Tablae 2 Bias’s, MSE’s, AIC’s and ACL’s for MLE’s for \(NBS(\alpha ,\beta )\).

\(\left({\alpha }_{N},{\beta }_{N}\right)=\left(\mathrm{1.25,3}\right)\), \({I}_{N}=\left(\mathrm{0.2,0.5}\right)\)

 50

(0.0093, 0.0153)

(0.0044, 0.0118)

(0.0235, 0.0295)

(0.0278, 0.0437)

(9922.846, 9945.161)

(0.1651, 0.1652)

(0.5118, 0.5128)

 100

(0.0038, 0.0068)

(0.0014, 0.0046)

(0.0115, 0.0145)

(0.0133, 0.0211)

(32,341.18, 32,385.81)

(0.1556, 0.1562)

(0.3754, 0.3767)

 200

(0.0020, 0.0035)

(0.0008, 0.0025)

(0.0059, 0.0074)

(0.0070, 0.0110)

(131,945.3, 132,034.5)

(0.0005, 0.0007)

(0.2492, 0.2653)

 500

(0.0008, 0.0014)

(0.0003, 0.0010)

(0.0024, 0.0030)

(0.0030, 0.0047)

(825,725.8, 825,949)

(0.0587, 0.0590)

(0.14831, 0.1524)

\(\left({\alpha }_{N},{\beta }_{N}\right)=\left(\mathrm{0.5,3}\right)\), \({I}_{N}=\left(\mathrm{0.2,0.5}\right)\)

 50

(− 0.0060 \(,\) − 2.8*\({10}^{-5})\)

(4.09*\({10}^{-8}\), 0.0006 \()\)

(0.0408, 0.0468)

(0.0834, 0.1097)

(8939.838, 8962.153)

(0.0740, 0.0743)

(0.1782, 0.1792)

 100

(− 0.0003, − 0.0001)

(2.6*\({10}^{-6}\), 0.0002)

(0.0189, 0.0219)

(0.0358, 0.0481)

(34,665.09, 34,709.72)

(0.0710, 0.0713)

(0.0162, 0.0163)

 200

(− 0.0002, − 0.0017)

(9.01*\({10}^{-6},\) 0.0001)

(0.0100, 0.0115)

(0.0201, 0.0266)

(128,668.4, 128,757.6)

(0.0450, 0.0452)

(0.0952, 0.0958)

 500

(− 7.8*\({10}^{-5},\) − 0.0006)

(3.08*\({10}^{-6},\) 0.0001)

(0.0040, 0.0046)

(0.0080, 0.0106)

(809,600.2, 809,823.4)

(0.0280, 0.0282)

(0.0613, 0.0614)

\(\left({\alpha }_{N},{\beta }_{N}\right)=\left(\mathrm{1,3}\right)\), \({I}_{N}=\left(\mathrm{0.2,0.5}\right)\)

 50

(0.0038, 0.0098)

(0.0007, 0.0048)

(0.0328, 0.0388)

(0.0538, 0.07534)

(8703.788, 8726.102)

(0.1338, 0.1339)

(0.3367, 0.3378)

 100

(0.0019, 0.0049)

(0.0003, 0.0024)

(0.0154, 0.0184)

(0.0239, 0.0341)

(36,033.54, 36,078.17)

(0.1156, 0.01159)

(0.2558, 0.2569)

 200

(0.0009, 0.0024)

(0.0001, 0.0011)

(0.0077, 0.0062)

(0.0118, 0.0169)

(139,354.3, 139,443.5)

(0.0740, 0.0742)

(0.1816, 0.1845)

 500

(0.0009, 0.0003)

(5.45*\({10}^{-5}\), 0.0004)

(0.0030, 0.0036)

(0.0045, 0.0065)

(811,728.7, 811,951.8)

(0.0512, 0.0514)

(0.1241, 0.1245)

\(\left({\alpha }_{N},{\beta }_{N}\right)=\left(\mathrm{1.25,3}\right)\), \({I}_{N}=\left(\mathrm{0.6,0.8}\right)\)

 50

(− 0.0001, 0.0038)

(8.32*\({10}^{-9},\) 0.0007)

(0.0154, 0.0190)

(0.0120, 0.0190)

(5667.594, 5679.373)

(0.0819, 0.0823)

(0.4594, 0.4646)

 100

(0.0009, 0.0029)

(8.46*\({10}^{-5},\) 0.0008)

(0.0101, 0.0121)

(0.0103, 0.0148)

(30,906.83, 30,930.38)

(0.0395, 0.0396)

(0.3249, 0.3290)

 200

(0.0005, 0.0006)

(5.15*\({10}^{-5},\) 0.0004)

(0.0044, 0.0054)

(0.0040, 0.0060)

(122,478.9, 122,526)

(0.0017, 0.0022)

(0.2528, 0.2529)

 500

(0.0002, 0.0006)

(2.84*\({10}^{-5},\) 0.0002)

(0.0018, 0.0022)

(0.0017, 0.0025)

(844,985.4, 845,573.2)

(0.0401, 0.0410)

(0.1425, 0.1513)

\(\left({\alpha }_{N},{\beta }_{N}\right)=\left(\mathrm{0.5,3}\right)\), \({I}_{N}=\left(\mathrm{0.6,0.8}\right)\)

 50

(− 1.0131, − 0.0091)

(0.0041, 0.0086)

(0.0341, 0.0381)

(0.0584, 0.0729)

(7632.389, 7644.168)

(0.0640, 0.0641)

(0.1620, 0.1622)

 100

(− 0.0063, − 0.0043)

(0.0019, 0.0040)

(0.0168, 0.0188)

(0.0284, 0.0355)

(32,526.3, 32,549.85)

(0.0656, 0.0658)

(0.1423, 0.1426)

 200

(− 0.0031, − 0.0021)

(0.0009, 0.0019)

(0.0082, 0.0092)

(0.0137, 0.0172)

(134,481.5, 134,528.6)

(0.0422, 0.0433)

(0.0051, 0.0054)

 500

(− 0.0012, − 0.0008)

(0.0003, 0.0007)

(0.0033, 0.0037)

(0.0057, 0.0071)

(840,294.8, 840,412.6)

(0.0281, 0.0282)

(0.0523, 0.0532)

\(\left({\alpha }_{N},{\beta }_{N}\right)=\left(\mathrm{1,3}\right)\), \({I}_{N}=\left(\mathrm{0.6,0.8}\right)\)

 50

(− 0.0025, 0.0014)

(0.0001, 0.0033)

(0.0255, 0.0295)

(0.0325, 0.0435)

(8349.03, 8360.808)

(0.1208, 0.1209)

(0.3618, 0.3647)

 100

(− 0.0019, 6.38*\({10}^{-5})\)

(4.07*\({10}^{-7},\) 0.0003)

(0.0126, 0.0146)

(0.0159, 0.0214)

(27,309.47, 27,333.03)

(0.1107, 0.1109)

(0.2479, 0.2499)

 200

(− 0.0007, 0.0002)

(1.60*\({10}^{-5},\) 0.0001)

(0.0060, 0.0070)

(0.0073, 0.0099

(129,253.6, 129,300.7)

(0.0630, 0.0632)

(0.1808, 0.1824)

 500

(− 0.0002, 0.0001)

(6.08*\({10}^{-6},\) 4.19*\({10}^{-5})\)

(0.0024, 0.0028)

(0.0030, 0.0041)

(800,043, 800,160.8)

(0.0513, 0.0515)

(0.1178, 0.1244)