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Licensed Unlicensed Requires Authentication Published by De Gruyter April 8, 2017

Computing with bivariate COM-Poisson model under different copulas

  • Naushad Mamode Khan EMAIL logo , Wasseem Rumjaun , Yuvraj Sunecher and Vandna Jowaheer

Abstract

Bivariate counts are collected in many sectors of research but the analysis of such data is often challenging because each series of counts may exhibit different levels and types of dispersion. This paper addresses this problem by proposing a flexible bivariate COM-Poisson model that may handle any combination of over-, equi- and under-dispersion at any levels. In this paper, the bivariate COM-Poisson is developed via Archimedean copulas. The Generalized Quasi-Likelihood (GQL) approach is used to estimate the unknown mean parameters in the copula-based bivariate COM-Poisson model while the dependence parameter is estimated using the copula likelihood. We further introduce a Monte Carlo experiment to generate bivariate COM-Poisson data under different dispersion levels. The performance of the GQL approach is assessed on the simulated data. The model is applied to analyze real-life epileptic seizures data.

A Appendix

The composite likelihood equations are the following:

(Frank)i=1Ii=1Ilogf(yi[1],yi[2];θ)=i=1Ii=1Ilog[-1θln[1+(e-θF1(yi[1])-1)(e-θF2(yi[2])-1)e-θ-1]
+1θln[1+(e-θF1(yi[1]-1)-1)(e-θF2(yi[2])-1)e-θ-1]
+1θln[1+(e-θF1(yi[1])-1)(e-θF2(yi[2]-1)-1)e-θ-1]
-1θln[1+(e-θF1(yi[1]-1)-1)(e-θF2(yi[2]-1)-1)e-θ-1]],
(Clayton)i=1Ii=1Ilogf(yi[1],yi[2];θ)=i=1Ii=1Ilog[(F1(yi[1])-θ+F2(yi[2])-θ-1)-1/θ
-(F1(yi[1]-1)-θ+F2(yi[2])-θ-1)-1/θ
-(F1(yi[1])-θ+F2(yi[2]-1)-θ-1)-1/θ
+(F1(yi[1]-1)-θ+F2(yi[2]-1)-θ-1)-1/θ],
(Gumbel–Hougaard)i=1Ii=1Ilogf(yi[1],yi[2];θ)=i=1Ii=1Ilog[exp(-[(-lnF1(yi[1]))θ+(-lnF2(yi[2]))θ]1/θ)
-exp(-[(-lnF1(yi[1]-1))θ+(-lnF2(yi[2]))θ]1/θ)
-exp(-[(-lnF1(yi[1]))θ+(-lnF2(yi[2]-1))θ]1/θ)
+exp(-[(-lnF1(yi[1]-1))θ+(-lnF2(yi[2]-1))θ]1/θ)],
(AMH)i=1Ii=1Ilogf(yi[1],yi[2];θ)=i=1Ii=1Ilog[F1(yi[1])F2(yi[2])1-θ(1-F1(yi[1]))(1-F2(yi[2]))
-F1(yi[1]-1)F2(yi[2])1-θ(1-F1(yi[1]-1))(1-F2(yi[2]))
-F1(yi[1])F2(yi[2]-1)1-θ(1-F1(yi[1]))(1-F2(yi[2]-1))
+F1(yi[1]-1)F2(yi[2]-1)1-θ(1-F1(yi[1]-1))(1-F2(yi[2]-1))].

