22:
2578:
2072:
The rationale behind twisting arguments does not change when parameters are vectors, though some complication arises from the management of joint inequalities. Instead, the difficulty of dealing with a vector of parameters proved to be the
Achilles heel of Fisher's approach to the
3120:
3080:
2289:
1631:
2823:
174:
In turn, parameter compatibility is a probability measure that we derive from the probability distribution of the random variable to which the parameter refers. In this way we identify a random parameter Θ compatible with an observed sample. Given a
732:
2201:
1082:
1295:
1201:
2422:
957:
906:
514:
803:
430:
2667:
1881:
1779:
246:. In more abstract terms, we speak about twisting properties of samples with properties of parameters and identify the former with statistics that are suitable for this exchange, so denoting a
2414:
1350:
582:
2904:
2353:
1486:
3174:
1448:
142:
2209:
366:
228:
1953:
287:
2715:
163:– of this parameter precisely on the basis of the sample. An estimate is suitable if replacing it with the unknown parameter does not cause major damage in next computations. In
3239:
2013:
2105:
858:
829:
3054:
1007:
594:
3113:
3000:
2938:
1678:
1376:
1108:
3263:
2707:
982:
3204:
2965:
2608:
2053:
1475:
2687:
2121:
3057:
1016:
242:
sample. Hence, we may derive the latter distribution directly from the former if we are able to relate domains of the sample space to subsets of Θ
1212:
2573:{\displaystyle F_{\Lambda ,K}(\lambda ,k)=F_{\Lambda \,\mid \,K=k}(\lambda )F_{K}(k)=F_{K\,\mid \,\Lambda =\lambda }(k)F_{\Lambda }(\lambda ).}
1124:
435:
835:, is a random variable since the master equation provides solutions that are feasible and independent of other (hidden) parameters.
914:
863:
748:
375:
2613:
81:
in general terms are associated with the properties of samples that identify with statistics that are suitable for exchange.
2115:, a twisting argument may be stated by following the below procedure. Given the meaning of these parameters we know that
1793:
1691:
2077:
of parameters. Also Fraser’s constructive probabilities devised for the same purpose do not treat this point completely.
2358:
3423:
168:
65:
43:
1626:{\displaystyle F_{\Theta }(\theta )\in \left(q_{1}(F_{S|\Theta =\theta }(s)),q_{2}(F_{S|\Theta =\theta }(s))\right)}
36:
251:
152:
1307:
519:
2831:
2297:
314:
176:
3418:
3065:
2284:{\displaystyle (\lambda \leq \lambda ')\leftrightarrow (s_{\lambda '}\leq s_{\lambda }){\text{ for fixed }}k,}
3134:
1401:
95:
2818:{\displaystyle F_{\Lambda \,\mid \,K=k}(\lambda )=1-{\frac {\Gamma (km,\lambda s_{\Lambda })}{\Gamma (km)}}}
319:
181:
1896:
256:
3208:
1962:
1114:
discretization grain, idem with the opposite monotony trend. Resuming these relations on all seeds, for
2088:
841:
812:
3009:
727:{\displaystyle s=h(g_{\theta }(z_{1}),\ldots ,g_{\theta }(z_{m}))=\rho (\theta ;z_{1},\ldots ,z_{m}).}
3003:
30:
3086:
2970:
2916:
1643:
1355:
1087:
3401:. International Series on Advanced Intelligence. Vol. 5 (2nd ed.). Adelaide: Magill.
742:
250:
w.r.t. the unknown parameters. The operational goal is to write the analytic expression of the
247:
47:
3248:
2692:
987:
243:
164:
156:
3179:
2943:
2586:
2026:
1460:
8:
2196:{\displaystyle (k\leq k')\leftrightarrow (s_{k}\leq s_{k'}){\text{ for fixed }}\lambda ,}
90:
962:
3385:
3350:
3061:
2672:
2108:
2074:
3377:
3354:
3346:
1077:{\displaystyle s\geq s'\rightarrow \theta \geq \theta '\rightarrow s\geq s'+\ell }
145:
2967:
are the observed statistics (hence with indices denoted by capital letters),
3412:
1290:{\displaystyle F_{\Theta \mid S=s}(\theta )=1-F_{S\mid \Theta =\theta }(s)}
745:
w.r.t the parameter, we are sure that a monotone relation exists for each
1196:{\displaystyle F_{\Theta \mid S=s}(\theta )=F_{S\mid \Theta =\theta }(s)}
3389:
3337:
Fisher, M.A. (1935). "The fiducial argument in statistical inference".
