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Table 2 Simulated data example

From: A flexible distribution class for count data

   Simulated data distribution
   Bin(b=3, p =0.667) Pois(μ =6) NB(n=3, p=0.333)
  \(\hat {\mu }_{*}\) (SE) 2.0000(0.1414) 6.1100(0.2472) 5.3000(0.2302)
Pois. log(L) -144.8109 -228.1811 -288.0710
  AIC 291.6218 458.3623 578.1419
  BIC 294.2270 460.9675 580.7471
  \(\hat {\mu }\) (SE) 1.9999(0.1419) 6.1100(0.2503) 5.3001(0.3632)
  \(\hat {\theta }\) (SE) 276.5396(394.3489) 239.7812(421.0122) 3.5599(0.8564)
NB log(L) -145.0563 -228.3236 -258.7486
  AIC 294.1126 460.6472 521.4971
  BIC 299.3229 465.8575 526.7075
  \(\hat {\lambda }\) (SE) 18.7071(8.9855) 6.9145(2.1193) 1.5576(0.2079)
  \(\hat {\nu }\) (SE) 3.3931(0.5024) 1.0653(0.1603) 0.3150(0.0708)
CMP/sCMP(m=1) log(L) -123.2624 -228.0950 -260.3649
  AIC 250.5248 460.1900 524.7298
  BIC 255.7351 465.4003 529.9402
  \(\hat {\lambda }\) (SE) 4.2531(0.9494) 3.4046(0.8152) 0.9309(0.1109)
  \(\hat {\nu }\) (SE) 4.2854(0.4998) 1.0838(0.1826) 0.1674(0.0825)
sCMP(m=2) log(L) -120.8816 -228.0722 -259.3193
  AIC 245.7632 460.1444 522.6386
  BIC 250.9735 465.3547 527.8489
  \(\hat {\lambda }\) (SE) 2.0000(0.2450) 2.2683(0.4822) 0.6709(0.0576)
  \(\hat {\nu }\) (SE) 33.6942(12536.57) 1.1093(0.2127) 0.0392(0.0698)
sCMP(m=3) log(L) -116.2486 -228.0418 -258.8683
  AIC 236.4972 460.0836 521.7366
  BIC 241.7075 465.2939 526.9469
  \(\hat {\lambda }\) (SE) 1.0000(0.1000) 1.7044(0.3382) 0.5700(0.0486)
  \(\hat {\nu }\) (SE) 32.5126(12322.57) 1.1381(0.2469) 0.0000(0.0826)
sCMP(m=4) log(L) -124.7123 -228.0175 -258.8470
  AIC 253.4246 460.0350 521.6940
  BIC 258.6349 465.2453 526.9043
  1. True model parameters versus estimated parameters (and associated standard errors provided in parentheses) for various assumed distributions. For model comparisons, the log-likelihood, Akaike and Bayes Information Criterions (AIC and BIC, respectively) are provided