A Supplement to Improved Likelihood Inferences for Weibull Regression Model by Yan Shen and Zhenlin Yang

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1 A Supplemet to Improved Likelihood Ifereces for Weibull Regressio Model by Ya She ad Zheli Yag More simulatio experimets were carried out to ivestigate the effect of differet cesorig percetages o the performace of the origial MLE ad the proposed methods. The experimets are desiged for Type-I cesored data, which are a special type of radomly cesored data but with a fixed cesorig time. Thus the cesorig percetage ca be cotrolled i simulatio. We cosider the same Weibull regressio model with a itercept as i the mauscript: log T = a 1 + a 2 X 2 + a 3 X 3 + Z/β. For all the Mote Carlo experimets, a = (a 1, a 2, a 3 ) is set at {5, 1, 1} or {5, 0, 1}, β takes values {0.5, 0.8, 1, 2, 5}, ad takes values {12, 20, 50}. The two covariates are geerated idepedetly, accordig to {x i2 } iid N(0, 1)/ 2 ad {x i3 } iid N(0, 1)/ 2. The cesorig percetage (cp) is set as {10%, 20%, 30%} respectively, ad the correspodig cesorig times for each β are obtaied by geeratig radom umbers from the model ad takig the upper 10%, 20%, 30% quatiles, which are summarized i the Table 1. Table 1. The fixed cesorig times β a = (5, 1, 1) a = (5, 0, 1) 10% 20% 30% 10% 20% 30% Performace of the 2d-order bias corrected MLEs Tables 2-4 summarize the empirical mea, root-mea-square-error (rmse) ad stadard error (se) of the origial ad 2d-order bias-corrected MLEs for the cesorig percetages of 10%, 20%, 30% respectively. The results i the three tables show that, (i) for the shaper parameter β, the 2di

2 order bias-corrected MLE ˆβ bc2 is able to greatly reduce the bias of the origial MLE ˆβ regardless of the cesorig percetage, ad also has smaller rmse s ad se s; (ii) although the improvemets of â bc2 i over â i, i = 1, 2, 3, are ot so sigificat as that of ˆβ bc2, â bc2 i is still geerally better tha â i with respect to the bias. Hece the 2dorder bias-corrected MLE parameter estimatio. ˆθ bc2 is much superior to the origial MLE ˆθ i terms of There are some other iterestig observatios. For example, whe the cesorig percetage icreases, the bias of ˆβ also icreases, but the other estimators, ˆβ bc2, â i, â bc2 i, behave i a irregular way. Moreover, the estimators â i ad â bc2 i with β < 1 are more sesitive to the chage of the cesorig percetage tha the estimators with β > 1. The differet performaces for β < 1 ad β > 1 also ca be foud i the followig simulatio results. 2. Performace of the sigificace tests To compare the performaces of the two t-ratios t ad t bc2, we set the covaraite coefficiets to a = (a 1, a 2, a 3 ) = (5, 0, 1) ad test H 0 : a 2 = 0. Table 5 reports the empirical sigificace levels of t ad t bc2 for the cesorig percetages of 10%, 20%, 30%. The variace estimate used i t bc2 J 1 (ˆθ bc2 ), which is the same settig as i the mauscript. is the iverse of the observed iformatio matrix, Based o the results i the tables, the mai observatio is that the test t bc2 could offer huge reductio i sigificace level distortios give by t, ad have the empirical levels gettig close to their omial levels faster tha t. At the same time, two exceptioal cases with = 12, β = 0.5 ad = 12, β = 1 are foud i Table 5 for the cesorig percetage 30%. Their empirical