3368:
Fraser, D. A. S. (1966). "Structural probability and generalization".
3359:
238:
distribution law for the given θ, and the Θ distribution law given an
160:
3381:
1393:
Generating a parameter distribution law through a twisting argument
3396:
3064:
again with proper parameters (for instance estimated through the
509:{\displaystyle \{g_{\theta }(Z_{1}),\ldots ,g_{\theta }(Z_{m})\}}
2111:, whose specification requires values for the parameters λ and
952:{\displaystyle s\geq s'\leftrightarrow \theta \leq \theta '}
901:{\displaystyle s\geq s'\leftrightarrow \theta \geq \theta '}
3119:
3079:
159:
problem consists of computing suitable values – call them
2416:. This leads to a joint cumulative distribution function
798:{\displaystyle {\boldsymbol {z}}=\{z_{1},\ldots ,z_{m}\}}
425:{\displaystyle {\boldsymbol {X}}=\{X_{1},\ldots ,X_{m}\}}
3241:, you may find the joint p.d.f. of the Gamma parameters
1013:
assumes discrete values the first relation changes into
3123:
Marginal cumulative distribution function of parameter
2662:{\displaystyle r_{k}={\frac {s_{k}}{s_{\lambda }^{m}}}}
3251:
3211:
3182:
3137:
3089:
3012:
2973:
2946:
2919:
2834:
2718:
2695:
2675:
2616:
2589:
2425:
2361:
2300:
2212:
2124:
2091:
2029:
1965:
1899:
1796:
1694:
1646:
1489:
1463:
1404:
1358:
1310:
1215:
1127:
1090:
1019:
990:
965:
917:
866:
844:
815:
751:
597:
522:
438:
378:
322:
259:
184:
98:
1876:{\displaystyle q_{1}(F_{S}(s))=q_{2}(F_{S}(s-\ell )}
1774:{\displaystyle q_{2}(F_{S}(s))=q_{1}(F_{S}(s-\ell )}
809:
and θ. We are also assured that Θ, as a function of
230:, the rationale of this operation lies in using the
3292:) will denote random variables and small letters (
3257:
3233:
3198:
3168:
3107:
3048:
2994:
2959:
2932:
2898:
2817:
2701:
2681:
2661:
2602:
2572:
2408:
2347:
2283:
2195:
2099:
2047:
2007:
1947:
1875:
1773:
1672:
1625:
1469:
1442:
1370:
1344:
1289:
1195:
1102:
1076:
1001:
976:
951:
900:
852:
823:
797:
726:
576:
508:
424:
360:
281:
222:
136:
3315:
3303:
3083:Joint probability density function of parameters
2409:{\displaystyle s_{\lambda }=\sum _{i=1}^{m}x_{i}}
1457:for the parameter θ and its discretization grain
1450:from a random variable with parameter θ unknown,
838:The direction of the monotony determines for any
3410:
1382:. A procedure implementing it is as follows.
167:, suitability of an estimate reads in terms of
3397:Apolloni, B; Malchiodi, D.; Gaito, S. (2006).
2583:Using the first factorization and replacing
1437:
1405:
1378:. The whole logical contrivance is called a
1345:{\displaystyle F_{\Theta \mid S=s}(\theta )}
792:
760:
577:{\displaystyle S=h_{1}(X_{1},\ldots ,X_{m})}
503:
439:
419:
387:
234:seed distribution law to determine both the
131:
99:
2899:{\displaystyle F_{K}(k)=1-F_{R_{k}}(r_{K})}
2348:{\displaystyle s_{k}=\prod _{i=1}^{m}x_{i}}
3265:on the left. The marginal distribution of
3399:Algorithmic Inference in Machine Learning
3358:
3269:is reported in the picture on the right.
2731:
2727:
2527:
2523:
2472:
2468:
66:Learn how and when to remove this message
3118:
3078:
984:is computed by the master equation with
29:This article includes a list of general
3169:{\displaystyle m=30,s_{\Lambda }=72.82}
2093:
1443:{\displaystyle \{x_{1},\ldots ,x_{m}\}}
846:
817:
753:
380:
137:{\displaystyle \{x_{1},\ldots ,x_{m}\}}
3411:
3367:
3336:
3321:
3309:
860:a relation between events of the type
3284:By default, capital letters (such as
361:{\displaystyle M_{X}=(g_{\theta },Z)}
223:{\displaystyle M_{X}=(g_{\theta },Z)}
1948:{\displaystyle q_{i}(F_{S})=1-F_{S}}
282:{\displaystyle F_{\Theta }(\theta )}
15:
3300:) their corresponding realizations.
