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August  2019, 13(4): 787-803. doi: 10.3934/ipi.2019036

Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation

Abraão D. C. Nascimento 1,, , Leandro C. Rêgo 2,3, and  Raphaela L. B. A. Nascimento 1,

1. 

Departamento de Estatística, Centro de Ciências Exatas e da Natureza, Universidade Federal de Pernambuco, Recife-PE, ZIP 50740-540, Brazil
2. 

Departamento de Estatística e Matemática Aplicada, Centro de Ciências, Universidade Federal do Ceará, Fortaleza-CE, ZIP 60440-900, Brazil
3. 

Programas de Pós-Graduação em Estatística e Engenharia de Produção, Universidade Federal de Pernambuco, Recife-PE, ZIP 50740-570, Brazil

* Corresponding author: Abraão D. C. Nascimento

Received  May 2018 Revised  February 2019 Published  May 2019

The proposal of efficient distributions is a crucial step for decision making in practice. Mixture models are adjustment tools which are often used to describe complex phenomena. However, as one disadvantage, such models impose hard inference procedures, submitted to a large number of parameters. To solve this issue, this paper proposes a new model which is able to describe multimodal, symmetric and asymmetric behaviors with only three parameters, called compound truncated Poisson normal (CTPN) distribution. Some properties of the CTPN law are derived and discussed: characteristic and cumulant functions and ordinary moments. A moment estimation procedure for CTPN parameters is also provided. This procedure consists of solving one nonlinear equation in terms of a single parameter. An application with images of synthetic aperture radar (SAR) is made. The results present evidence that the CTPN can outperform the $ \mathcal{G}^0 $, $ \mathcal{K} $ and BGN (laws commonly used in SAR literature), as well as GBGL models.

Keywords: Compound generator, radar data, multimodal behaviour, ordinary moments.
Mathematics Subject Classification: Primary: 62F10; Secondary: 62P30.
Citation: Abraão D. C. Nascimento, Leandro C. Rêgo, Raphaela L. B. A. Nascimento. Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation. Inverse Problems and Imaging, 2019, 13 (4) : 787-803. doi: 10.3934/ipi.2019036
References:
[1]

V. Barnett, Comparative Statistical Inference, Second edition. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester, 1982.

[2]

D. Blacknell, Comparison of parameter estimators for K-distribution, IEE Proceedings-Radar, Sonar and Navigation, 141 (1994), 45-52.  doi: 10.1049/ip-rsn:19949885.

[3]

G. Casella and R. Berger, Statistical Inference, Duxbury advanced series in statistics and decision sciences, Duxbury Pacific Grove, CA, United States of America, 2002.

[4]

R. Cintra, L. Rêgo, G. Cordeiro and A. Nascimento, Beta generalized normal distribution with an application for SAR image processing, Statistics, 48 (2014), 279-294.  doi: 10.1080/02331888.2012.748776.

[5]

L. Cobb, Stochastic catastrophe models and multimodal distributions, Behavioral Science, 23 (1978), 360-374.  doi: 10.1002/bs.3830230407.

[6]

G. M. Cordeiro and A. J. Lemonte, The McDonald inverted beta distribution, Journal of the Franklin Institute, 349 (2012), 1174-1197.  doi: 10.1016/j.jfranklin.2012.01.006.

[7]

G. M. Cordeiro, E. M. Ortega and S. Nadarajah, The Kumaraswamy Weibull distribution with application to failure data, Journal of the Franklin Institute, 347 (2010), 1399-1429.  doi: 10.1016/j.jfranklin.2010.06.010.

[8]

B. Efron, Bootstrap methods: Another look at the jackknife, Annals of Statistics, 7 (1979), 1–26, URL https://projecteuclid.org/euclid.aos/1176344552. doi: 10.1214/aos/1176344552.

[9]

A. El-Zaart and D. Ziou, Statistical modelling of multimodal SAR images, Int. J. Remote Sens., 28 (2007), 2277-2294.  doi: 10.1080/01431160600933997.

[10]

B. S. Everitt and D. J. Hand, Finite Mixture Distributions, Chapman and Hall, London, 1981.

[11]

T. L. Fine, Probability and Probabilistic Reasoning for Electrical Engineering, Prentice Hall, 2006.

[12]

A. C. Frery, A. D. C. Nascimento and R. J. Cintra, Analytic expressions for stochastic distances between relaxed complex Wishart distributions, IEEE Transactions on Geoscience and Remote Sensing, 52 (2014), 1213-1226.  doi: 10.1109/TGRS.2013.2248737.

