Table 5 Simulated estimations of Bias, Mean Squared Error (MSE), and Mean Relative Error (MRE) for \(\gamma _{1}=2.0,~\gamma _{2}=3.0\).
n | Est. | Est.Par. | \(M_1\) | \(M_2\) | \(M_3\) | \(M_4\) | \(M_5\) | \(M_6\) |
|---|---|---|---|---|---|---|---|---|
50 | BIAS | \(\gamma _{1}\) | \(0.230928^{\{ 2\}}\) | \(0.234645^{\{ 4\}}\) | \(0.247321^{\{ 5\}}\) | \(0.229099^{\{ 1\}}\) | \(0.252045^{\{ 6\}}\) | \(0.231302^{\{ 3\}}\) |
\(\gamma _{2}\) | \(0.289417^{\{ 2 \}}\) | \(0.282640^{\{ 1\}}\) | \(0.352821^{\{ 6\}}\) | \(0.310913^{\{ 4 \}}\) | \(0.337261^{\{ 5\}}\) | \(0.306237^{\{ 3\}}\) | ||
MSE | \(\gamma _{1}\) | \(0.086902^{\{ 3 \}}\) | \(0.088558^{\{ 4\}}\) | \(0.098639^{\{ 5\}}\) | \(0.082574^{\{ 1\}}\) | \(0.103528^{\{ 6\}}\) | \(0.084082^{\{ 2\}}\) | |
\(\gamma _{2}\) | \(0.136893^{\{ 2\}}\) | \(0.136630^{\{ 1\}}\) | \(0.206084^{\{ 6\}}\) | \(0.144337^{\{ 3\}}\) | \(0.180385^{\{ 5 \}}\) | \(0.148247^{\{ 4\}}\) | ||
MRE | \(\gamma _{1}\) | \(0.115464^{\{ 2\}}\) | \(0.117323^{\{ 4\}}\) | \(0.123661 ^{\{ 5 \}}\) | \(0.114549^{\{ 1 \}}\) | \(0.126022^{\{ 6\}}\) | \(0.115651^{\{ 3\}}\) | |
\(\gamma _{2}\) | \(0.096472^{\{ 2\}}\) | \(0.094213^{\{ 1\}}\) | \(0.117607^{\{ 6\}}\) | \(0.103638^{\{ 4 \}}\) | \(0.112420^{\{ 5\}}\) | \(0.102079^{\{ 3\}}\) | ||
\(\sum Ranks\) | Â | \(13^{\{ 1\}}\) | \(15^{\{ 3\}}\) | \(33^{\{ 5.5\}}\) | \(14^{\{ 2 \}}\) | \(33^{\{ 5.5\}}\) | \(18^{\{ 4\}}\) | |
100 | BIAS | \(\gamma _{1}\) | \(0.169685^{\{ 3\}}\) | \(0.158887^{\{ 1\}}\) | \(0.180695^{\{ 6\}}\) | \(0.160702^{\{ 2\}}\) | \(0.176395^{\{ 4\}}\) | \(0.178196^{\{ 5\}}\) |
\(\gamma _{2}\) | \(0.180370^{\{ 1 \}}\) | \(0.215468^{\{ 3\}}\) | \(0.244161^{\{ 6\}}\) | \(0.180655^{\{ 2\}}\) | \(0.231288^{\{ 5\}}\) | \(0.218165^{\{ 4\}}\) | ||
MSE | \(\gamma _{1}\) | \(0.044601^{\{ 3\}}\) | \(0.042435^{\{ 2\}}\) | \(0.053684^{\{ 6\}}\) | \(0.042285^{\{ 1 \}}\) | \(0.049978^{\{ 4\}}\) | \(0.050704^{\{ 5\}}\) | |
\(\gamma _{2}\) | \(0.052093^{\{ 2\}}\) | \(0.072910^{\{ 3\}}\) | \(0.103650^{\{ 6\}}\) | \(0.051139^{\{ 1\}}\) | \(0.082185^{\{ 5\}}\) | \(0.075798^{\{ 4\}}\) | ||
MRE | \(\gamma _{1}\) | \(0.084843^{\{ 3\}}\) | \(0.079443^{\{ 1\}}\) | \(0.090347^{\{ 6\}}\) | \(0.080351^{\{ 2\}}\) | \(0.088198^{\{ 4\}}\) | \(0.089098^{\{ 5\}}\) | |
\(\gamma _{2}\) | \(0.060123^{\{ 1\}}\) | \(0.071823^{\{ 3\}}\) | \(0.081387^{\{ 6\}}\) | \(0.060218^{\{ 2\}}\) | \(0.077096^{\{ 5\}}\) | \(0.072722^{\{ 4\}}\) | ||