sigificace levels idicate that whe cesorig percetage is high, t has a better performace tha t bc2 for the sceario of small sample size ad β < 1. We ca fid that for β < 1 ad = 12, the empirical sigificace levels decrease rather fast with the icrease of the cesorig percetage, which agai implies the sesitiveess of the estimators for β < 1. I geeral, the t bc2 based o the 2d-order bias-corrected MLE outperforms the asymptotic test t greatly. If the effective data is few (i.g. small sample size ad high cesorig percetage) ad β < 1, t is a better choice. ii

3 3. Cofidece itervals for the shape parameter Tables 6 summarizes the simulatios results for the cofidece iterval for the shape parameter β. The 2d-order variace estimate V2 (ˆθ bc2 ) is used for the costructio of cofidece iterval. The results show that, (i) for cesorig percetage 10%, 1 γ = 0.90, 0.95 ad cesorig percetage 20%, 1 γ = 0.90, CI 2 (β) with V2 (ˆθ bc2 ) has the coverage probabilities much closer to the omial levels compared to CI 1 (β); (ii) for almost all cesored situatios with 1 γ = 0.99, CI 1 (β) has better performaces i terms of both coverage probability ad iterval legth; (iii) for other cesored cases, CI 2 (β) has the advatage i costructig cofidece itervals for β > 1 while CI 1 (β) is preferred whe β < 1. I overall, the two CIs are comparable to each other. 4. Cofidece itervals for percetiles with certai covariates We also cosider the cofidece itervals for 50% percetile with give covariates x reg = (1, 0, 0), that is y p = 5 + z p /β or T p = exp(y p ) with the probability p = 0.5. The results are give i Tables 7. It is show that, (i) for the cesorig percetages 10%, 20%, the cofidece iterval based o the improved t-ratio, CI 2 (y p ), has a overwhelmig superior performace compared to the regular CI 1 (y p ), with the coverage probabilities much closer to the omial levels; (ii) the thigs differ whe the cesorig percetage is 30%, as CI 1 (y p ) outperforms CI 2 (y p ) i terms of coverage probability for β < 1. From the simulatio results, we fid agai that the chage of cesorig percetage has differet effects o the performace of the ifereces based o either the origial or the bias-corrected MLEs. Coclusio Based o all the simulatio results for Type-I cesorig, it ca be cocluded that the 2d-order bias-corrected MLE ad the improved ifereces are geerally better tha the origial MLE ad the asymptotic ifereces. Of great iterest is the the differet effects of cesorig percetage o the performaces of the two types of ifereces. It is foud that whe cesorig percetage is iii

4 high, the improved ifereces still have a rather satisfactory performace for β > 1, but for β < 1, the asymptotic ifereces based o the origial MLE would become better. iv

5 Table 2. Empirical mea [rmse](se) of the estimators of all parameters, Type-I cesored data with differet cesorig percetages: = 12 β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.2445](.2045) [.1745](.1724) 10% [.7792](.7789) [.8341](.8335) % [.2597](.2200) [.1852](.1839) % [.7668](.7668) [.7902](.7902) 30% [.2987](.2549) [.2071](.2059) 30% [.8639](.8625) [.8460](.8461) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [1.607](1.552) [1.819](1.794) 10% [1.294](1.227) [1.450](1.409) % [1.426](1.347) [1.572](1.519) % [1.139](1.046) [1.180](1.100) 