2669:in order to have a distribution of
1453:Identify a well behaving statistic
1304:discrete we have an interval where
516:. Focusing on a relevant statistic
13:
3351:10.1111/j.1469-1809.1935.tb02120.x
3252:
3234:{\displaystyle 4.5\times 10^{-46}}
3155:
3099:
2974:
2925:
2797:
2787:
2764:
2724:
2696:
2553:
2528:
2465:
2431:
2008:{\displaystyle q_{i}(F_{S})=F_{S}}
1595:
1543:
1495:
1316:
1267:
1221:
1173:
1133:
265:
35:it lacks sufficient corresponding
14:
3435:
2100:{\displaystyle {\boldsymbol {x}}}
853:{\displaystyle {\boldsymbol {z}}}
824:{\displaystyle {\boldsymbol {Z}}}
289:, in light of the observed value
3403:Advanced Knowledge International
3060:that can be approximated with a
3049:{\displaystyle F_{R_{k}}(r_{K})}
252:cumulative distribution function
20:
3278:
3102:
3090:
3043:
3030:
2989:
2977:
2893:
2880:
2851:
2845:
2809:
2800:
2792:
2767:
2749:
2743:
2564:
2558:
2545:
2539:
2509:
2503:
2490:
2484:
2454:
2442:
2267:
2236:
2233:
2230:
2213:
2179:
2148:
2145:
2142:
2125:
1989:
1976:
1923:
1910:
1870:
1858:
1845:
1829:
1826:
1820:
1807:
1768:
1756:
1743:
1727:
1724:
1718:
1705:
1615:
1612:
1606:
1591:
1579:
1563:
1560:
1554:
1539:
1527:
1506:
1500:
1339:
1333:
1284:
1278:
1244:
1238:
1190:
1184:
1156:
1150:
1051:
1034:
932:
881:
718:
680:
671:
668:
655:
633:
620:
607:
571:
539:
500:
487:
465:
452:
355:
336:
276:
270:
217:
198:
84:
1:
3330:
1959:does not decrease with θ and
155:with a non-set parameter, a
3108:{\displaystyle (K,\Lambda )}
2995:{\displaystyle \Gamma (a,b)}
2933:{\displaystyle s_{\Lambda }}
1385:
588:, the master equation reads
7:
3127:of a Gamma random variable.
3115:of a Gamma random variable.
1673:{\displaystyle q_{1}=q_{2}}
1480:decide the monotony versus;
1118:continuous we have either
171:with the observed sample.
10:
3440:
2913:denoting the sample size,
2080:
1371:{\displaystyle \ell >0}
1103:{\displaystyle \ell >0}
301:distribution law when the
3004:incomplete gamma function
2067:
368:for the random variable
308:
3424:Computational statistics
3272:
3258:{\displaystyle \Lambda }
2702:{\displaystyle \Lambda }
1002:{\displaystyle \theta '}
305:parameter is exactly θ.
297:, as a function of the
2689:that is independent of
2019:does not increase with
1887:does not increase with
1785:does not decrease with
584:for the parameter
50:more precise citations.