[13]

A. Golubev, Exponentially modified Gaussian (emg) relevance to distributions related to cell proliferation and differentiation, Journal of Theoretical Biology, 262 (2010), 257-266.  doi: 10.1016/j.jtbi.2009.10.005.

[14]

E. Grushka, Characterization of exponentially modified Gaussian peaks in chromatography, Analytical Chemistry, 44 (1972), 1733-1738.  doi: 10.1021/ac60319a011.

[15]

D. Karlis and E. Xekalaki, Mixed Poisson distributions, International Statistical Review, 73 (2005), 35-58.  doi: 10.1111/j.1751-5823.2005.tb00250.x.

[16]

M. C. S. Lima, G. M. Cordeiro, A. D. C. Nascimento and K. F. Silva, A new model for describing remission times: The generalized beta-generated Lindley distribution, Anais da Academia Brasileira de Ciências, 89 (2017), 1343–1367, URL http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652017000401343&nrm=iso. doi: 10.1590/0001-3765201720160455.

[17]

K. V. Muller, SAR correlation imaging and anisotropic scattering, Inverse Problems and Imaging, 12 (2018), 697-731.  doi: 10.3934/ipi.2018030.

[18]

M. M. Najafabadi, T. M. Khoshgoftaar, F. Villanustre and J. Holt, Large-scale distributed l-bfgs, Journal of Big Data, 4 (2017), 22.  doi: 10.1186/s40537-017-0084-5.

[19]

A. K. Nandi and D. Mämpel, An extension of the generalized Gaussian distribution to include asymmetry, Journal of the Franklin Institute, 332 (1995), 67-75.  doi: 10.1016/0016-0032(95)00029-W.

[20]

A. D. C. Nascimento, R. J. Cintra and A. C. Frery, Hypothesis testing in speckled data with stochastic distances, IEEE Transactions on Geoscience and Remote Sensing, 48 (2010), 373-385.  doi: 10.1109/TGRS.2009.2025498.

[21]

N. B. Norman and L. Johnson Samuel Kotz, Continuous Univariate Distributions, Wiley Series in Probability and Statistics, Wiley-Interscience, 1995.

[22]

H. H. Panjer, Recursive evaluation of a family of compound distributions, Astin Bulletin, 12 (1981), 22-26.  doi: 10.1017/S0515036100006796.

[23]

D. Povey, L. Burget, M. Agarwal, P. Akyazi, F. Kai, A. Ghoshal, O. Glembek, N. Goel, M. Karafiát, A. Rastrow, R. C. Rose, P. Schwarz and S. Thomas, The subspace Gaussian mixture model–a structured model for speech recognition, Computer Speech & Language, 25 (2011), 404–439, URL http://www.sciencedirect.com/science/article/pii/S088523081000063X, Language and speech issues in the engineering of companionable dialogue systems. doi: 10.1016/j.csl.2010.06.003.

[24]

S. I. Resnick, Adventures in Stochastic Processes, Birkhauser, Boston, 1992.

[25]

K. Revfeim, An initial model of the relationship between rainfall events and daily rainfalls, Journal of Hydrology, 75 (1984), 357-364.  doi: 10.1016/0022-1694(84)90059-3.

[26]

M. C. Teich and P. Diament, Multiply stochastic representations for K distributions and their Poisson transforms, Journal of the Optical Society of America A, 6 (1989), 80-91.  doi: 10.1364/JOSAA.6.000080.

[27]

C. Thompson, Homogeneity analysis of rainfall series: An application of the use of a realistic rainfall model, Journal of Climatology, 4 (1984), 609-619. 

[28]

T. S. Wirjanto and D. Xu, The Applications of Mixtures of Normal Distributions in Empirical Finance: A Selected Survey, Technical report, Working paper, 2009.

show all references

References:
[1]

V. Barnett, Comparative Statistical Inference, Second edition. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Ltd., Chichester, 1982.

[2]

D. Blacknell, Comparison of parameter estimators for K-distribution, IEE Proceedings-Radar, Sonar and Navigation, 141 (1994), 45-52.  doi: 10.1049/ip-rsn:19949885.

