Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation

Abraão D. C. Nascimento; Leandro C. Rêgo; Raphaela L. B. A. Nascimento

doi:10.3934/ipi.2019036

Article Contents

2019, Volume 13, Issue 4: 787-803. Doi: 10.3934/ipi.2019036

This issue Previous Article Total generalized variation regularization in data assimilation for Burgers' equation Next Article Image segmentation via

$ L_1 $

Monge-Kantorovich problem

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
^* Corresponding author: Abraão D. C. Nascimento

Received: May 2018

Revised: February 2019

Published: May 2019

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Abstract

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:

Mathematics Subject Classification: Primary: 62F10; Secondary: 62P30.

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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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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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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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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
Goodness-of-fit measures	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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