
515:.
204:)
100:.
20:,
4391:.
4385::
4376:.
4364::
4354::
4337:.
4323::
4292:.
4288::
4280::
4270::
4244:.
4232::
4199:.
4179::
4151:.
4147::
4139::
4109:.
4097::
4070:.
4056::
4029:.
3987:.
3983::
3975::
3795:,
3792:)
3789:)
3782:/
3766:(
3749:1
3746:(
3741:1
3727:+
3712:=
3707:p
3704:u
3661:1
3636:-
3614:2
3610:H
3586:)
3577:1
3574:(
3549:1
3545:H
3524:)
3518:(
3496:1
3492:H
3469:2
3465:H
3433:)
3424:1
3421:(
3417:/
3389:)
3380:1
3377:(
3373:/
3347:1
3343:H
3318:)
3313:2
3309:H
3305:(
3302:,
3297:1
3293:H
3272:)
3266:(
3258:,
3240:1
3220:)
3211:1
3208:(
3200:,
3140:)
3131:1
3128:(
3124:/
3072:)
3066:(
3061:)
3057:(
3053:.
2999:.
2993:)
2990:0
2986:|
2976:q
2969:)
2966:X
2963:(
2960:q
2957:(
2953:P
2947:)
2940:|
2930:q
2923:)
2920:X
2917:(
2914:q
2911:(
2907:P
2900:=
2888:0
2884:p
2879:)
2872:|
2862:q
2855:)
2852:X
2849:(
2846:q
2843:(
2839:P
2832:=
2826:)
2819:|
2808:0
2804:p
2791:p
2787:(
2783:P
2777:)
2770:|
2760:q
2753:)
2750:X
2747:(
2744:q
2741:(
2737:P
2730:=
2727:)
2721:,
2711:0
2707:p
2694:p
2689:|
2679:q
2672:)
2669:X
2666:(
2663:q
2660:(
2656:P
2630:b
2626:n
2595:0
2591:p
2584:)
2581:0
2577:|
2567:q
2560:)
2557:X
2554:(
2551:q
2548:(
2544:P
2531:p
2504:q
2438:p
2414:q
2391:)
2384:|
2374:q
2367:)
2364:X
2361:(
2358:q
2355:(
2351:P
2347:=
2338:p
2310:)
2307:X
2304:(
2301:q
2267:n
2258:b
2254:n
2241:n
2234:n
2207:.
2201:)
2198:b
2194:|
2184:n
2177:n
2174:(
2170:P
2164:)
2161:b
2158:+
2155:s
2151:|
2141:n
2134:n
2131:(
2127:P
2120:=
2114:)
2111:b
2108:+
2105:s
2101:|
2091:n
2082:b
2078:n
2074:(
2070:P
2064:)
2061:b
2058:+
2055:s
2051:|
2041:n
2032:b
2028:n
2024:,
2015:n
2008:n
2005:(
2001:P
1994:=
1991:)
1988:b
1985:+
1982:s
1979:,
1970:n
1961:b
1957:n
1952:|
1942:n
1935:n
1932:(
1928:P
1904:s
1882:b
1878:n
1855:b
1851:n
1828:b
1824:n
1801:b
1797:n
1770:n
1761:b
1757:n
1730:n
1723:n
1702:s
1682:s
1662:0
1656:s
1636:3
1630:b
1627:+
1624:s
1604:0
1601:=
1592:n
1571:3
1568:=
1565:b
1539:1
1513:)
1510:b
1507:+
1504:s
1500:|
1490:n
1483:n
1480:(
1476:P
1455:s
1429:n
1408:s
1388:s
1368:b
1348:)
1345:b
1342:+
1339:s
1336:(
1325:n
1305:b
1285:s
1261:n
1202:q
1178:.
1167:=
1161:)
1158:0
1154:|
1139:q
1132:)
1129:X
1126:(
1117:q
1113:(
1109:P
1103:)
1096:|
1081:q
1074:)
1071:X
1068:(
1059:q
1055:(
1051:P
1004:)
1001:X
998:(
989:q
962:0
959:=
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