[3]

G. Casella and R. Berger, Statistical Inference, Duxbury advanced series in statistics and decision sciences, Duxbury Pacific Grove, CA, United States of America, 2002.

[4]

R. Cintra, L. Rêgo, G. Cordeiro and A. Nascimento, Beta generalized normal distribution with an application for SAR image processing, Statistics, 48 (2014), 279-294.  doi: 10.1080/02331888.2012.748776.

[5]

L. Cobb, Stochastic catastrophe models and multimodal distributions, Behavioral Science, 23 (1978), 360-374.  doi: 10.1002/bs.3830230407.

[6]

G. M. Cordeiro and A. J. Lemonte, The McDonald inverted beta distribution, Journal of the Franklin Institute, 349 (2012), 1174-1197.  doi: 10.1016/j.jfranklin.2012.01.006.

[7]

G. M. Cordeiro, E. M. Ortega and S. Nadarajah, The Kumaraswamy Weibull distribution with application to failure data, Journal of the Franklin Institute, 347 (2010), 1399-1429.  doi: 10.1016/j.jfranklin.2010.06.010.

[8]

B. Efron, Bootstrap methods: Another look at the jackknife, Annals of Statistics, 7 (1979), 1–26, URL https://projecteuclid.org/euclid.aos/1176344552. doi: 10.1214/aos/1176344552.

[9]

A. El-Zaart and D. Ziou, Statistical modelling of multimodal SAR images, Int. J. Remote Sens., 28 (2007), 2277-2294.  doi: 10.1080/01431160600933997.

[10]

B. S. Everitt and D. J. Hand, Finite Mixture Distributions, Chapman and Hall, London, 1981.

[11]

T. L. Fine, Probability and Probabilistic Reasoning for Electrical Engineering, Prentice Hall, 2006.

[12]

A. C. Frery, A. D. C. Nascimento and R. J. Cintra, Analytic expressions for stochastic distances between relaxed complex Wishart distributions, IEEE Transactions on Geoscience and Remote Sensing, 52 (2014), 1213-1226.  doi: 10.1109/TGRS.2013.2248737.

[13]

A. Golubev, Exponentially modified Gaussian (emg) relevance to distributions related to cell proliferation and differentiation, Journal of Theoretical Biology, 262 (2010), 257-266.  doi: 10.1016/j.jtbi.2009.10.005.

[14]

E. Grushka, Characterization of exponentially modified Gaussian peaks in chromatography, Analytical Chemistry, 44 (1972), 1733-1738.  doi: 10.1021/ac60319a011.

[15]

D. Karlis and E. Xekalaki, Mixed Poisson distributions, International Statistical Review, 73 (2005), 35-58.  doi: 10.1111/j.1751-5823.2005.tb00250.x.

[16]

M. C. S. Lima, G. M. Cordeiro, A. D. C. Nascimento and K. F. Silva, A new model for describing remission times: The generalized beta-generated Lindley distribution, Anais da Academia Brasileira de Ciências, 89 (2017), 1343–1367, URL http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652017000401343&nrm=iso. doi: 10.1590/0001-3765201720160455.

[17]

K. V. Muller, SAR correlation imaging and anisotropic scattering, Inverse Problems and Imaging, 12 (2018), 697-731.  doi: 10.3934/ipi.2018030.

[18]

M. M. Najafabadi, T. M. Khoshgoftaar, F. Villanustre and J. Holt, Large-scale distributed l-bfgs, Journal of Big Data, 4 (2017), 22.  doi: 10.1186/s40537-017-0084-5.

[19]

A. K. Nandi and D. Mämpel, An extension of the generalized Gaussian distribution to include asymmetry, Journal of the Franklin Institute, 332 (1995), 67-75.  doi: 10.1016/0016-0032(95)00029-W.

[20]

A. D. C. Nascimento, R. J. Cintra and A. C. Frery, Hypothesis testing in speckled data with stochastic distances, IEEE Transactions on Geoscience and Remote Sensing, 48 (2010), 373-385.  doi: 10.1109/TGRS.2009.2025498.

[21]

N. B. Norman and L. Johnson Samuel Kotz, Continuous Univariate Distributions, Wiley Series in Probability and Statistics, Wiley-Interscience, 1995.

[22]

H. H. Panjer, Recursive evaluation of a family of compound distributions, Astin Bulletin, 12 (1981), 22-26.  doi: 10.1017/S0515036100006796.

