# American Institute of Mathematical Sciences

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

## Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation

 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.

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 & Imaging, 2019, 13 (4) : 787-803. doi: 10.3934/ipi.2019036
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##### References:
CTPN pdf and hrf curves at several parametric points
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
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)
 $n$ $\widehat{\lambda}$ $\widehat{\mu}$ $\widehat{\sigma^2}$ $\widehat{\lambda}_\text{ML}$ $\widehat{\mu}_{\text{ML}}$ $\widehat{\sigma^2}_{\text{ML}}$ $\text{MSE}(\widehat{\lambda})$ $\text{MSE}(\widehat{\mu})$ $\text{MSE}(\widehat{\sigma^2})$ $\text{MSE}(\widehat{\lambda}_{\text{ML}})$ $\text{MSE}(\widehat{\mu}_{\text{ML}})$ $\text{MSE}(\widehat{\sigma^2}_{\text{ML}})$ $\underline{ \lambda=0.5, \quad \mu=0.0 \quad\text{ and } \quad \sigma^2=1.0 }$ $100$ 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)
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
 Mean Median CV % K S Size 0.0026170 0.0031120 69.24582 10.79774 2.221156 1248
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) $\bullet$ $\bullet$
 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) $\bullet$ $\bullet$
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
 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