150:
489:
169:
to measure failure risk. It was further developed and defined as the inverse CVaR by
Matthew Norton, Stan Uryasev, and Alexander Mafusalov. Similar to CVaR, bPOE considers not only the probability that outcomes (losses) exceed the threshold
652:
700:
545:
272:
322:
329:
740:
805:
720:
227:
188:
144:
124:
100:
76:
56:
166:
555:
858:
893:"Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"
78:(marked in red) as the blue shaded area. Therefore, by definition, bPOE is equal to one minus the confidence level at which the
892:
970:
965:
660:
496:
960:
760:
975:
232:
484:{\displaystyle {\bar {p}}_{x}(X)=\min _{a\geq 0}E^{+}=\min _{\gamma <x}{\frac {E^{+}}{x-\gamma }}}
955:
277:
103:
162:
725:
8:
925:
907:
828:
783:
705:
212:
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129:
109:
85:
79:
61:
41:
929:
832:
917:
873:
820:
787:
775:
31:
921:
779:
35:
859:"Buffered Probability of Exceedance: Mathematical Properties and Optimization"
824:
949:
761:"On Buffered Failure Probability in Design and Optimization of Structures"
647:{\displaystyle {\bar {p}}_{x}(X)=\{1-\alpha |{\bar {q}}_{\alpha }(X)=x\}}
877:
27:
16:
Explains the buffered probability of exceedance (bPOE), a risk measure
149:
912:
126:, but the tail is defined by its mean rather than the lowest point
198:
There are two slightly different definitions of bPOE, so called
806:"Maximization of AUC and Buffered AUC in binary classification"
38:. The bPOE is the probability of a tail with known mean value
891:
Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019).
153:
bPOE shaded in blue for the threshold, x, marked in red.
550:
bPOE can be expressed as the inverse function of CVaR:
890:
728:
708:
663:
558:
499:
332:
280:
235:
215:
190:, but also the magnitude of these outcomes (losses).
176:
132:
112:
88:
64:
44:
759:Rockafellar, R. Tyrrell; Royset, Johannes (2010).
734:
714:
694:
646:
539:
483:
316:
266:
221:
182:
138:
118:
94:
70:
50:
758:
947:
856:
519:
424:
365:
305:
803:
26:) is a function of a random variable used in
857:Mafusalov, Alexander; Uryasev, Stan (2018).
641:
590:
534:
522:
852:
850:
848:
911:
799:
797:
768:Reliability Engineering and System Safety
58:. The figure shows the bPOE at threshold
845:
148:
804:Norton, Matthew; Uryasev, Stan (2018).
695:{\displaystyle {\bar {q}}_{\alpha }(X)}
157:bPOE has its origins in the concept of
948:
794:
159:buffered probability of failure (bPOF)
884:
540:{\displaystyle ^{+}=\max\{\cdot ,0\}}
193:
13:
20:Buffered probability of exceedance
14:
987:
267:{\displaystyle {\bar {p}}_{x}(X)}
80:Conditional Value at Risk (CVaR)
752:
689:
683:
671:
632:
626:
614:
603:
584:
578:
566:
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500:
458:
445:
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389:
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311:
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287:
261:
255:
243:
1:
900:Annals of Operations Research
745:
317:{\displaystyle x\in ,\sup X]}
906:(1–2). Springer: 1281–1315.
866:SIAM Journal on Optimization
7:
10:
992:
922:10.1007/s10479-019-03373-1
819:(1–2). Springer: 575–612.
780:10.1016/j.ress.2010.01.001
102:. bPOE is similar to the
825:10.1007/s10107-018-1312-2
104:probability of exceedance
813:Mathematical Programming
735:{\displaystyle \alpha }
209:For a random variable,
736:
722:with confidence level
716:
696:
648:
541:
485:
318:
268:
223:
184:
163:R. Tyrrell Rockafellar
154:
140:
120:
96:
72:
52:
737:
717:
697:
649:
542:
486:
319:
269:
224:
185:
152:
141:
121:
97:
73:
53:
971:Stochastic processes
966:Reliability analysis
726:
706:
661:
556:
497:
330:
278:
233:
213:
174:
130:
110:
86:
62:
42:
961:Extreme value data
878:10.1137/15M1042644
732:
712:
692:
644:
537:
481:
438:
379:
314:
264:
219:
180:
155:
136:
116:
92:
68:
48:
976:Survival analysis
715:{\displaystyle X}
674:
617:
569:
479:
423:
364:
343:
246:
229:the Lower bPOE,
222:{\displaystyle X}
194:Formal definition
183:{\displaystyle x}
139:{\displaystyle x}
119:{\displaystyle x}
106:of the threshold
95:{\displaystyle x}
71:{\displaystyle x}
51:{\displaystyle x}
983:
940:
939:
937:
936:
915:
897:
888:
882:
881:
872:(5): 1077–1103.