\(\sum Ranks\) | Â | \(13^{\{ 2.5\}}\) | \(13^{\{ 2.5 \}}\) | \(36^{\{ 6 \}}\) | \(10^{\{ 1 \}}\) | \(27^{\{ 4.5 \}}\) | \(27^{\{ 4.5\}}\) | |
250 | BIAS | \(\gamma _{1}\) | \(0.108062^{\{ 4 \}}\) | \(0.114106^{\{ 6 \}}\) | \(0.106762^{\{ 2 \}}\) | \(0.104958^{\{ 1\}}\) | \(0.110339^{\{ 5\}}\) | \(0.106781^{\{ 3\}}\) |
\(\gamma _{2}\) | \(0.122374^{\{ 1\}}\) | \(0.125788^{\{ 3\}}\) | \(0.153247^{\{ 6 \}}\) | \(0.122483^{\{ 2\}}\) | \(0.140219^{\{ 5 \}}\) | \(0.128473^{\{ 4\}}\) | ||
MSE | \(\gamma _{1}\) | \(0.018023^{\{ 2\}}\) | \(0.019974^{\{ 6\}}\) | \(0.018292^{\{ 3\}}\) | \(0.017245^{\{ 1\}}\) | \(0.019926^{\{ 5 \}}\) | \(0.018706^{\{ 4\}}\) | |
\(\gamma _{2}\) | \(0.023231^{\{ 2\}}\) | \(0.024978^{\{ 3\}}\) | \(0.037406^{\{ 6\}}\) | \(0.022719^{\{ 1\}}\) | \(0.032791^{\{ 5\}}\) | \(0.026610^{\{ 4\}}\) | ||
MRE | \(\gamma _{1}\) | \(0.054031^{\{ 4\}}\) | \(0.057053^{\{ 6\}}\) | \(0.053381^{\{ 2 \}}\) | \(0.052479^{\{ 1\}}\) | \(0.055170^{\{ 5\}}\) | \(0.053390^{\{ 3\}}\) | |
\(\gamma _{2}\) | \(0.040791^{\{ 1 \}}\) | \(0.041929^{\{ 3\}}\) | \(0.051082^{\{ 6\}}\) | \(0.040828^{\{ 2 \}}\) | \(0.046740^{\{ 5 \}}\) | \(0.042824^{\{ 4\}}\) | ||
\(\sum Ranks\) | Â | \(14^{\{ 2\}}\) | \(27^{\{ 5 \}}\) | \(25^{\{ 4 \}}\) | \(8^{\{ 1 \}}\) | \(30^{\{ 6 \}}\) | \(22^{\{ 3 \}}\) | |
350 | BIAS | \(\gamma _{1}\) | \(0.090942^{\{ 4\}}\) | \(0.091396^{\{ 5\}}\) | \(0.086886^{\{ 1\}}\) | \(0.087569^{\{ 2 \}}\) | \(0.091495^{\{ 6 \}}\) | \(0.090590^{\{ 3\}}\) |
\(\gamma _{2}\) | \(0.112887^{\{ 2\}}\) | \(0.120108^{\{ 4\}}\) | \(0.121644^{\{ 6\}}\) | \(0.105371^{\{ 1\}}\) | \(0.120591^{\{ 5\}}\) | \(0.117723^{\{ 3\}}\) | ||
MSE | \(\gamma _{1}\) | \(0.015190^{\{ 6\}}\) | \(0.012995^{\{ 5\}}\) | \(0.011850^{\{ 1 \}}\) | \(0.012033^{\{ 2 \}}\) | \(0.012927^{\{ 4\}}\) | \(0.012774^{\{ 3\}}\) | |
\(\gamma _{2}\) | \(0.024789^{\{ 6\}}\) | \(0.022187^{\{ 3\}}\) | \(0.023382^{\{ 4 \}}\) | \(0.017213^{\{ 1 \}}\) | \(0.023564^{\{ 5 \}}\) | \(0.020724^{\{2 \}}\) | ||
MRE | \(\gamma _{1}\) | \(0.045471^{\{ 4\}}\) | \(0.045698^{\{ 5\}}\) | \(0.043443^{\{ 1\}}\) | \(0.043785^{\{ 2\}}\) | \(0.045747^{\{ 6\}}\) | \(0.045295^{\{ 3\}}\) | |
\(\gamma _{2}\) | \(0.037629^{\{ 2\}}\) | \(0.040036^{\{ 4 \}}\) | \(0.040548^{\{ 6 \}}\) | \(0.035124^{\{ 1\}}\) | \(0.040197^{\{ 5\}}\) | \(0.039241^{\{ 3\}}\) | ||