Table 2

Estimates of the parameters and standard errors (s.e.) under different copulas for theCOM-Poisson model for ν1=1, ν2=1 and θ=0.3.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98460.98571.98221.98410.98790.98710.2841
(0.0701)(0.0709)(0.0708)(0.0706)(0.0607)(0.0614)(0.0821)
C20.98620.98591.98751.98570.98510.98710.2830
(0.0608)(0.0682)(0.0632)(0.0613)(0.0600)(0.0610)(0.0805)
C30.98470.98911.98941.98470.98130.98890.2821
(0.0766)(0.0778)(0.0711)(0.0704)(0.0612)(0.0674)(0.0882)
C40.98560.98101.98261.98420.98290.98310.3117
(0.0754)(0.0752)(0.0726)(0.0789)(0.0610)(0.0627)(0.0857)
500C10.99730.99201.99571.99860.99010.99550.2984
(0.0428)(0.0451)(0.0462)(0.0493)(0.0411)(0.0490)(0.0645)
C20.99840.99051.99271.99560.99080.99520.2974
(0.0425)(0.0442)(0.0412)(0.0492)(0.0407)(0.0483)(0.0609)
C30.99960.99411.99181.99180.99010.99600.2931
(0.0499)(0.0527)(0.0508)(0.0551)(0.0549)(0.0495)(0.0687)
C40.99250.99161.99031.99300.99200.99110.2972
(0.0471)(0.0485)(0.0500)(0.0523)(0.0430)(0.0547)(0.0658)
900C10.99830.99561.99731.99960.99790.99750.2990
(0.0258)(0.0270)(0.0290)(0.0299)(0.0222)(0.0264)(0.0425)
C20.99980.99901.99481.99720.99720.99580.2988
(0.0217)(0.0263)(0.0225)(0.0292)(0.0211)(0.0260)(0.0411)
C30.99980.99681.99571.99770.99750.99730.2955
(0.0291)(0.0287)(0.0361)(0.0328)(0.0326)(0.0360)(0.0475)
C40.99300.99311.99071.99410.99490.99810.2984
(0.0246)(0.0311)(0.0242)(0.0381)(0.0317)(0.0369)(0.0470)
Table 3

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=0.5, ν2=0.9 and θ=0.3.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98770.98901.98921.98060.48210.88830.2896
(0.0702)(0.0701)(0.0730)(0.0728)(0.0718)(0.0701)(0.0838)
C20.98890.98131.98931.98430.48030.88440.2883
(0.0655)(0.0605)(0.0611)(0.0613)(0.0619)(0.0666)(0.0817)
C30.98970.98431.98541.98150.48350.88260.3167
(0.0707)(0.0747)(0.0739)(0.0764)(0.0785)(0.0709)(0.0878)
C40.98200.98191.98641.98950.48920.88500.2864
(0.0729)(0.0703)(0.0760)(0.0793)(0.0734)(0.0711)(0.0866)
500C10.99010.99271.99531.99900.49930.89510.2907
(0.0469)(0.0479)(0.0449)(0.0438)(0.0458)(0.0472)(0.0628)
C20.99050.99511.99171.99250.49540.89190.2956
(0.0457)(0.0473)(0.0412)(0.0401)(0.0408)(0.0467)(0.0620)
C30.99210.99031.99341.99050.49160.89520.2975
(0.0487)(0.0483)(0.0458)(0.0434)(0.0420)(0.0545)(0.0679)
C40.99080.99021.99301.99170.49350.89010.2992
(0.0549)(0.0489)(0.0501)(0.0504)(0.0502)(0.0475)(0.0660)
900C10.99130.99501.99541.99980.49970.89690.2931
(0.0259)(0.0227)(0.0272)(0.0219)(0.0270)(0.0288)(0.0431)
C20.99180.99691.99341.99470.49610.89560.2983
(0.0255)(0.0224)(0.0270)(0.0222)(0.0264)(0.0282)(0.0422)
C30.99650.99821.99351.99720.49240.89960.2985
(0.0301)(0.0325)(0.0380)(0.0311)(0.0323)(0.0366)(0.0486)
C40.99110.99071.99611.99260.49480.89770.2995
(0.0321)(0.0335)(0.0363)(0.0376)(0.0329)(0.0380)(0.0466)
Table 4