[23]

D. Povey, L. Burget, M. Agarwal, P. Akyazi, F. Kai, A. Ghoshal, O. Glembek, N. Goel, M. Karafiát, A. Rastrow, R. C. Rose, P. Schwarz and S. Thomas, The subspace Gaussian mixture model–a structured model for speech recognition, Computer Speech & Language, 25 (2011), 404–439, URL http://www.sciencedirect.com/science/article/pii/S088523081000063X, Language and speech issues in the engineering of companionable dialogue systems. doi: 10.1016/j.csl.2010.06.003.

[24]

S. I. Resnick, Adventures in Stochastic Processes, Birkhauser, Boston, 1992.

[25]

K. Revfeim, An initial model of the relationship between rainfall events and daily rainfalls, Journal of Hydrology, 75 (1984), 357-364.  doi: 10.1016/0022-1694(84)90059-3.

[26]

M. C. Teich and P. Diament, Multiply stochastic representations for K distributions and their Poisson transforms, Journal of the Optical Society of America A, 6 (1989), 80-91.  doi: 10.1364/JOSAA.6.000080.

[27]

C. Thompson, Homogeneity analysis of rainfall series: An application of the use of a realistic rainfall model, Journal of Climatology, 4 (1984), 609-619. 

[28]

T. S. Wirjanto and D. Xu, The Applications of Mixtures of Normal Distributions in Empirical Finance: A Selected Survey, Technical report, Working paper, 2009.