3259:
3235:
3200:
3199:{\displaystyle r_{K}=}
3170:
3128:
3116:
3109:
3050:
2996:
2961:
2934:
2900:
2819:
2703:
2683:
2663:
2604:
2574:
2410:
2395:
2349:
2334:
2285:
2197:
2101:
2049:
2009:
1949:
1877:
1775:
1674:
1627:
1471:
1444:
1372:
1346:
1291:
1197:
1104:
1078:
1003:
978:
953:
902:
854:
825:
799:
743:well-behaved statistic
728:
578:
510:
426:
362:
283:
224:
138:
3419:Algorithmic inference
3260:
3236:
3201:
3171:
3122:
3110:
3082:
3051:
2997:
2962:
2960:{\displaystyle r_{K}}
2935:
2901:
2820:
2704:
2684:
2664:
2605:
2603:{\displaystyle s_{k}}
2575:
2411:
2375:
2350:
2314:
2286:
2272: for fixed
2198:
2184: for fixed
2102:
2075:fiducial distribution
2050:
2048:{\displaystyle i=1,2}
2010:
1950:
1878:
1776:
1675:
1628:
1472:
1470:{\displaystyle \ell }
1445:
1373:
1347:
1292:
1198:
1105:
1079:
1004:
979:
954:
903:
855:
826:
800:
729:
579:
511:
427:
363:
284:
225:
165:algorithmic inference
139:
3249:
3209:
3180:
3135:
3087:
3010:
2971:
2944:
2917:
2832:
2716:
2693:
2673:
2614:
2587:
2423:
2359:
2298:
2210:
2122:
2089:
2027:
1963:
1897:
1794:
1692:
1644:
1487:
1461:
1402:
1356:
1308:
1213:
1125:
1110:is the size of the
1088:
1017:
988:
963:
915:
864:
842:
813:
749:
595:
520:
436:
376:
320:
257:
182:
157:parametric inference
96:
3131:With a sample size
3068:) as a function of
2656:
1009:. In the case that
79:Twisting properties
3339:Annals of Eugenics
3255:
3231:
3196:
3166:
3129:
3117:
3105:
3062:gamma distribution
3046:
2992:
2957:
2930:
2896:
2815:
2699:
2679:
2659:
2642:
2600:
2570:
2406:
2345:
2281:
2193:
2109:gamma distribution
2097:
2045:
2005:
1945:
1873:
1771:
1670:
1623:
1467:
1440:
1368:
1342:
1287:
1193:
1100:
1074:
999:
977:{\displaystyle s'}
974:
949:
898:
850:
821:
795:
724:
574:
506:
422:
358:
315:sampling mechanism
279:
220:
177:sampling mechanism
134:
3066:method of moments
2813:
2682:{\displaystyle K}
2657:
2273:
2185:
2065:
2064:
1380:twisting argument
1352:lies, because of
76:
75:
68:
3431:
3405:
3393:
3364:
3362:
3325:
3319:
3313:
3307:
3301:
3282:
3264:
3262:
3261:
3256:
3240:
3238:
3237:
3232:
3230:
3229:
3205:
3203:
3202:
3197:
3192:
3191:
3175:
3173:
3172:
3167:
3159:
3158:
3114:
3112:
3111:
3106:
3058:Fox's H function
3055:
3053:
3052:
3047:
3042:
3041:
3029:
3028:
3027:
3026:
3001:
2999:
2998:
2993:
2966:
2964:
2963:
2958:
2956:
2955:
2939:
2937:
2936:
2931:
2929:
2928:
2905:
2903:
2902:
2897:
2892:
2891:
2879:
2878:
2877:
2876:
2844:
2843:
2824:
2822:
2821:
2816:
2814:
2812:
2795:
2791:
2790:
2762:
2742:
2741:
2708:
2706:
2705:
2700:
2688:
2686:
2685:
2680:
2668:
2666:
2665:
2660:
2658:
2655:
2650:
2641:
2640:
2631:
2626:
2625:
2609:
2607:
2606:
2601:
2599:
2598:
2579:
2577:
2576:
2571:
2557:
2556:
2538:
2537:
2502:
2501:
2483:
2482:
2441:
2440:
2415:
2413:
2412:
2407:
2405:
2404:
2394:
2389:
2371:
2370:
2354:
2352:
2351:
2346:
2344:
2343:
2333:
2328:
2310:
2309:
2290:
2288:
2287:
2282:
2274:
2271:
2266:
2265:
2253:
2252:
2251:
2229:
2202:
2200:
2199:
2194:
2186:
2183:
2178:
2177:
2176:
2160:
2159:
2141:
2106:
2104:
2103:
2098:
2096:
2054:
2052:
2051:
2046:
2014:
2012:
2011:
2006:
2004:
2003:
1988:
1987:
1975:
1974:
1954:
1952:
1951:
1946:
1944:
1943:
1922:
1921:
1909:
1908:
1882:
1880:
1879:
1874:
1857:
1856:
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1843:
1819:
1818:
1806:
1805:
1780:
1778:
1777:
1772:
1755:
1754:
1742:
1741:
1717:
1716:
1704:
1703:
1679:
1677:
1676:
1671:
1669:
1668:
1656:
1655:
1632:
1630:
1629:
1624:
1622:
1618:
1605:
1604:
1594:
1578:
1577:
1553:
1552:
1542:
1526:
1525:
1499:
1498:
1476:
1474:
1473:
1468:
1449:
1447:
1446:
1441:
1436:
1435:
1417:
1416:
1390:
1389:
1377:
1375:
1374:
1369:
1351:
1349:
1348:
1343:
1332:
1331:
1296:
1294:
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1288:
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1236:
1202:
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1199:
1194:
1183:
1182:
1149:
1148:
1109:
1107:
1106:
1101:
1083:
1081:
1080:
1075:
1067:
1050:
1033:
1008:
1006:
1005:
1000:
998:
983:
981:
980:
975:
973:
958:
956:
955:
950:
948:
931:
907:
905:
904:
899:
897:
880:
859:
857:
856:
851:
849:
830:
828:
827:
822:
820:
804:
802:
801:
796:
791:
790:
772:
771:
756:
733:
731:
730:
725:
717:
716:
698:
697:
667:
666:
654:
653:
632:
631:
619:
618:
583:
581:
580:
575:
570:
569:
551:
550:
538:
537:
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513:
512:
507:
499:
498:
486:
485:
464:
463:
451:
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431:
429:
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423:
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398:
383:
367:
365:
364:
359:
348:
347:
332:
331:
288:
286:
285:
280:
269:
268:
229:
227:
226:
221:
210:
209:
194:
193:
153:distribution law
144:observed from a
143:
141:
140:
135:
130:
129:
111:
110:
89:Starting with a
71:
64:
60:
57:
51:
46:this article by
37:inline citations
24:
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2296:
2295:
2270:
2261:
2257:
2244:
2243:
2239:
2222:
2211:
2208:
2207:
2182:
2169:
2168:
2164:
2155:
2151:
2134:
2123:
2120:
2119:
2092:
2090:
2087:
2086:
2083:
2070:
2028:
2025:
2024:
1999:
1995:
1983:
1979:
1970:
1966:
1964:
1961:
1960:
1939:
1935:
1917:
1913:
1904:
1900:
1898:
1895:
1894:
1852:
1848:
1839:
1835:
1814:
1810:
1801:
1797:
1795:
1792:
1791:
1750:
1746:
1737:
1733:
1712:
1708:
1699:
1695:
1693:
1690:
1689:
1664:
1660:
1651:
1647:
1645:
1642:
1641:
1590:
1586:
1582:
1573:
1569:
1538:
1534:
1530:
1521:
1517:
1516:
1512:
1494:
1490:
1488:
1485:
1484:
1462:
1459:
1458:
1431:
1427:
1412:
1408:
1403:
1400:
1399:
1398:Given a sample
1388:
1357:
1354:
1353:
1315:
1311:
1309:
1306:
1305:
1260:
1256:
1220:
1216:
1214:
1211:
1210:
1166:
1162:
1132:
1128:
1126:
1123:
1122:
1089:
1086:
1085:
1060:
1043:
1026:
1018:
1015:
1014:
991:
989:
986:
985:
966:
964:
961:
960:
941:
924:
916:
913:
912:
890:
873:
865:
862:
861:
845:
843:
840:
839:
816:
814:
811:
810:
786:
782:
767:
763:
752:
750:
747:
746:
712:
708:
693:
689:
662:
658:
649:
645:
627:
623:
614:
610:
596:
593:
592:
565:
561:
546:
542:
533:
529:
521:
518:
517:
494:
490:
481:
477:
459:
455:
446:
442:
437:
434:
433:
432:to be equal to
413:
409:
394:
390:
379:
377:
374:
373:
343:
339:
327:
323:
321:
318:
317:
311:
293:of a statistic
264:
260:
258:
255:
254:
205:
201:
189:
185:
183:
180:
179:
151:having a given
146:random variable
125:
121:
106:
102:
97:
94:
93:
87:
72:
61:
55:
52:
42:Please help to
41:
25:
21:
12:
11:
5:
3437:
3427:
3426:
3421:
3407:
3406:
3394:
3365:
3345:(4): 391–398.