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=0.5, ν2=1.2 and θ=0.3.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98710.98271.98121.98710.48301.88460.2823
(0.0613)(0.0611)(0.0600)(0.0612)(0.0601)(0.0610)(0.0844)
C20.98380.98961.98151.98650.48731.18770.2887
(0.0502)(0.0535)(0.0525)(0.0511)(0.0528)(0.0518)(0.0830)
C30.98980.98851.98251.98090.48551.18100.2804
(0.0673)(0.0618)(0.0627)(0.0647)(0.0617)(0.0633)(0.0883)
C40.98630.98051.98751.98140.48291.18830.2830
(0.0658)(0.0608)(0.0601)(0.0696)(0.0686)(0.0608)(0.0880)
500C10.99060.99021.99571.99690.49081.19360.2932
(0.0508)(0.0504)(0.0515)(0.0513)(0.0512)(0.0510)(0.0616)
C20.99010.99131.99411.99690.49021.19270.2971
(0.0469)(0.0488)(0.0413)(0.0498)(0.0468)(0.0439)(0.0612)
C30.99020.99471.99871.99030.49251.19310.2927
(0.0524)(0.0585)(0.0581)(0.0509)(0.0576)(0.0577)(0.0696)
C40.99270.99071.99111.99310.49421.19120.2930
(0.0559)(0.0548)(0.0580)(0.0535)(0.0525)(0.0541)(0.0648)
900C10.99100.99351.99711.99700.49291.19400.2950
(0.0414)(0.0410)(0.0416)(0.0415)(0.0407)(0.0409)(0.0415)
C20.99620.99371.99851.99560.49581.19810.2983
(0.0325)(0.0324)(0.0383)(0.0392)(0.0328)(0.0344)(0.0409)
C30.99400.99881.99951.99800.49501.19380.2947
(0.0460)(0.0463)(0.0458)(0.0439)(0.0423)(0.0459)(0.0456)
C40.99890.99081.99131.99600.49691.19400.2944
(0.0407)(0.0454)(0.0473)(0.0435)(0.0489)(0.0404)(0.0463)
Table 5

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=0.9, ν2=0.9 and θ=0.3.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98100.98921.98191.98180.88460.88580.2854
(0.0604)(0.0608)(0.0611)(0.0616)(0.0610)(0.0612)(0.0834)
C20.98190.98901.98091.98530.88800.91210.2819
(0.0547)(0.0532)(0.0585)(0.0586)(0.0564)(0.0515)(0.0817)
C30.98410.98271.98841.98280.88770.91900.2890
(0.0683)(0.0665)(0.0679)(0.0644)(0.0663)(0.0610)(0.0873)
C40.98620.98731.98521.98100.88720.88730.2874
(0.0605)(0.0685)(0.0670)(0.0614)(0.0613)(0.0636)(0.0842)
500C10.99900.99581.99071.99670.89020.89400.2934
(0.0510)(0.0509)(0.0525)(0.0507)(0.0508)(0.0504)(0.0618)
C20.99140.99231.99181.99200.89240.89590.2957
(0.0496)(0.0406)(0.0405)(0.0439)(0.0443)(0.0447)(0.0608)
C30.99010.99271.99661.99770.89280.89900.2972
(0.0574)(0.0515)(0.0566)(0.0532)(0.0522)(0.0513)(0.0676)
C40.99070.99131.99081.99120.89550.89210.2918
(0.0513)(0.0539)(0.0524)(0.0597)(0.0590)(0.0511)(0.0641)
900C10.99300.99511.99921.99810.89620.89620.2961
(0.0405)(0.0414)(0.0415)(0.0418)(0.0403)(0.0419)(0.0409)
C20.99690.99561.99251.99640.89710.89850.2978
(0.0336)(0.0335)(0.0393)(0.0317)(0.0314)(0.0357)(0.0401)
C30.99570.99721.99861.99810.89350.90020.2984
(0.0482)(0.0442)(0.0459)(0.0477)(0.0453)(0.0489)(0.0453)
C40.99170.99211.99321.99380.90090.89550.2925
(0.0425)(0.0498)(0.0487)(0.0444)(0.0453)(0.0495)(0.0431)
Table 6