Figure 1.  CTPN pdf and hrf curves at several parametric points
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Figure 2.  Real SAR image and plots of empirical densities (gray curve) vs. fitted pdf and cdf of CTPN (solid curves), BGN (dashed curves), GBGL (long dashed curves), $ \mathcal{G}^0 $ (dot curves) and $ \mathcal{K} $ (dashes and dot curves) distributions
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Table 1.  Performance under synthetic data from ML and MM estimates
$ n $ $ \widehat{\lambda} $ $ \widehat{\mu} $ $ \widehat{\sigma^2} $ $ \widehat{\lambda}_\text{ML} $ $ \widehat{\mu}_{\text{ML}} $ $ \widehat{\sigma^2}_{\text{ML}} $
$ \text{MSE}(\widehat{\lambda}) $ $ \text{MSE}(\widehat{\mu}) $ $ \text{MSE}(\widehat{\sigma^2}) $ $ \text{MSE}(\widehat{\lambda}_{\text{ML}}) $ $ \text{MSE}(\widehat{\mu}_{\text{ML}}) $ $ \text{MSE}(\widehat{\sigma^2}_{\text{ML}}) $
$\underline{ \lambda=0.5, \quad \mu=0.0 \quad\text{ and } \quad \sigma^2=1.0 }$
$ 100 $ 0.52180 -0.00120 0.98868 0.30940 -0.00031 1.04032
(0.32178) (0.43337) (0.40898) (0.37080) (0.43669) (0.46791)
$ 500 $ 0.48627 0.00517 0.98751 0.51697 -0.00215 0.99796
(0.30808) (0.42401) (0.41713) (0.28531) (0.42028) (0.42127)
$ 1000 $ 0.49767 0.00086 0.98671 0.51001 0.00007 0.99977
(0.30462) (0.42893) (0.41342) (0.23593) (0.41714) (0.41957)
$\underline{ \lambda=1.0, \quad \mu=0.0 \quad \text{ and } \quad \sigma^2=1.0} $
$ 100 $ 1.29115 0.00151 0.98319 1.13192 0.00392 0.99745
(2.74401) (0.67237) (0.37800) (1.91113) (0.67445) (0.38453)
$ 500 $ 1.04780 0.00364 1.01459 1.21035 -0.00048 0.98445
(2.00933) (0.68142) (0.38981) (1.32994) (0.66745) (0.34096)
$ 1000 $ 1.00027 0.00511 1.02289 1.14585 0.00100 0.99060
(1.87405) (0.66863) (0.39364) (1.06556) (0.66538) (0.34668)
$\underline{ \lambda=2.0, \quad \mu=0.0\quad \text{ and } \quad \sigma^2=1.0} $
$ 100 $ 2.46275 -0.00678 1.04992 2.02837 -0.00339 1.05890
(8.67409) (1.66999) (0.80539) (4.75657) (1.66939) (0.74710)
$ 500 $ 2.28215 -0.00509 1.07334 2.10954 -0.00261 1.01838
(8.09059) (1.68661) (0.76915) (3.48636) (1.67472) (0.69420)
$ 1000 $ 2.23503 -0.00410 1.07847 2.27558 -0.00020 0.98418
(7.93255) (1.68785) (0.76165) (3.55280) (1.67065) (0.67192)
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$ n $ $ \widehat{\lambda} $ $ \widehat{\mu} $ $ \widehat{\sigma^2} $ $ \widehat{\lambda}_\text{ML} $ $ \widehat{\mu}_{\text{ML}} $ $ \widehat{\sigma^2}_{\text{ML}} $
$ \text{MSE}(\widehat{\lambda}) $ $ \text{MSE}(\widehat{\mu}) $ $ \text{MSE}(\widehat{\sigma^2}) $ $ \text{MSE}(\widehat{\lambda}_{\text{ML}}) $ $ \text{MSE}(\widehat{\mu}_{\text{ML}}) $ $ \text{MSE}(\widehat{\sigma^2}_{\text{ML}}) $
$\underline{ \lambda=0.5, \quad \mu=0.0 \quad\text{ and } \quad \sigma^2=1.0 }$
$ 100 $ 0.52180 -0.00120 0.98868 0.30940 -0.00031 1.04032
(0.32178) (0.43337) (0.40898) (0.37080) (0.43669) (0.46791)
$ 500 $ 0.48627 0.00517 0.98751 0.51697 -0.00215 0.99796
(0.30808) (0.42401) (0.41713) (0.28531) (0.42028) (0.42127)
$ 1000 $ 0.49767 0.00086 0.98671 0.51001 0.00007 0.99977
(0.30462) (0.42893) (0.41342) (0.23593) (0.41714) (0.41957)
$\underline{ \lambda=1.0, \quad \mu=0.0 \quad \text{ and } \quad \sigma^2=1.0} $
$ 100 $ 1.29115 0.00151 0.98319 1.13192 0.00392 0.99745
(2.74401) (0.67237) (0.37800) (1.91113) (0.67445) (0.38453)
$ 500 $ 1.04780 0.00364 1.01459 1.21035 -0.00048 0.98445
(2.00933) (0.68142) (0.38981) (1.32994) (0.66745) (0.34096)
$ 1000 $ 1.00027 0.00511 1.02289 1.14585 0.00100 0.99060
(1.87405) (0.66863) (0.39364) (1.06556) (0.66538) (0.34668)
$\underline{ \lambda=2.0, \quad \mu=0.0\quad \text{ and } \quad \sigma^2=1.0} $
$ 100 $ 2.46275 -0.00678 1.04992 2.02837 -0.00339 1.05890
(8.67409) (1.66999) (0.80539) (4.75657) (1.66939) (0.74710)
$ 500 $ 2.28215 -0.00509 1.07334 2.10954 -0.00261 1.01838