30% [1.354](1.239) [1.387](1.289) 30% [1.374](1.257) [1.408](1.314) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.4026](.3345) [.2841](.2797) 10% [.4076](.4070) [.4065](.4064) % [.4367](.3700) [.3085](.3061) % [.4594](.4594) [.4536](.4535) 30% [.4832](.4145) [.3331](.3315) 30% [.5298](.5287) [.5153](.5151) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.6673](.6516) [.6853](.6723) 10% [.6980](.6769) [.7105](.6930) % [.8037](.7783) [.8158](.7970) % [.7990](.7698) [.8281](.8018) 30% [.8772](.8415) [.8972](.8665) 30% [.9274](.8864) [.9410](.9061) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.5036](.4153) [.3528](.3469) 10% [.3378](.3362) [.3391](.3385) % [.5117](.4290) [.3589](.3558) % [.3824](.3824) [.3767](.3767) 30% [.5936](.5048) [.4032](.4013) 30% [.4457](.4448) [.4362](.4360) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.5268](.5207) [.5493](.5430) 10% [.5829](.5710) [.6037](.5911) % [.6637](.6465) [.6778](.6634) % [.6169](.6040) [.6275](.6162) 30% [.7259](.7069) [.7495](.7340) 30% [.7277](.7079) [.7584](.7409) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [1.072](.8762) [.7481](.7302) 10% [.1827](.1818) [.1870](.1866) % [1.239](1.025) [.8330](.8173) % [.2126](.2124) [.2204](.2203) 30% [1.289](1.073) [.8339](.8244) 30% [.2380](.2380) [.2386](.2387) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.3158](.3144) [.3272](.3261) 10% [.3376](.3352) [.3439](.3418) % [.3295](.3267) [.3593](.3568) % [.3396](.3371) [.3454](.3438) 30% [.3850](.3786) [.3963](.3919) 30% [.4151](.4091) [.4289](.4246) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [2.877](2.316) [1.943](1.886) 10% [.0763](.0754) [.0785](.0779) % [3.113](2.513) [2.057](2.004) % [.0854](.0849) [.0870](.0866) 30% [3.653](2.934) [2.232](2.179) 30% [.1030](.1027) [.1045](.1042) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.1243](.1239) [.1292](.1288) 10% [.1258](.1253) [.1314](.1308) % [.1525](.1517) [.1607](.1600) % [.1383](.1378) [.1432](.1428) 30% [.1630](.1624) [.1709](.1703) 30% [.1655](.1647) [.1720](.1710) v

6 Table 3. Empirical mea [rmse](se) of the estimators of all parameters, Type-I cesored data with differet cesorig percetages: = 20 β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.1469](.1285) [.1173](.1164) 10% [.5260](.5243) [.5296](.5293) % [.1551](.1376) [.1246](.1240) % [.5582](.5575) [.5548](.5542) 30% [.1721](.1543) [.1382](.1378) 30% [.6457](.6458) [.6381](.6380) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.8171](.7922) [.8480](.8306) 10% [.7721](.7424) [.7884](.7628) % [.9831](.9366) [.9973](.9661) % [.8820](.8382) [.8758](.8409) 30% [.9791](.9446) [1.031](1.009) 30% [.8095](.7737) [.8124](.7830) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.2343](.2030) [.1851](.1833) 10% [.3266](.3248) [.3229](.3226) % [.2492](.2196) [.1991](.1980) % [.3435](.3430) [.3379](.3378) 30% [.2739](.2457) [.2188](.2184) 30% [.3982](.3981) [.3875](.3875) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.4977](.4929) [.4990](.4949) 10% [.5024](.4978) [.5025](.4991) % [.5455](.5384) [.5397](.5355) % [.5277](.5209) [.5273](.5221) 30% [.6196](.6066) [.6158](.6074) 30% [.5997](.5889) [.5935](.5869) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.2875](.2487) [.2267](.2248) 10% [.2630](.2612) [.2601](.2597) % [.3021](.2657) [.2395](.2386) % [.2832](.2828) [.2789](.2788) 