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=1.2, ν2=3 and θ=0.3

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98520.98621.98831.98341.18502.98980.2894
(0.0628)(0.0605)(0.0626)(0.0610)(0.0617)(0.0616)(0.0818)
C20.98870.98631.98701.98001.18892.98280.2853
(0.0571)(0.0505)(0.0534)(0.0551)(0.0562)(0.0519)(0.0810)
C30.98290.98661.98461.98061.18122.98100.2802
(0.0664)(0.0690)(0.0619)(0.0682)(0.0658)(0.0629)(0.0869)
C40.98140.98181.98711.98641.18162.98430.2872
(0.0681)(0.0640)(0.0607)(0.0626)(0.0654)(0.0624)(0.0860)
500C10.99160.99251.99111.99121.19512.99230.2977
(0.0511)(0.0501)(0.0508)(0.0509)(0.0514)(0.0504)(0.0629)
C20.99240.99851.99221.99151.19092.99150.2922
(0.0454)(0.0480)(0.0468)(0.0422)(0.0429)(0.0485)(0.0624)
C30.99090.99051.99791.99781.19882.99770.2932
(0.0562)(0.0567)(0.0542)(0.0570)(0.0504)(0.0560)(0.0677)
C40.99500.99121.99641.99801.19012.99100.2992
(0.0563)(0.0554)(0.0531)(0.0530)(0.0579)(0.0592)(0.0654)
900C10.99510.99591.99281.99271.19802.99640.2980
(0.0408)(0.0410)(0.0400)(0.0415)(0.0413)(0.0412)(0.0436)
C20.99300.99941.99311.99161.19262.99840.2935
(0.0331)(0.0302)(0.0385)(0.0369)(0.0335)(0.0352)(0.0430)
C30.99580.99701.99871.99891.19942.99900.2988
(0.0414)(0.0439)(0.0455)(0.0417)(0.0415)(0.0410)(0.0451)
C40.99970.99411.99981.99911.19052.99340.2995
(0.0449)(0.0410)(0.0499)(0.0434)(0.0423)(0.0482)(0.0457)
Table 7

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=1.2, ν2=0.5 and θ=0.3.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98520.98151.98281.98031.18060.48290.2871
(0.0615)(0.0612)(0.0617)(0.0619)(0.0615)(0.0612)(0.0841)
C20.98420.98311.98021.98101.18300.48990.2869
(0.0567)(0.0501)(0.0552)(0.0521)(0.0513)(0.0529)(0.0820)
C30.98900.98201.98431.98241.18080.48580.2824
(0.0693)(0.0629)(0.0689)(0.0603)(0.0640)(0.0686)(0.0866)
C40.98240.98701.98651.98331.18680.48170.2898
(0.0645)(0.0613)(0.0630)(0.0620)(0.0692)(0.0618)(0.0855)
500C10.99680.99201.99571.99361.19350.49600.2911
(0.0514)(0.0517)(0.0503)(0.0516)(0.0519)(0.0517)(0.0611)
C20.99170.99551.99611.99291.19910.49100.2914
(0.0475)(0.0408)(0.0414)(0.0408)(0.0447)(0.0481)(0.0605)
C30.99130.99181.99651.99771.19190.49830.2984
(0.0593)(0.0560)(0.0511)(0.0550)(0.0542)(0.0575)(0.0659)
C40.99510.99561.99111.99121.19100.49370.2970
(0.0543)(0.0524)(0.0513)(0.0558)(0.0554)(0.0565)(0.0635)
900C10.99790.99641.99701.99481.19920.49620.2950
(0.0412)(0.0416)(0.0410)(0.0419)(0.0401)(0.0404)(0.0444)
C20.99740.99611.99721.99631.19920.49840.2927
(0.0339)(0.0399)(0.0351)(0.0341)(0.0398)(0.0342)(0.0430)
C30.99800.99601.99741.99801.19610.49910.2990
(0.0441)(0.0428)(0.0440)(0.0490)(0.0446)(0.0469)(0.0491)
C40.99710.99571.99271.99241.19160.49650.2981
(0.0470)(0.0481)(0.0412)(0.0446)(0.0465)(0.0459)(0.0467)
Table 8