863:
854:
843:
842:
840:
839:
810:
801:
792:
791:
765:
756:
741:
739:
738:
733:
721:
719:
718:
713:
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682:
681:
676:
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653:
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650:
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619:
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577:
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570:
562:
546:
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515:
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490:
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378:
351:
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145:
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137:
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101:
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77:
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69:
57:
55:
54:
49:
991:
990:
986:
985:
984:
982:
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980:
956:Risk management
946:
945:
944:
943:
934:
932:
895:
889:
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861:
855:
846:
837:
835:
808:
802:
795:
763:
757:
753:
748:
727:
724:
723:
707:
704:
703:
702:is the CVaR of
677:
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665:
664:
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659:
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620:
609:
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602:
572:
561:
560:
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510:
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468:
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410:
368:
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279:
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274:, at threshold
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196:
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167:Johannes Royset
161:, developed by
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107:
87:
84:
83:
63:
60:
59:
43:
40:
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32:risk management
17:
12:
11:
5:
989:
979:
978:
973:
968:
963:
958:
942:
941:
883:
844:
793:
774:(5): 499–510.
750:
749:
747:
744:
731:
711:
691:
688:
685:
680:
673:
670:
643:
640:
637:
634:
631:
628:
623:
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589:
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583:
580:
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568:
565:
536:
533:
530:
527:
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518:
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509:
505:
502:
477:
474:
471:
464:
460:
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450:
447:
444:
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430:
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417:
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397:
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388:
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377:
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367:
363:
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349:
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195:
192:
179:
135:
115:
91:
67:
47:
36:financial risk
15:
9:
6:
4:
3:
2:
988:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
953:
951:
931:
927:
923:
919:
914:
909:
905:
901:
894:
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875:
871:
867:
860:
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851:
849:
834:
830:
826:
822:
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807:
800:
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789:
785:
781:
777:
773:
769:
762:
755:
751:
743:
729:
709:
686:
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655:
638:
635:
629:
621:
611:
599:
596:
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587:
581:
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563:
551:
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531:
528:
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511:
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491:
475:
472:
469:
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442:
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407:
404:
398:
395:
392:
386:
380:
375:
372:
369:
361:
355:
347:
337:
325:
324:is given by:
308:
302:
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284:
281:
258:
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216:
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191:
177:
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133:
113:
105:
89:
81:
65:
45:
37:
33:
29:
25:
21:
933:. Retrieved
903:
899:
886:
869:
865:
836:. Retrieved
816:
812:
771:
767:
754:
656:
552:
549:
492:
326:
208:
203:
199:
197:
158:
156:
146:of the tail.
82:is equal to
34:, including
23:
19:
18:
950:Categories
935:2023-02-27
913:1811.11301
838:2023-03-12
746:References
204:Upper bPOE
200:Lower bPOE
28:statistics
930:254231768
833:254145122
730:α
679:α
672:¯
622:α
615:¯
600:α
597:−
567:¯
526:⋅
504:⋅
476:γ
473:−
455:γ
452:−
429:γ
396:−
373:≥
341:¯
285:∈
244:¯
788:1653873
928:
831:
786:
657:where
493:where
926:S2CID
908:arXiv
896:(PDF)
862:(PDF)
829:S2CID
809:(PDF)
784:S2CID
764:(PDF)
432:<
202:and
165:and
30:and
24:bPOE
918:doi
904:299
874:doi
821:doi
817:174
776:doi
547:.
520:max
425:min
366:min
306:sup
952::
924:.
916:.
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870:95
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