Figure 1.
GeTy2PrHCS. Case (a): , if , . Case (b): , if . Case (c): , if .
Figure 1.
GeTy2PrHCS. Case (a): , if , . Case (b): , if . Case (c): , if .
Figure 2.
The CondPDFs of and for example.
Figure 3.
Relative rMSEs for and
Figure 4.
Relative ConfL for and .
Table 1.
Full data for illustrative example.
xi | 11 | 35 | 49 | 170 | 329 | 381 | 708 | 958 | 1062 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1167 | 1594 | 1925 | 1990 | 2223 | 2327 | 2400 | 2451 | 2471 |
| 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 1 |
| 2551 | 2565 | 2831 | 2568 | 2694 | 2702 | 2761 | 3034 | 3059 |
| 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
| 3112 | 3214 | 3478 | 3504 | 4329 | 6367 | 6976 | 7846 | 13403 |
| 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 |
Table 2.
PrTy2CS data for illustrative example.
| 11 | 35 | 49 | 170 | 329 | 381 | 708 | 958 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1062 | 1167 | 1594 | 1925 | 1990 | 2223 | 2327 | 2400 |
| 0 | 1 | 0 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2451 | 2471 | 2551 | 2565 | 2568 | 2694 | 2702 | 2761 |
| 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 |
Table 3.
Inference of and for illustrative example.
| | | | | SE() | 95% ConfI for |
---|
| SE() | 95% ConfI for |
---|
3000 | 4000 | 9 | 16 | 6221.000 | 2073.667 | (3236.839, 11,956.370) |
3499.312 | 874.828 | (2143.771, 5711.985) |
2000 | 3000 | 9 | 15 | 6080.444 | 2026.815 | (3163.707, 11,686.230) |
3648.267 | 941.978 | (2199.395, 6051.596) |
2000 | 2500 | 6 | 12 | 8719.500 | 3559.721 | (3917.271, 19,408.840) |
4359.750 | 1258.551 | (2475.919, 7676.917) |
Table 4.
Relative rMSEs and biases for and .
n | | m | Scheme | | | |
---|
| | | | | |
---|
20 | 0.3 | 18 | (a) | 0.1498(0.0233) | 0.3436(0.0678) | 0.1424(0.0198) | 0.3030(0.0606) | 0.1415(0.0195) | 0.3028(0.0601) |
(b) | 0.1473(0.0224) | 0.3914(0.0792) | 0.1428(0.0203) | 0.3250(0.0616) | 0.1431(0.0205) | 0.3060(0.0560) |
(c) | 0.1500(0.0240) | 0.3939(0.0891) | 0.1394(0.0196) | 0.3708(0.0797) | 0.1373(0.0182) | 0.3710(0.0790) |
(d) | 0.1468(0.0217) | 0.3694(0.0733) | 0.1421(0.0198) | 0.3249(0.0610) | 0.1424(0.0201) | 0.3036(0.0555) |
16 | (a) | 0.1622(0.0276) | 0.3767(0.0819) | 0.1596(0.0268) | 0.3483(0.0786) | 0.1596(0.0268) | 0.3483(0.0786) |
(b) | 0.1704(0.0313) | 0.4404(0.0958) | 0.1611(0.0256) | 0.4065(0.0813) | 0.1554(0.0232) | 0.3961(0.0762) |