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=1, ν2=1 and θ=0.9.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98960.98381.98121.98130.98920.98080.8864
(0.0776)(0.0782)(0.0742)(0.0780)(0.0626)(0.06981)(0.0824)
C20.98630.98121.98601.98430.98340.98420.8861
(0.0641)(0.0633)(0.0625)(0.0639)(0.0620)(0.0670)(0.0816)
C30.98250.98181.98981.98220.98210.98010.8866
(0.0791)(0.0798)(0.0763)(0.0796)(0.0657)(0.0694)(0.0848)
C40.98780.98741.98751.98930.98110.98490.9150
(0.0780)(0.0790)(0.0755)(0.0789)(0.0641)(0.0689)(0.0839)
500C10.99150.99181.99421.99160.99600.99210.8919
(0.0444)(0.0425)(0.0450)(0.0447)(0.0422)(0.0480)(0.0685)
C20.99120.99171.99231.99200.99490.99130.8973
(0.0430)(0.0411)(0.0442)(0.0439)(0.0411)(0.0477)(0.0673)
C30.99140.99361.99851.99660.99790.99140.8927
(0.0467)(0.0515)(0.0526)(0.0530)(0.0516)(0.0520)(0.0697)
C40.99120.99191.99811.99250.99620.99150.8985
(0.0455)(0.0433)(0.0509)(0.0511)(0.0435)(0.0515)(0.0690)
900C10.99900.99431.99151.99250.99100.99610.8970
(0.0220)(0.0270)(0.0260)(0.0291)(0.0227)(0.0220)(0.0427)
C20.91790.99531.99491.99510.99320.99100.8977
(0.0217)(0.0263)(0.0250)(0.0282)(0.0218)(0.0209)(0.0412)
C30.99240.99091.99111.99660.99440.99930.8927
(0.0250)(0.0297)(0.0307)(0.0326)(0.0330)(0.0333)(0.0467)
C40.99530.99341.99331.99640.99700.99380.8961
(0.0229)(0.0307)(0.0275)(0.0312)(0.0315)(0.0321)(0.0445)
Table 9

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=1.2, ν2=3 and θ=0.3.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98950.98101.98621.98301.18262.98690.8880
(0.0622)(0.0646)(0.0614)(0.0685)(0.0632)(0.0655)(0.0828)
C20.98360.98741.98501.98251.18452.98100.8853
(0.0518)(0.0556)(0.0545)(0.0511)(0.0539)(0.0568)(0.0814)
C30.98980.98221.98971.98441.18802.98400.8870
(0.0651)(0.0667)(0.0644)(0.0696)(0.0661)(0.0687)(0.0850)
C40.98450.98531.98181.98471.18952.98750.8853
(0.0636)(0.0655)(0.0630)(0.0690)(0.0647)(0.0674)(0.0841)
500C10.99280.99661.99371.99231.19652.99160.8913
(0.0543)(0.0547)(0.0518)(0.0550)(0.0516)(0.0510)(0.0671)
C20.99900.99611.99581.99091.19922.99670.8939
(0.0475)(0.0403)(0.0415)(0.0435)(0.0481)(0.0419)(0.0618)
C30.99910.99591.99621.99651.19152.99950.8955
(0.0566)(0.0578)(0.0544)(0.0571)(0.0555)(0.0530)(0.0695)
C40.99310.99081.99181.99121.19292.99280.8903
(0.0557)(0.0561)(0.0530)(0.0560)(0.0533)(0.0525)(0.0683)
900C10.99110.99351.99081.99941.19612.99540.8952
(0.0478)(0.0416)(0.0417)(0.0457)(0.0437)(0.0423)(0.0440)
C20.99950.99051.99891.99781.19102.99170.8975
(0.0375)(0.0318)(0.0313)(0.0321)(0.0347)(0.0323)(0.0413)
C30.99570.99391.99361.99831.19282.99780.8961
(0.0495)(0.0440)(0.0447)(0.0475)(0.0470)(0.0450)(0.0466)
C40.99250.99361.99471.99501.19352.99540.8910
(0.0488)(0.0426)(0.0431)(0.0467)(0.0455)(0.0439)(0.0457)
Table 10