30% [.3564](.3148) [.2773](.2757) 30% [.3299](.3299) [.3227](.3226) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.4215](.4194) [.4233](.4219) 10% [.4129](.4107) [.4143](.4130) % [.4504](.4466) [.4473](.4454) % [.4235](.4203) [.4227](.4209) 30% [.5193](.5127) [.5146](.5110) 30% [.5156](.5093) [.5102](.5069) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.5884](.5073) [.4607](.4561) 10% [.1359](.1351) [.1353](.1352) % [.6497](.5596) [.5047](.4990) % [.1467](.1461) [.1460](.1458) 30% [.7612](.6624) [.5781](.5732) 30% [.1686](.1684) [.1663](.1662) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.2176](.2169) [.2196](.2190) 10% [.2137](.2132) [.2168](.2164) % [.2403](.2393) [.2427](.2418) % [.2493](.2477) [.2515](.2507) 30% [.2918](.2894) [.2925](.2910) 30% [.2864](.2840) [.2874](.2860) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [1.573](1.351) [1.228](1.211) 10% [.0562](.0555) [.0558](.0556) % [1.800](1.520) [1.349](1.325) % [.0633](.0628) [.0633](.0631) 30% [1.963](1.661) [1.453](1.426) 30% [.0708](.0705) [.0707](.0707) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.0908](.0906) [.0919](.0916) 10% [.0891](.0890) [.0903](.0902) % [.1070](.1064) [.1087](.1082) % [.1055](.1052) [.1067](.1064) 30% [.1184](.1179) [.1209](.1204) 30% [.1123](.1116) [.1142](.1134) vi

7 Table 4. Empirical mea [rmse](se) of the estimators of all parameters, Type-I cesored data with differet cesorig percetages: = 50 β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.0762](.0720) [.0701](.0700) 10% [.3860](.3845) [.3838](.3835) % [.0770](.0735) [.0706](.0705) % [.3317](.3316) [.3284](.3284) 30% [.0834](.0800) [.0766](.0766) 30% [.3647](.3648) [.3595](.3596) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.4393](.4381) [.4357](.4352) 10% [.4552](.4530) [.4519](.4506) % [.4737](.4715) [.4685](.4675) % [.4866](.4829) [.4811](.4790) 30% [.5184](.5146) [.5106](.5087) 30% [.5246](.5208) [.5162](.5143) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.1170](.1094) [.1053](.1051) 10% [.2998](.2988) [.2990](.2987) % [.1247](.1174) [.1126](.1125) % [.2143](.2140) [.2126](.2125) 30% [.1349](.1280) [.1224](.1224) 30% [.2330](.2329) [.2305](.2305) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.3030](.3025) [.3013](.3012) 10% [.3059](.3053) [.3043](.3040) % [.3248](.3240) [.3221](.3220) % [.3198](.3190) [.3174](.3172) 30% [.3500](.3489) [.3469](.3466) 30% [.3433](.3417) [.3404](.3398) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.1434](.1345) [.1288](.1287) 10% [.2774](.2760) [.2766](.2760) % [.1533](.1443) [.1382](.1381) % [.1695](.1693) [.1682](.1682) 30% [.1683](.1590) [.1520](.1519) 30% [.1876](.1875) [.1856](.1856) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.2334](.2330) [.2330](.2329) 10% [.2311](.2307) [.2303](.2303) % [.2628](.2621) [.2615](.2614) % [.2619](.2610) [.2602](.2600) 30% [.2969](.2962) [.2951](.2950) 30% [.2836](.2825) [.2812](.2809) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.3040](.2892) [.2764](.2764) 10% [.2404](.2395) [.2402](.2397) % [.3147](.2944) [.2816](.2812) % [.0889](.0886) [.0885](.0884) 30% [.3376](.3188) [.3027](.3026) 30% [.1002](.1000) [.0998](.0997) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.1261](.1259) [.1259](.1259) 