(c) | 0.1494(0.0211) | 0.4200(0.0794) | 0.1459(0.0189) | 0.3672(0.0659) | 0.1460(0.0187) | 0.3683(0.0661) |
(d) | 0.1661(0.0300) | 0.4231(0.0891) | 0.1559(0.0234) | 0.4076(0.0799) | 0.1516(0.0219) | 0.3926(0.0743) |
14 | (a) | 0.1737(0.0346) | 0.4694(0.1370) | 0.1731(0.0344) | 0.4694(0.1037) | 0.1731(0.0344) | 0.4694(0.1037) |
(b) | 0.2018(0.0393) | 0.4634(0.1251) | 0.1858(0.0342) | 0.4092(0.1059) | 0.1763(0.0302) | 0.4006(0.1004) |
(c) | 0.1650(0.0279) | 0.3720(0.0814) | 0.1633(0.0258) | 0.4081(0.0906) | 0.1630(0.0257) | 0.4048(0.0897) |
(d) | 0.2000(0.0365) | 0.4396(0.1154) | 0.1824(0.0329) | 0.4040(0.1038) | 0.1763(0.0301) | 0.4017(0.1003) |
12 | (a) | 0.1984(0.0437) | 0.4519(0.1122) | 0.1984(0.0437) | 0.4519(0.1122) | 0.1984(0.0437) | 0.4519(0.1122) |
(b) | 0.2373(0.0486) | 0.5328(0.1392) | 0.2262(0.0441) | 0.5088(0.1223) | 0.2154(0.0401) | 0.5107(0.1183) |
(c) | 0.1861(0.0338) | 0.5030(0.1113) | 0.1855(0.0336) | 0.5027(0.1114) | 0.1860(0.0336) | 0.5021(0.1111) |
(d) | 0.2286(0.0443) | 0.5196(0.1302) | 0.2162(0.0415) | 0.5128(0.1228) | 0.2165(0.0401) | 0.5103(0.1180) |
30 | 0.3 | 28 | (a) | 0.1134(0.0126) | 0.2278(0.0358) | 0.1092(0.0103) | 0.2228(0.0315) | 0.1079(0.0097) | 0.2121(0.0301) |
(b) | 0.1173(0.0182) | 0.3166(0.0582) | 0.1123(0.0161) | 0.3123(0.0517) | 0.1101(0.0146) | 0.2461(0.0463) |
(c) | 0.1229(0.0144) | 0.2485(0.0483) | 0.1145(0.0116) | 0.2306(0.0430) | 0.1108(0.0103) | 0.2265(0.0404) |
(d) | 0.1167(0.0182) | 0.3176(0.0581) | 0.1120(0.0158) | 0.3121(0.0515) | 0.1101(0.0145) | 0.2457(0.0462) |
26 | (a) | 0.1141(0.0111) | 0.2349(0.0390) | 0.1122(0.0101) | 0.2336(0.0381) | 0.1118(0.0100) | 0.2334(0.0380) |
(b) | 0.1177(0.0177) | 0.2364(0.0398) | 0.1128(0.0150) | 0.2259(0.0358) | 0.1103(0.0141) | 0.2183(0.0333) |
(c) | 0.1176(0.0172) | 0.3164(0.0565) | 0.1143(0.0152) | 0.3190(0.0517) | 0.1142(0.0149) | 0.3183(0.0505) |
(d) | 0.1163(0.0166) | 0.2352(0.0387) | 0.1121(0.0148) | 0.2253(0.0353) | 0.1102(0.0140) | 0.2176(0.0328) |
24 | (a) | 0.1183(0.0130) | 0.2418(0.0433) | 0.1176(0.0127) | 0.2412(0.0430) | 0.1176(0.0127) | 0.2412(0.0430) |
(b) | 0.1256(0.0165) | 0.2789(0.0582) | 0.1185(0.0140) | 0.2693(0.0518) | 0.1168(0.0129) | 0.2514(0.0465) |
(c) | 0.1184(0.0151) | 0.2385(0.0382) | 0.1152(0.0137) | 0.2328(0.0356) | 0.1147(0.0135) | 0.2326(0.0353) |