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=0.5, ν2=0.9 and θ=0.9.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98700.98161.98191.98700.48620.88690.8891
(0.0737)(0.0765)(0.0707)(0.0710)(0.0788)(0.0736)(0.0805)
C20.98770.98841.98091.98110.48050.88220.8815
(0.0622)(0.0651)(0.0671)(0.0625)(0.0621)(0.0661)(0.0801)
C30.98050.98011.98401.98540.48310.88380.9122
(0.0761)(0.0788)(0.0729)(0.0750)(0.0797)(0.0756)(0.0825)
C40.98930.98681.98801.98430.48220.88070.8810
(0.0755)(0.0776)(0.0720)(0.0735)(0.0792)(0.0747)(0.0816)
500C10.99930.99181.99131.99370.49200.89190.8912
(0.0429)(0.0458)(0.0444)(0.0450)(0.0475)(0.0430)(0.0643)
C20.99320.99631.99531.99420.49890.89490.8986
(0.0413)(0.0447)(0.0438)(0.0427)(0.0461)(0.0419)(0.0634)
C30.99870.99521.99251.99110.49320.89760.8921
(0.0517)(0.0478)(0.0587)(0.0520)(0.0530)(0.0515)(0.0675)
C40.99330.99221.99171.99260.49850.89750.8939
(0.0507)(0.0468)(0.0521)(0.0519)(0.0524)(0.0443)(0.0659)
900C10.99080.99631.99411.99250.49730.89410.8905
(0.0273)(0.0282)(0.0224)(0.0254)(0.0285)(0.0244)(0.0440)
C20.99750.99701.99311.99530.49860.89780.8953
(0.0261)(0.0274)(0.0211)(0.0240)(0.0275)(0.0237)(0.0431)
C30.99730.99941.99621.99790.49300.89310.8915
(0.0336)(0.0324)(0.0343)(0.0333)(0.0319)(0.0339)(0.0479)
C40.99520.99271.99581.99110.49840.89170.8963
(0.0317)(0.0305)(0.0331)(0.0319)(0.0327)(0.0322)(0.0464)
Table 11

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=0.5, ν2=1.2 and θ=0.3.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98190.98411.98511.98910.48121.88710.8849
(0.0676)(0.0612)(0.0631)(0.0611)(0.0650)(0.0638)(0.0837)
C20.98920.98101.98901.98460.48181.18200.8841
(0.0529)(0.0540)(0.0523)(0.0581)(0.0513)(0.0561)(0.0820)
C30.98450.98791.98251.98110.48781.18430.8884
(0.0694)(0.0640)(0.0663)(0.0637)(0.0679)(0.0671)(0.0880)
C40.98110.98661.98161.98830.48201.18460.8881
(0.0684)(0.0623)(0.0649)(0.0625)(0.0666)(0.0653)(0.0858)
500C10.99480.99231.99441.99530.49731.19930.8927
(0.0541)(0.0543)(0.0521)(0.0572)(0.0567)(0.0555)(0.0651)
C20.99100.99171.99321.99180.49731.19780.8985
(0.0415)(0.0438)(0.0493)(0.0402)(0.0411)(0.0415)(0.0628)
C30.99030.99731.99131.99040.49651.19900.8958
(0.0576)(0.0573)(0.0547)(0.0592)(0.0588)(0.0578)(0.0679)
C40.99240.99061.99501.99810.49281.19500.8943
(0.0559)(0.0560)(0.0536)(0.0580)(0.0581)(0.0561)(0.0670)
900C10.99650.99981.99221.99050.49441.19720.8911
(0.0428)(0.0441)(0.0435)(0.0423)(0.0471)(0.0429)(0.0445)
C20.99580.99151.99391.99190.49431.19310.8944
(0.0331)(0.0314)(0.0362)(0.0324)(0.0335)(0.0315)(0.0432)
C30.99500.99341.99691.99530.49221.19850.8907
(0.0455)(0.0463)(0.0466)(0.0441)(0.0493)(0.0450)(0.0469)
C40.99230.99881.99551.99670.49131.19310.8922
(0.0440)(0.0450)(0.0451)(0.0430)(0.0480)(0.0443)(0.0460)
Table 12