10% [.1227](.1225) [.1225](.1224) % [.1391](.1388) [.1388](.1388) % [.1362](.1359) [.1362](.1361) 30% [.1508](.1505) [.1508](.1507) 30% [.1552](.1549) [.1549](.1549) β cp ˆβ ˆβbc2 a 1 cp â 1 â bc2 1 10% [.8098](.7755) [.7398](.7398) 10% [.2286](.2281) [.2285](.2282) % [.8462](.7945) [.7546](.7541) % [.0376](.0375) [.0375](.0375) 30% [.8922](.8340) [.7871](.7866) 30% [.0428](.0428) [.0429](.0429) a 2 cp â 2 â bc2 2 a 3 cp â 3 â bc2 3 10% [.0505](.0504) [.0506](.0506) 10% [.0505](.0505) [.0507](.0507) % [.0598](.0596) [.0600](.0599) % [.0577](.0576) [.0578](.0578) 30% [.0700](.0698) [.0704](.0702) 30% [.0659](.0658) [.0662](.0661) vii

8 Table 5. Empirical sigificace levels: two-sided tests of H 0 : a 2 = 0, Type-I cesored data with differet cesorig percetages β cp Test 10% 5% 1% 10% 5% 1% 10% 5% 1% = 12 = 20 = 50 10% (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) % (1) (2) viii

9 Table 6. Empirical coverage probability (average legth) of cofidece itervals for β, Type-I cesored data with differet cesorig percetages: ˆβ with variace J 1 (ˆθ ), ˆβ bc2 with variace V 2 (ˆθ bc2) β 0 cp ˆβ 1 γ = γ = γ = 0.99 ˆβ ˆβbc2 ˆβ ˆβbc2 = % (0.5249) (0.5375) (0.6255) (0.6405) (0.8220) (0.8417) 20% (0.5699) (0.5680) (0.6791) (0.6768) (0.8925) (0.8894) 30% (0.6430) (0.6175) (0.7661) (0.7358) (1.0069) (0.9670) 10% (0.8499) (0.8707) (1.0127) (1.0375) (1.3309) (1.3635) % (0.9078) (0.9179) (1.0817) (1.0937) (1.4215) (1.4374) 30% (1.0587) (1.0320) (1.2616) (1.2297) (1.6580) (1.6161) 10% (1.0616) (1.0917) (1.2649) (1.3009) (1.6624) (1.7096) % (1.1443) (1.1596) (1.3636) (1.3818) (1.7920) (1.8160) 30% (1.3049) (1.2744) (1.5549) (1.5186) (2.0434) (1.9957) 10% (2.1289) (2.2137) (2.5367) (2.6378) (3.3338) (3.4667) % (2.3318) (2.4162) (2.7785) (2.8790) (3.6516) (3.7837) 30% (2.6784) (2.7201) (3.1915) (3.2411) (4.1944) (4.2596) % (5.4360) (5.7231) (6.4774) (6.8195) (8.5128) (8.9623) 20% (6.1445) (6.4614) (7.3217) (7.6992) (9.6223) (10.119) 30% (7.5104) (7.7271) (8.9492) (9.2074) (11.761) (12.101) = % (0.3672) (0.3665) (0.4376) (0.4367) (0.5751) (0.5740) 20% (0.3979) (0.3915) (0.4741) (0.4665) (0.6231) (0.6131) 30% (0.4390) (0.4207) (0.5231) (0.5013) (0.6875) (0.6589) 10% (0.5874) (0.5877) (0.7000) (0.7003) (0.9199) (0.9204) % (0.6342) (0.6270) (0.7557) (0.7472) (0.9932) (0.9820) 30% (0.6972) (0.6768) (0.8307) (0.8065) (1.0918) (1.0599) 10% (0.7252) (0.7277) (0.8641) (0.8671) (1.1356) (1.1396) % (0.7857) (0.7804) (0.9362) (0.9298) (1.2304) (1.2220) 30% (0.8685) (0.8509) (1.0349) (1.0139) (1.3601) (1.3325) 10% (1.4361) (1.4525) (1.7112) (1.7307) (2.2489) (2.2746) % (1.5572) (1.5649) (1.8555) (1.8647) (2.4385) (2.4506) 30% (1.6962) (1.6985) (2.0212) (2.0239) (2.6563) (2.6598) % (3.6353) (3.6892) (4.3318) (4.3960) (5.6929) (5.7773) 20% (3.9881) (4.0669) (4.7521) (4.8460) (6.2453) (6.3687) 30% (4.4099) (4.5259) (5.2547) (5.3930) (6.9058) (7.0876) = % (0.2135) (0.2118) (0.2538) (0.2517) (0.3325) (0.3298) 20% (0.2312) (0.2279) (0.2755) (0.2716) (0.3621) (0.3570) 30% (0.2546) (0.2482) (0.3034) (0.2957) (0.3988) (0.3887) 10% (0.3363) (0.3333) (0.4001) (0.3965) (0.5249) (0.5201) % (0.3619) (0.3574) (0.4313) (0.4259) (0.5668) (0.5597) 30% (0.3975) (0.3895) (0.4736) (0.4641) (0.6224) (0.6099) 10% (0.4182) (0.4127) (0.4977) (0.4911) (0.6531) (0.6445) % (0.4493) (0.4426) (0.5354) (0.5273) (0.7036) (0.6931) 30% (0.4911) (0.4811) (0.5852) (0.5732) (0.7691) (0.7534) 10% (0.8174) (0.8041) (0.9734) (0.9575) (1.2782) (1.2574) % (0.8724) (0.8585) (1.0396) (1.0230) (1.3662) (1.3444) 30% (0.9443) (0.9286) (1.1252) (1.1065) (1.4787) (1.4542) % (2.0274) (2.0003) (2.4152) (2.3828) (3.1731) (3.1306) 20% (2.1581) (2.1306) (2.5715) (2.5388) (3.3796) (3.3365) 30% (2.3028) (2.2761) (2.7439) (2.7121) (3.6061) (3.5643) ix