(d) | 0.1237(0.0160) | 0.2738(0.0555) | 0.1175(0.0134) | 0.2668(0.0502) | 0.1158(0.0126) | 0.2503(0.0462) |
22 | (a) | 0.1205(0.0138) | 0.2667(0.0547) | 0.1205(0.0138) | 0.2667(0.0547) | 0.1205(0.0138) | 0.2667(0.0547) |
(b) | 0.1381(0.0208) | 0.2747(0.0479) | 0.1285(0.0148) | 0.2584(0.0423) | 0.1277(0.0139) | 0.2577(0.0411) |
(c) | 0.1216(0.0183) | 0.2472(0.0396) | 0.1201(0.0178) | 0.2472(0.0394) | 0.1202(0.0179) | 0.2474(0.0394) |
(d) | 0.1338(0.0186) | 0.2587(0.0432) | 0.1283(0.0147) | 0.2588(0.0424) | 0.1259(0.0133) | 0.2571(0.0403) |
40 | 10.0 | 38 | (a) | 0.0924(0.0084) | 0.1825(0.0237) | 0.0906(0.0072) | 0.1766(0.0191) | 0.0892(0.0065) | 0.1744(0.0179) |
(b) | 0.0952(0.0102) | 0.1832(0.0183) | 0.0919(0.0081) | 0.1782(0.0161) | 0.0893(0.0070) | 0.1745(0.0145) |
(c) | 0.0934(0.0076) | 0.1971(0.0300) | 0.0901(0.0059) | 0.1837(0.0237) | 0.0892(0.0049) | 0.1791(0.0207) |
(d) | 0.0945(0.0099) | 0.1822(0.0173) | 0.0912(0.0077) | 0.1766(0.0156) | 0.0893(0.0070) | 0.1746(0.0143) |
36 | (a) | 0.0937(0.0079) | 0.1837(0.0230) | 0.0921(0.0068) | 0.1810(0.0208) | 0.0922(0.0069) | 0.1807(0.0206) |
(b) | 0.0968(0.0101) | 0.2052(0.0293) | 0.0942(0.0093) | 0.1971(0.0253) | 0.0932(0.0096) | 0.1891(0.0222) |
(c) | 0.0952(0.0097) | 0.1837(0.0180) | 0.0926(0.0081) | 0.1797(0.0149) | 0.0914(0.0078) | 0.1761(0.0142) |
(d) | 0.0960(0.0098) | 0.2052(0.0297) | 0.0941(0.0093) | 0.1957(0.0242) | 0.0929(0.0095) | 0.1892(0.0221) |
34 | (a) | 0.0949(0.0076) | 0.1866(0.0239) | 0.0940(0.0072) | 0.1859(0.0235) | 0.0940(0.0072) | 0.1859(0.0235) |
(b) | 0.0998(0.0100) | 0.2257(0.0370) | 0.0952(0.0083) | 0.2026(0.0293) | 0.0945(0.0081) | 0.1958(0.0271) |
(c) | 0.0932(0.0095) | 0.1900(0.0239) | 0.0895(0.0070) | 0.1876(0.2019) | 0.0888(0.0067) | 0.1861(0.0214) |
(d) | 0.0974(0.0088) | 0.2158(0.0350) | 0.0953(0.0086) | 0.2022(0.0293) | 0.0944(0.0081) | 0.1962(0.0274) |
32 | (a) | 0.0973(0.0081) | 0.1897(0.0257) | 0.0972(0.0080) | 0.1896(0.0258) | 0.0972(0.0080) | 0.1896(0.0258) |
(b) | 0.1099(0.0130) | 0.2079(0.0236) | 0.1047(0.0104) | 0.1957(0.0212) | 0.1020(0.0104) | 0.1896(0.0197) |
(c) | 0.0976(0.0080) | 0.2075(0.0261) | 0.0958(0.0072) | 0.2009(0.0231) | 0.0956(0.0071) | 0.2004(0.0229) |
(d) | 0.1086(0.0121) | 0.2069(0.0232) | 0.1036(0.0103) | 0.1922(0.0199) | 0.1015(0.0102) | 0.1893(0.0197) |
Table 5.
Relative ConfL and CovP for and .