3332:
3329:
3327:
3326:
3314:
3302:
3276:
3274:
3271:
3254:
3228:
3225:
3221:
3217:
3214:
3195:
3190:
3186:
3165:
3162:
3157:
3153:
3149:
3146:
3143:
3140:
3104:
3101:
3098:
3095:
3092:
3045:
3040:
3036:
3032:
3025:
3021:
3016:
2991:
2988:
2985:
2982:
2979:
2976:
2954:
2950:
2927:
2923:
2907:
2906:
2895:
2890:
2886:
2882:
2875:
2871:
2866:
2862:
2859:
2856:
2853:
2850:
2847:
2842:
2838:
2826:
2825:
2811:
2808:
2805:
2802:
2799:
2794:
2789:
2785:
2781:
2778:
2775:
2772:
2769:
2766:
2760:
2757:
2754:
2751:
2748:
2745:
2740:
2737:
2734:
2730:
2726:
2722:
2698:
2678:
2654:
2649:
2645:
2639:
2635:
2629:
2624:
2620:
2597:
2593:
2581:
2580:
2569:
2566:
2563:
2560:
2555:
2551:
2547:
2544:
2541:
2536:
2533:
2530:
2526:
2522:
2518:
2514:
2511:
2508:
2505:
2500:
2496:
2492:
2489:
2486:
2481:
2478:
2475:
2471:
2467:
2463:
2459:
2456:
2453:
2450:
2447:
2444:
2439:
2436:
2433:
2429:
2403:
2399:
2393:
2388:
2385:
2382:
2378:
2374:
2369:
2365:
2342:
2338:
2332:
2327:
2324:
2321:
2317:
2313:
2308:
2304:
2292:
2291:
2280:
2277:
2269:
2264:
2260:
2256:
2250:
2247:
2242:
2238:
2235:
2232:
2228:
2225:
2221:
2218:
2215:
2204:
2203:
2192:
2189:
2181:
2175:
2172:
2167:
2163:
2158:
2154:
2150:
2147:
2144:
2140:
2137:
2133:
2130:
2127:
2095:
2082:
2079:
2069:
2066:
2063:
2062:
2061:
2060:
2059:
2058:
2057:
2056:
2044:
2041:
2038:
2035:
2032:
2002:
1998:
1994:
1991:
1986:
1982:
1978:
1973:
1969:
1942:
1938:
1934:
1931:
1928:
1925:
1920:
1916:
1912:
1907:
1903:
1892:
1872:
1869:
1866:
1863:
1860:
1855:
1851:
1847:
1842:
1838:
1834:
1831:
1828:
1825:
1822:
1817:
1813:
1809:
1804:
1800:
1789:
1770:
1767:
1764:
1761:
1758:
1753:
1749:
1745:
1740:
1736:
1732:
1729:
1726:
1723:
1720:
1715:
1711:
1707:
1702:
1698:
1680:
1667:
1663:
1659:
1654:
1650:
1640:is continuous
1621:
1617:
1614:
1611:
1608:
1603:
1600:
1597:
1593:
1589:
1585:
1581:
1576:
1572:
1568:
1565:
1562:
1559:
1556:
1551:
1548:
1545:
1541:
1537:
1533:
1529:
1524:
1520:
1515:
1511:
1508:
1505:
1502:
1497:
1493:
1481:
1478:
1466:
1439:
1434:
1430:
1426:
1423:
1420:
1415:
1411:
1407:
1395:
1394:
1387:
1384:
1367:
1364:
1361:
1341:
1338:
1335:
1330:
1327:
1324:
1321:
1318:
1314:
1298:
1297:
1286:
1283:
1280:
1275:
1272:
1269:
1266:
1263:
1259:
1255:
1252:
1249:
1246:
1243:
1240:
1235:
1232:
1229:
1226:
1223:
1219:
1204:
1203:
1192:
1189:
1186:
1181:
1178:
1175:
1172:
1169:
1165:
1161:
1158:
1155:
1152:
1147:
1144:
1141:
1138:
1135:
1131:
1099:
1096:
1093:
1073:
1070:
1066:
1063:
1059:
1056:
1053:
1049:
1046:
1042:
1039:
1036:
1032:
1029:
1025:
1022:
997:
994:
972:
969:
947:
944:
940:
937:
934:
930:
927:
923:
920:
896:
893:
889:
886:
883:
879:
876:
872:
869:
848:
819:
794:
789:
785:
781:
778:
775:
770:
766:
762:
759:
755:
735:
734:
723:
720:
715:
711:
707:
704:
701:
696:
692:
688:
685:
682:
679:
676:
673:
670:
665:
661:
657:
652:
648:
644:
641:
638:
635:
630:
626:
622:
617:
613:
609:
606:
603:
600:
573:
568:
564:
560:
557:
554:
549:
545:
541:
536:
532:
528:
525:
505:
502:
497:
493:
489:
484:
480:
476:
473:
470:
467:
462:
458:
454:
449:
445:
441:
421:
416:
412:
408:
405:
402:
397:
393:
389:
386:
382:
357:
354:
351:
346:
342:
338:
335:
330:
326:
310:
307:
278:
275:
272:
267:
263:
219:
216:
213:
208:
204:
200:
197:
192:
188:
133:
128:
124:
120:
117:
114:
109:
105:
101:
86:
83:
74:
73:
56:September 2009
28:
26:
19:
9:
6:
4:
3:
2:
3436:
3425:
3422:
3420:
3417:
3416:
3414:
3404:
3400:
3395:
3391:
3387:
3383:
3379:
3375:
3371:
3366:
3361:
3356:
3352:
3348:
3344:
3340:
3335:
3334:
3323:
3318:
3311:
3306:
3299:
3295:
3291:
3287:
3281:
3277:
3270:
3268:
3244:
3226:
3223:
3219:
3215:
3212:
3193:
3188:
3184:
3163:
3160:
3151:
3147:
3144:
3141:
3138:
3126:
3121:
3096:
3093:
3081:
3077:
3075:
3071:
3067:
3063:
3059:
3038:
3034:
3023:
3019:
3014:
3005:
2986:
2983:
2980:
2952:
2948:
2921:
2912:
2888:
2884:
2873:
2869:
2864:
2860:
2857:
2854:
2848:
2840:
2836:
2828:
2827:
2806:
2803:
2783:
2779:
2776:
2773:
2770:
2758:
2755:
2752:
2746:
2738:
2735:
2732:
2728:
2720:
2712:
2711:
2710:
2676:
2652:
2647:
2643:
2637:
2633:
2627:
2622:
2618:
2595:
2591:
2567:
2561:
2549:
2542:
2534:
2531:
2524:
2520:
2516:
2512:
2506:
2498:
2494:
2487:
2479:
2476:
2473:
2469:
2461:
2457:
2451:
2448:
2445:
2437:
2434:
2427:
2419:
2418:
2417:
2401:
2397:
2391:
2386:
2383:
2380:
2376:
2372:
2367:
2363:
2340:
2336:
2330:
2325:
2322:
2319:
2315:
2311:
2306:
2302:
2278:
2275:
2262:
2258:
2254:
2248:
2245:
2240:
2226:
2223:
2219:
2216:
2206:
2205:
2190:
2187:
2173:
2170:
2165:
2161:
2156:
2152:
2138:
2135:
2131:
2128:
2118:
2117:
2116:
2114:
2110:
2107:drawn from a
2078:
2076:
2042:
2039:
2036:
2033:
2030:
2022:
2018:
2000:
1996:
1992:
1984:
1980:
1971:
1967:
1958:
1940:
1936:
1932:
1929:
1926:
1918:
1914:
1905:
1901:
1893:
1890:
1886:
1867:
1864:
1861:
1853:
1849:
1840:
1836:
1832:
1823:
1815:
1811:
1802:
1798:
1790:
1788:
1784:
1765:
1762:
1759:
1751:
1747:
1738:
1734:
1730:
1721:
1713:
1709:
1700:
1696:
1688:
1687:
1685:
1681:
1665:
1661:
1657:
1652:
1648:
1639:
1635:
1634:
1619:
1609:
1601:
1598:
1587:
1583:
1574:
1570:
1566:
1557:
1549:
1546:
1535:
1531:
1522:
1518:
1513:
1509:
1503:
1491:
1482:
1479:
1464:
1456:
1452:
1451:
1432:
1428:
1424:
1421:
1418:
1413:
1409:
1397:
1396:
1392:
1391:
1383:
1381:
1365:
1362:
1359:
1336:
1328:
1325:
1322:
1319:
1312:
1303:
1281:
1273:
1270:
1264:
1261:
1257:
1253:
1250:
1247:
1241:
1233:
1230:
1227:
1224:
1217:
1209:
1208:
1207:
1187:
1179:
1176:
1170:
1167:
1163:
1159:
1153:
1145:
1142:
1139:
1136:
1129:
1121:
1120:
1119:
1117:
1113:
1097:
1094:
1091:
1071:
1068:
1064:
1061:
1057:
1054:
1047:
1044:
1040:
1037:
1030:
1027:
1023:
1020:
1012:
995:
992:
970:
967:
945:
942:
938:
935:
928:
925:
921:
918:
911:
894:
891:
887:
884:
877:
874:
870:
867:
836:
834:
808:
787:
783:
779:
776:
773:
768:
764:
757:
744:
740:
721:
713:
709:
705:
702:
699:
694:
690:
686:
683:
677:
674:
663:
659:
650:
646:
642:
639:
636:
628:
624:
615:
611:
604:
601:
598:
591:
590:
589:
587:
566:
562:
558:
555:
552:
547:
543:
534:
530:
526:
523:
495:
491:
482:
478:
474:
471:
468:
460:
456:
447:
443:
414:
410:
406:
403:
400:
395:
391:
384:
371:
352:
349:
344:
340:
333:
328:
324:
316:
306:
304:
300:
296:
292:
273:
261:
253:
249:
248:well behavior
245:
241:
237:
233:
214:
211:
206:
202:
195:
190:
186:
178:
172:
170:
169:compatibility
166:
162:
158:
154:
150:
147:
126:
122:
118:
115:
112:
107:
103:
92:
82:
80:
70:
67:
59:
49:
45:
39:
38:
32:
27:
18:
17:
3402:
3398:
3376:(1/2): 1–9.