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=1.2, ν2=0.5 and θ=0.9.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98130.98061.98581.98121.18710.48100.8891
(0.0657)(0.0634)(0.0665)(0.0629)(0.0664)(0.0666)(0.0874)
C20.98600.98881.98521.98821.18790.48170.8897
(0.0529)(0.0590)(0.0594)(0.0514)(0.0560)(0.0576)(0.0848)
C30.98850.98511.98721.98081.18630.48770.8839
(0.0674)(0.0650)(0.0688)(0.0645)(0.0679)(0.0688)(0.0892)
C40.98070.98141.98901.98291.18630.48100.8832
(0.0666)(0.0640)(0.0679)(0.0637)(0.0670)(0.0680)(0.0885)
500C10.99640.99371.99651.99231.19520.49660.8970
(0.0546)(0.0513)(0.0521)(0.0543)(0.0565)(0.0564)(0.0675)
C20.99550.99591.99181.99801.19120.49150.8937
(0.0413)(0.0485)(0.0417)(0.0444)(0.0496)(0.0436)(0.0669)
C30.99690.99801.99211.99411.19570.49370.8952
(0.0560)(0.0525)(0.0541)(0.0558)(0.0580)(0.0579)(0.0691)
C40.99910.99541.99841.99521.19080.49750.8999
(0.0558)(0.0526)(0.0530)(0.0550)(0.0577)(0.0575)(0.0686)
900C10.99500.99771.99311.99841.19830.49710.8958
(0.0453)(0.0411)(0.0415)(0.0420)(0.0410)(0.0431)(0.0440)
C20.99500.99271.99651.99511.19530.49230.8938
(0.0368)(0.0324)(0.0320)(0.0305)(0.0364)(0.0339)(0.0420)
C30.99320.99311.99811.99851.19600.49720.8919
(0.0475)(0.0495)(0.0466)(0.0470)(0.0461)(0.0478)(0.0485)
C40.99250.99111.99591.99391.19170.49760.8936
(0.0460)(0.0428)(0.0447)(0.0455)(0.0449)(0.0460)(0.0475)
Table 13

Estimates of the parameters and s.e. under different copulas for the COM-Poissonmodel for ν1=0.9, ν2=0.9 and θ=0.9.