\(\sum Ranks\) | Â | \(24^{\{ 4\}}\) | \(26^{\{ 5\}}\) | \(19^{\{ 3 \}}\) | \(9^{\{ 1 \}}\) | \(31^{\{ 6 \}}\) | \(17^{\{ 2 \}}\) | |
400 | BIAS | \(\gamma _{1}\) | \(0.087201^{\{ 6\}}\) | \(0.081290^{\{ 2\}}\) | \(0.086522^{\{ 5\}}\) | \(0.077015^{\{ 1\}}\) | \(0.086365^{\{ 4 \}}\) | \(0.085863^{\{ 3\}}\) |
\(\gamma _{2}\) | \(0.111927^{\{ 4\}}\) | \(0.103777^{\{ 3\}}\) | \(0.115053^{\{ 5\}}\) | \(0.096729^{\{ 1 \}}\) | \(0.115590^{\{ 6 \}}\) | \(0.101615^{\{ 2\}}\) | ||
MSE | \(\gamma _{1}\) | \(0.013335^{\{ 6\}}\) | \(0.010690^{\{ 2\}}\) | \(0.011360^{\{ 3\}}\) | \(0.009199^{\{ 1\}}\) | \(0.011993^{\{ 5\}}\) | \(0.011401^{\{ 4\}}\) | |
\(\gamma _{2}\) | \(0.028492^{\{ 6\}}\) | \(0.017162^{\{ 3\}}\) | \(0.021500^{\{ 5\}}\) | \(0.015096^{\{ 1\}}\) | \(0.020853^{\{ 4\}}\) | \(0.016740^{\{ 2\}}\) | ||
MRE | \(\gamma _{1}\) | \(0.043600^{\{ 6\}}\) | \(0.040645^{\{ 2\}}\) | \(0.043261^{\{ 5\}}\) | \(0.038508^{\{ 1\}}\) | \(0.043183^{\{ 4\}}\) | \(0.042931^{\{ 3\}}\) | |
\(\gamma _{2}\) | \(0.037309^{\{ 4\}}\) | \(0.034592^{\{ 3\}}\) | \(0.038351^{\{ 5\}}\) | \(0.032243^{\{ 1\}}\) | \(0.038530^{\{ 6\}}\) | \(0.033872^{\{ 2\}}\) | ||
\(\sum Ranks\) | Â | \(32^{\{ 6 \}}\) | \(15^{\{ 2 \}}\) | \(28^{\{ 4 \}}\) | \(6^{\{ 1\}}\) | \(29^{\{ 5\}}\) | \(16^{\{ 3\}}\) | |
450 | BIAS | \(\gamma _{1}\) | \(0.081370^{\{ 5\}}\) | \(0.080797^{\{ 4 \}}\) | \(0.078583^{\{ 2\}}\) | \(0.080534^{\{ 3\}}\) | \(0.082487^{\{ 6\}}\) | \(0.076552^{\{ 1\}}\) |
\(\gamma _{2}\) | \(0.097757^{\{ 3\}}\) | \(0.095652^{\{ 2 \}}\) | \(0.101122^{\{ 5\}}\) | \(0.094022^{\{ 1 \}}\) | \(0.118376^{\{ 6 \}}\) | \(0.100267^{\{ 4\}}\) | ||
MSE | \(\gamma _{1}\) | \(0.011836^{\{ 6\}}\) | \(0.010042^{\{ 4\}}\) | \(0.009881^{\{ 3\}}\) | \(0.009828^{\{ 2\}}\) | \(0.010897^{\{ 5\}}\) | \(0.009319^{\{ 1\}}\) | |
\(\gamma _{2}\) | \(0.017388^{\{ 4\}}\) | \(0.014305^{\{ 2\}}\) | \(0.017393^{\{ 5\}}\) | \(0.013602^{\{ 1\}}\) | \(0.022223^{\{ 6\}}\) | \(0.015303^{\{ 3\}}\) | ||
MRE | \(\gamma _{1}\) | \(0.040685^{\{ 5\}}\) | \(0.040399^{\{ 4\}}\) | \(0.039292^{\{ 2\}}\) | \(0.040267^{\{ 3\}}\) | \(0.041243^{\{ 6\}}\) | \(0.038276^{\{1 \}}\) | |
\(\gamma _{2}\) | \(0.032586^{\{ 3\}}\) | \(0.031884^{\{ 2\}}\) | \(0.033707^{\{ 5\}}\) | \(0.031341^{\{ 1 \}}\) | \(0.039459^{\{ 6 \}}\) | \(0.033422^{\{ 4\}}\) | ||
\(\sum Ranks\) | Â | \(26^{\{ 5\}}\) | \(18^{\{ 3 \}}\) | \(22^{\{ 4\}}\) | \(11^{\{ 1\}}\) | \(35^{\{ 6 \}}\) | \(14^{\{ 2\}}\) |