(8.09059) (1.68661) (0.76915) (3.48636) (1.67472) (0.69420)
$ 1000 $ 2.23503 -0.00410 1.07847 2.27558 -0.00020 0.98418
(7.93255) (1.68785) (0.76165) (3.55280) (1.67065) (0.67192)
Table 2.  Descriptive analysis of real intensity data (CV, $ K $ and $ S $ represent sample coefficient of variation, kurtosis and skewness, respectively)
Mean Median CV % K S Size
0.0026170 0.0031120 69.24582 10.79774 2.221156 1248
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Mean Median CV % K S Size
0.0026170 0.0031120 69.24582 10.79774 2.221156 1248
Table 3.  ML estimates for the $ \operatorname{CTPN}(\lambda, \mu, \sigma) $, $ \operatorname{BGN}(s, \mu, \sigma, \alpha, \beta) $, $ \operatorname{GBGL}(\lambda, a, b, c) $, $ \mathcal{G}^0(\alpha, \gamma, L) $ and $ \mathcal{K}(\alpha, \lambda, L) $ distributions. Standard errors are in parenthesis
Model Estimated Parameters
BGN 0.928 $ 0.112 \times 10^{-2} $ $ 0.036 \times 10^{-2} $ 1.945 0.224
($2.158 \times 10^{-4} $) ($ 1.455 \times 10^{-5} $) ($ 3.109\times 10^{-6} $) (0.077) ($6.617 \times 10^{-3} $)
GBGL 9.802 28.242 41.872 0.242 $ \bullet $
(0.755) (0.160) (0.204) ($0.559 \times 10^{-2} $) $ \bullet $
CTPN 1.576 $ 0.149 \times 10^{-2} $ $ 0.034 \times 10^{-2} $ $ \bullet $ $ \bullet $
(0.107) ($1.151 \times 10^{-2} $) ($ 1.661 \times 10^{-3} $) $ \bullet $ $ \bullet $
$ \mathcal{G}^0 $ -4.210 0.010 11.818 $ \bullet $ $ \bullet $
(0.186) (0.007) (4.443) $ \bullet $ $ \bullet $
$ \mathcal{K} $ 0.959 192.416 12.567 $ \bullet $ $ \bullet $
(0.049) (21.964) (0.828) $ \bullet $ $ \bullet $
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Model Estimated Parameters
BGN 0.928 $ 0.112 \times 10^{-2} $ $ 0.036 \times 10^{-2} $ 1.945 0.224
($2.158 \times 10^{-4} $) ($ 1.455 \times 10^{-5} $) ($ 3.109\times 10^{-6} $) (0.077) ($6.617 \times 10^{-3} $)
GBGL 9.802 28.242 41.872 0.242 $ \bullet $
(0.755) (0.160) (0.204) ($0.559 \times 10^{-2} $) $ \bullet $
CTPN 1.576 $ 0.149 \times 10^{-2} $ $ 0.034 \times 10^{-2} $ $ \bullet $ $ \bullet $
(0.107) ($1.151 \times 10^{-2} $) ($ 1.661 \times 10^{-3} $) $ \bullet $ $ \bullet $
$ \mathcal{G}^0 $ -4.210 0.010 11.818 $ \bullet $ $ \bullet $
(0.186) (0.007) (4.443) $ \bullet $ $ \bullet $
$ \mathcal{K} $ 0.959 192.416 12.567 $ \bullet $ $ \bullet $
(0.049) (21.964) (0.828) $ \bullet $ $ \bullet $
Table 4.  Goodness-of-fit measures for the fitted CTPN, BGN, GBGL, $ \mathcal{G}^0 $ and $ \mathcal{K} $ models on EMISAR real data
Goodness-of-fit measures Performance for different models
CTPN BGN $ \mathcal{G}^0 $ $ \mathcal{K} $ GBGL
$ \text{d}_\text{KS} $ 0.039778 0.065079 0.072632 0.21423 0.079914
$ \text{p-value}_\text{KS} $ 0.03853 $5.12 \times 10^{-5} $ $3.8 \times 10^{-6} $ $ < 2.2 \times 10^{-6} $ $2.4 \times 10^{-7} $
$ \text{W}^* $ 0.5869176 0.8601945 1.600842 1.961929 1.692876
$ \text{A}^* $ 4.6622426 4.6381167 9.733052 14.299778 12.226518
$ \text{AIC} $ -12675.79 -12726.81 -12646.98 -11635.48 -12557.52
$ \text{AIC}_c $ -12675.77 -12726.76 -12646.96 -12675.77 -12557.49
$ \text{BIC} $ -12660.41 -12701.16 -12631.59 -12660.41 -12537.01
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Goodness-of-fit measures Performance for different models
CTPN BGN $ \mathcal{G}^0 $ $ \mathcal{K} $ GBGL
$ \text{d}_\text{KS} $ 0.039778 0.065079 0.072632 0.21423 0.079914
$ \text{p-value}_\text{KS} $ 0.03853 $5.12 \times 10^{-5} $ $3.8 \times 10^{-6} $ $ < 2.2 \times 10^{-6} $ $2.4 \times 10^{-7} $
$ \text{W}^* $ 0.5869176 0.8601945 1.600842 1.961929 1.692876
$ \text{A}^* $ 4.6622426 4.6381167 9.733052 14.299778 12.226518
$ \text{AIC} $ -12675.79 -12726.81 -12646.98 -11635.48 -12557.52
$ \text{AIC}_c $ -12675.77 -12726.76 -12646.96 -12675.77 -12557.49
$ \text{BIC} $ -12660.41 -12701.16 -12631.59 -12660.41 -12537.01
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