n | | m | Scheme | | | |
---|
| | | | | |
---|
20 | 0.3 | 18 | (a) | 0.5708(940.1) † | 10.2596(960.1) | 0.5567(940.1) | 10.2036(960.0) | 0.5553(940.0) | 10.2010(950.9) |
(b) | 0.5949(940.7) | 10.3765(940.7) | 0.5723(940.8) | 10.2353(950.1) | 0.5655(940.5) | 10.1928(950.3) |
(c) | 0.5817(940.9) | 10.3657(950.3) | 0.5590(950.2) | 10.3001(950.6) | 0.5530(950.2) | 10.2922(950.2) |
(d) | 0.5909(940.7) | 10.3323(950.0) | 0.5700(940.8) | 10.2307(950.1) | 0.5641(940.5) | 10.1896(950.3) |
16 | (a) | 0.6122(940.0) | 10.3998(950.5) | 0.6096(940.1) | 10.3694(950.5) | 0.6096(940.1) | 10.3694(950.5) |
(b) | 0.6623(940.8) | 10.6005(950.7) | 0.6275(940.5) | 10.4725(950.4) | 0.6133(940.5) | 10.4249(950.5) |
(c) | 0.6082(950.1) | 10.4458(940.6) | 0.5977(950.1) | 10.3503(950.2) | 0.5967(950.2) | 10.3503(950.2) |
(d) | 0.6527(940.7) | 10.5460(950.4) | 0.6194(940.7) | 10.4641(950.5) | 0.6090(940.6) | 10.4145(950.6) |
14 | (a) | 0.6630(940.8) | 10.6396(950.7) | 0.6625(940.8) | 10.6396(950.7) | 0.6625(940.8) | 10.6396(950.7) |
(b) | 0.7425(950.0) | 10.8444(950.4) | 0.7013(950.0) | 10.6625(950.2) | 0.6794(950.3) | 10.6101(950.1) |
(c) | 0.6686(940.5) | 10.5124(950.1) | 0.6613(930.7) | 10.5683(940.9) | 0.6609(930.8) | 10.5634(950.0) |
(d) | 0.7261(940.9) | 10.7664(950.2) | 0.6931(940.7) | 10.6427(950.0) | 0.6777(950.3) | 10.6064(950.1) |
12 | (a) | 0.7125(940.5) | 10.6972(960.1) | 0.7125(940.5) | 10.6972(960.1) | 0.7125(940.5) | 10.6972(960.1) |
(b) | 0.8725(940.8) | 20.2903(940.6) | 0.8275(950.0) | 20.1114(940.6) | 0.8001(940.8) | 20.0659(940.4) |
(c) | 0.7447(940.5) | 10.9367(960.0) | 0.7434(940.6) | 10.9358(960.0) | 0.7435(940.6) | 10.9347(960.0) |
(d) | 0.8440(940.8) | 20.1976(940.7) | 0.8108(940.9) | 20.1065(940.6) | 0.7982(940.8) | 20.0598(940.6) |
30 | 0.3 | 28 | (a) | 0.4304(950.5) | 0.8608(950.7) | 0.4204(950.7) | 0.8390(950.6) | 0.4184(950.6) | 0.8316(950.6) |
(b) | 0.4521(940.6) | 0.9736(950.7) | 0.4355(940.2) | 0.9288(950.2) | 0.4280(940.6) | 0.8769(950.2) |
(c) | 0.4407(940.0) | 0.9051(950.0) | 0.4254(940.1) | 0.8676(940.5) | 0.4202(940.0) | 0.8563(940.7) |
(d) | 0.4506(940.4) | 0.9705(950.4) | 0.4346(940.3) | 0.9270(950.4) | 0.4276(940.7) | 0.8762(950.3) |
26 | (a) | 0.4395(950.5) | 0.8943(950.2) | 0.4364(950.4) | 0.8895(950.2) | 0.4362(950.4) | 0.8893(950.3) |
(b) | 0.4709(950.6) | 0.9399(950.9) | 0.4534(950.4) | 0.9007(950.6) | 0.4467(950.4) | 0.8824(950.7) |
(c) | 0.4540(940.5) | 0.9772(960.1) | 0.4439(940.2) | 0.9527(950.6) | 0.4426(940.2) | 0.9483(950.7) |
(d) | 0.4664(950.5) | 0.9320(950.8) | 0.4521(950.5) | 0.8975(950.8) | 0.4460(950.3) | 0.8802(950.7) |