ICβ^1[1]β^2[1]β^1[2]β^2[2]ν^1ν^2θ^
100C10.98310.98061.98481.98790.88820.88070.8893
(0.0652)(0.0616)(0.0620)(0.0627)(0.0631)(0.0626)(0.0875)
C20.98800.98761.98301.98870.88270.91700.8841
(0.0542)(0.0596)(0.0538)(0.0560)(0.0584)(0.0525)(0.0851)
C30.98230.98201.98051.98880.88600.91240.8873
(0.0675)(0.0645)(0.0650)(0.0656)(0.0661)(0.0647)(0.0898)
C40.98210.98851.98751.98340.88440.88980.8801
(0.0660)(0.0629)(0.0633)(0.0640)(0.0644)(0.0637)(0.0886)
500C10.99140.99621.99821.99610.89850.89320.8946
(0.0531)(0.0552)(0.0562)(0.0512)(0.0513)(0.0562)(0.0688)
C20.99560.99401.99221.99170.89270.89330.8936
(0.0414)(0.0495)(0.0449)(0.0436)(0.0487)(0.0427)(0.0680)
C30.99780.99801.99331.99880.89540.89120.8971
(0.0551)(0.0570)(0.0574)(0.0530)(0.0528)(0.0595)(0.0697)
C40.99100.99361.99501.99740.89010.89090.8908
(0.0540)(0.0563)(0.0570)(0.0524)(0.0521)(0.0580)(0.0695)
900C10.99400.99031.99521.99120.89240.89710.8969
(0.0461)(0.0414)(0.0431)(0.0444)(0.0476)(0.0438)(0.0446)
C20.99020.99301.99741.99570.89420.89090.8923
(0.0397)(0.0364)(0.0371)(0.0352)(0.0358)(0.0393)(0.0422)
C30.99150.99591.99121.99040.89580.90810.8982
(0.0489)(0.0440)(0.0450)(0.0464)(0.0492)(0.0464)(0.0471)
C40.99710.99611.99351.99670.90530.89400.8911
(0.0479)(0.0429)(0.0440)(0.0458)(0.0488)(0.0455)(0.0460)
Table 14

Number of non-convergent simulations under BINAR(1)NB process under different combinations.

ν^1ν^2θ^IC1C2C3C4
110.3100400360440425
110.3500375325410400
110.3900320290370360
0.50.90.3100380340410400
0.50.90.3500350310375370
0.50.90.3900300275335325
1.230.3100425400450440
1.230.3500400360425415
1.230.3900350315390375
0.51.20.3100390375420410
0.51.20.3500350340385380
0.51.20.3900315300350340
1.20.50.3100395370415405
1.20.50.3500360330385380
1.20.50.3900320300355345
0.90.90.3100385350430420
0.90.90.3500370320400405
0.90.90.3900325295375370
ν^1ν^2θ^IC1C2C3C4
110.9100415375460440
110.9500385340425410
110.9900330310390375
0.50.90.9100400360430425
0.50.90.9500365325400390
0.50.90.9900315290350340
1.230.9100440415470460
1.230.9500410375435425
1.230.9900365325400390
0.51.20.9100415390440425
0.51.20.9500360350400390
0.51.20.9900330315365350
1.20.50.9100370390445430
1.20.50.9500420355410395
1.20.50.9900355320365360
0.90.90.9100395360440430
0.90.90.9500375325415410
0.90.90.9900330300385375
Table 15

Estimates of the regression and over-dispersion parameters using CMP model.

SeriesCopulasINTCTRBRAgeINTAν^1
Yi[1]AMH0.3522-0.31990.03030.02370.00380.3138
(0.4320)(0.2550)(0.0044)(0.0121)(0.0062)(0.3761)
Frank0.3542-0.31920.03040.02530.00040.3121
(0.4317)(0.2547)(0.0041)(0.0119)(0.0061)(0.3760)
Clayton0.3627-0.31890.03290.02330.00390.3197
(0.4340)(0.2555)(0.0049)(0.0127)(0.0068)(0.3780)
Gumbel0.3597-0.31450.03180.02640.00380.3133
(0.4334)(0.2559)(0.0048)(0.0130)(0.0069)(0.3772)
Yi[2]AMH0.2237-0.20510.01970.01720.00280.3015
(0.4165)(0.2252)(0.0055)(0.0115)(0.0059)(0.3984)
Frank0.2258-0.19620.01710.01500.00030.2966
(0.4161)(0.2251)(0.0052)(0.0111)(0.0055)(0.3979)
Clayton0.2313-0.20610.01410.01290.00270.3113
(0.4194)(0.2278)(0.0059)(0.0120)(0.0061)(0.3987)
Gumbel0.2299-0.19750.01520.01440.00290.2933
(0.4170)(0.2265)(0.0058)(0.0122)(0.0064)(0.3990)

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Received: 2016-7-10
Accepted: 2017-2-15
Published Online: 2017-4-8
Published in Print: 2017-6-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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