10 Table 7. Empirical coverage probability (average legth) of cofidece itervals for y 0.5, Type-I cesored data with differet cesorig percetages: ŷ,0.5 with variace J 1 (ˆθ ), ŷ,0.5 bc2 2 (ˆθ bc2) β 0 cp ˆβ 1 γ = γ = γ = 0.99 ˆβ ˆβbc2 ˆβ ˆβbc2 = 12 10% (2.1291) (2.7500) (2.5370) (3.2768) (3.3341) (4.3065) % (2.2494) (3.0216) (2.6804) (3.6004) (3.5226) (4.7317) 30% (2.4396) (3.3913) (2.9070) (4.0410) (3.8204) (5.3108) 10% (1.2982) (1.6849) (1.5469) (2.0077) (2.0330) (2.6386) % (1.4281) (1.9354) (1.7017) (2.3062) (2.2364) (3.0309) 30% (1.6122) (2.2531) (1.9211) (2.6848) (2.5247) (3.5284) 10% (1.0598) (1.3780) (1.2629) (1.6420) (1.6597) (2.1579) % (1.1770) (1.5572) (1.4024) (1.8556) (1.8431) (2.4386) 30% (1.2659) (1.7831) (1.5084) (2.1247) (1.9823) (2.7923) 10% (0.5414) (1.3328) (0.6451) (1.5881) (0.8478) (2.0871) % (0.6048) (0.8630) (0.7206) (1.0284) (0.9471) (1.3515) 30% (0.6732) (0.9431) (0.8021) (1.1237) (1.0542) (1.4768) 10% (0.2127) (0.2858) (0.2535) (0.3405) (0.3331) (0.4475) % (0.2345) (0.3234) (0.2794) (0.3854) (0.3672) (0.5065) 30% (0.2681) (0.3930) (0.3195) (0.4683) (0.4199) (0.6155) = 20 10% (1.6871) (1.9167) (2.0097) (2.2833) (2.6402) (2.9997) % (1.7543) (2.0210) (2.0904) (2.4081) (2.7473) (3.1648) 30% (1.8863) (2.2182) (2.2477) (2.6431) (2.9539) (3.4736) 10% (1.0606) (1.2071) (1.2631) (1.4377) (1.6590) (1.8885) % (1.0852) (1.2434) (1.2931) (1.4817) (1.6994) (1.9472) 30% (1.2177) (1.4278) (1.4509) (1.7013) (1.9068) (2.2359) 10% (0.8450) (0.9548) (1.0063) (1.1371) (1.3214) (1.4935) % (0.8925) (1.0205) (1.0634) (1.2160) (1.3976) (1.5982) 30% (0.9504) (1.1079) (1.1325) (1.3202) (1.4883) (1.7350) 10% (0.4238) (0.4772) (0.5043) (0.5680) (0.6618) (0.7454) % (0.4508) (0.5124) (0.5372) (0.6106) (0.7060) (0.8024) 30% (0.5008) (0.5797) (0.5968) (0.6908) (0.7843) (0.9078) 10% (0.1738) (0.1958) (0.2065) (0.2327) (0.2704) (0.3048) % (0.1868) (0.2133) (0.2226) (0.2542) (0.2925) (0.3340) 30% (0.2151) (0.2499) (0.2564) (0.2978) (0.3369) (0.3914) = 50 10% (1.0833) (1.1390) (1.2903) (1.3566) (1.6947) (1.7819) % (1.1131) (1.1731) (1.3263) (1.3979) (1.7431) (1.8371) 30% (1.1645) (1.2348) (1.3876) (1.4713) (1.8236) (1.9336) 10% (0.6811) (0.7161) (0.8109) (0.8527) (1.0648) (1.1196) % (0.7048) (0.7422) (0.8398) (0.8843) (1.1037) (1.1622) 30% (0.7543) (0.7988) (0.8988) (0.9519) (1.1812) (1.2510) 10% (0.5474) (0.5758) (0.6517) (0.6856) (0.8555) (0.9000) % (0.5652) (0.5950) (0.6734) (0.7090) (0.8851) (0.9318) 30% (0.6100) (0.6454) (0.7269) (0.7691) (0.9553) (1.0107) 10% (0.2761) (0.2901) (0.3284) (0.3451) (0.4306) (0.4525) % (0.2907) (0.3050) (0.3464) (0.3634) (0.4553) (0.4776) 30% (0.3207) (0.3360) (0.3822) (0.4004) (0.5022) (0.5262) 10% (0.1155) (0.1210) (0.1370) (0.1436) (0.1791) (0.1877) % (0.1217) (0.1273) (0.1450) (0.1517) (0.1905) (0.1994) 30% (0.1376) (0.1431) (0.1640) (0.1706) (0.2156) (0.2242) Note: {4.2670, , , , } correspods to y 0.5 with {β = 0.5, 0.8, 1.0, 2.0, 5.0}. x