24 | (a) | 0.4596(950.7) | 0.9419(940.7) | 0.4590(950.8) | 0.9410(940.8) | 0.4590(950.8) | 0.9410(940.8) |
(b) | 0.4903(940.1) | 10.0400(940.7) | 0.4715(940.7) | 0.9919(940.7) | 0.4639(940.7) | 0.9605(940.9) |
(c) | 0.4686(940.7) | 0.9386(940.7) | 0.4628(940.7) | 0.9252(940.6) | 0.4623(940.7) | 0.9245(940.5) |
(d) | 0.4851(940.2) | 10.0230(940.8) | 0.4688(940.6) | 0.9837(940.8) | 0.4624(940.6) | 0.9580(940.9) |
22 | (a) | 0.4782(950.7) | 10.0145(950.1) | 0.4782(950.7) | 10.0145(950.1) | 0.4782(950.7) | 10.0145(950.1) |
(b) | 0.5275(940.0) | 10.0762(940.1) | 0.5006(930.8) | 10.0271(940.8) | 0.4932(930.8) | 10.0124(940.4) |
(c) | 0.4939(960.2) | 0.9895(950.9) | 0.4922(960.2) | 0.9874(950.8) | 0.4922(960.2) | 0.9875(950.8) |
(d) | 0.5174(930.9) | 10.0478(940.6) | 0.4984(930.8) | 10.0232(940.5) | 0.4912(930.9) | 10.0082(940.5) |
40 | 10.0 | 38 | (a) | 0.3606(940.9) | 0.7008(940.2) | 0.3526(950.0) | 0.6792(940.0) | 0.3505(940.9) | 0.6749(940.0) |
(b) | 0.3732(950.0) | 0.7115(930.8) | 0.3603(940.9) | 0.6882(940.2) | 0.3549(940.9) | 0.6772(940.3) |
(c) | 0.3639(960.0) | 0.7232(930.8) | 0.3530(940.7) | 0.6928(940.6) | 0.3494(940.7) | 0.6827(940.5) |
(d) | 0.3719(940.6) | 0.7081(930.9) | 0.3595(940.6) | 0.6866(940.1) | 0.3547(940.8) | 0.6767(940.2) |
36 | (a) | 0.3650(940.1) | 0.7096(930.6) | 0.3614(940.0) | 0.7013(930.5) | 0.3614(930.9) | 0.7008(930.4) |
(b) | 0.3836(940.8) | 0.7556(930.9) | 0.3718(940.5) | 0.7264(930.6) | 0.3678(940.7) | 0.7106(930.7) |
(c) | 0.3736(940.8) | 0.7134(930.8) | 0.3650(950.1) | 0.6954(930.7) | 0.3633(940.8) | 0.6912(930.8) |
(d) | 0.3813(950.0) | 0.7530(930.3) | 0.3712(940.6) | 0.7229(930.9) | 0.3674(940.8) | 0.7099(930.8) |
34 | (a) | 0.3739(940.5) | 0.7304(940.4) | 0.3729(940.7) | 0.7288(940.6) | 0.3729(940.7) | 0.7288(940.6) |
(b) | 0.3957(940.2) | 0.8007(940.8) | 0.3824(950.3) | 0.7595(950.1) | 0.3777(940.8) | 0.7454(950.0) |
(c) | 0.3799(950.6) | 0.7376(940.4) | 0.3729(960.1) | 0.7271(940.5) | 0.3723(960.0) | 0.7252(940.6) |
(d) | 0.3913(940.4) | 0.7893(940.6) | 0.3819(950.3) | 0.7574(950.0) | 0.3772(940.9) | 0.7453(950.1) |
32 | (a) | 0.3857(940.7) | 0.7571(940.3) | 0.3856(940.7) | 0.7571(940.3) | 0.3856(940.7) | 0.7571(940.3) |
(b) | 0.4137(940.3) | 0.7976(940.4) | 0.3984(940.5) | 0.7684(940.3) | 0.3937(940.3) | 0.7555(940.6) |
(c) | 0.3890(950.1) | 0.7676(930.9) | 0.3859(940.9) | 0.7573(940.0) | 0.3857(950.0) | 0.7567(940.0) |
(d) | 0.4090(940.2) | 0.7901(940.3) | 0.3969(940.3) | 0.7630(940.5) | 0.3929(940.2) | 0.7543(940.6) |