3373:
3369:
3342:
3338:
3317:
3305:
3297:
3293:
3289:
3285:
3280:
3266:
3242:
3130:
3124:
3073:
3069:
2910:
2908:
2582:
2293:
2112:
2084:
2071:
2020:
2016:
1956:
1888:
1884:
1786:
1782:
1686:is discrete
1683:
1637:
1454:
1379:
1301:
1299:
1205:
1115:
1111:
1010:
909:
837:
832:
806:
738:
736:
585:
369:
312:
302:
298:
294:
290:
239:
235:
231:
173:
148:
88:
78:
77:
62:
53:
34:
3322:Fraser 1966
3310:Fisher 1935
372:, we model
85:Description
48:introducing
3413:Categories
3370:Biometrika
3360:2440/15222
3331:References
2709:, we have
910:vice versa
831:for given
31:references
3253:Λ
3224:−
3216:×
3156:Λ
3100:Λ
2975:Γ
2926:Λ
2861:−
2798:Γ
2788:Λ
2780:λ
2765:Γ
2759:−
2747:λ
2729:∣
2725:Λ
2697:Λ
2648:λ
2562:λ
2554:Λ
2535:λ
2529:Λ
2525:∣
2488:λ
2470:∣
2466:Λ
2446:λ
2432:Λ
2377:∑
2368:λ
2316:∏
2263:λ
2255:≤
2246:λ
2234:↔
2224:λ
2220:≤
2217:λ
2188:λ
2162:≤
2146:↔
2132:≤
1933:−
1868:ℓ
1865:−
1766:ℓ
1763:−
1602:θ
1596:Θ
1550:θ
1544:Θ
1510:∈
1504:θ
1496:Θ
1477:(if any);
1465:ℓ
1422:…
1386:Algorithm
1360:ℓ
1337:θ
1320:∣
1317:Θ
1274:θ
1268:Θ
1265:∣
1254:−
1242:θ
1225:∣
1222:Θ
1180:θ
1174:Θ
1171:∣
1154:θ
1137:∣
1134:Θ
1092:ℓ
1072:ℓ
1058:≥
1052:→
1045:θ
1041:≥
1038:θ
1035:→
1024:≥
993:θ
959:, where
943:θ
939:≤
936:θ
933:↔
922:≥
892:θ
888:≥
885:θ
882:↔
871:≥
777:…
703:…
684:θ
678:ρ
651:θ
640:…
616:θ
556:…
483:θ
472:…
448:θ
404:…
345:θ
274:θ
266:Θ
207:θ
161:estimates
116:…
2249:′
2227:′
2174:′
2139:′
1483:compute
1065:′
1048:′
1031:′
996:′
971:′
946:′
929:′
895:′
878:′
805:between
313:Given a
3390:2334048
2081:Example
1633:where:
244:support
44:improve
3388:
2294:where
2068:Remark
1084:where
309:Method
91:sample
33:, but
3386:JSTOR
3273:Notes
3164:72.82
2909:with
2610:with
741:is a
737:When
3245:and
3176:and
3072:and
3056:the
3006:and
3002:the
2940:and
2355:and
2085:For
2023:for
1363:>
1300:For
1206:or
1095:>
3378:doi
3355:hdl
3347:doi
3213:4.5
2015:if
1955:if
1891:and
1883:if
1781:if
1682:if
1636:if
908:or
3415::
3384:.
3374:53
3372:.
3353:.
3341:.
3296:,
3288:,
3227:46
3220:10
3145:30
3076:.
3392:.
3380::
3363:.
3357::
3349::
3343:6
3324:.
3312:.
3298:x
3294:u
3290:X
3286:U
3267:K
3243:K
3194:=
3189:K
3185:r
3161:=
3152:s
3148:,
3142:=
3139:m
3125:K
3103:)
3097:,
3094:K
3091:(
3074:m
3070:k
3044:)
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