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American Journal of  Applied 
Statistics and Economics (AJASE)

Advancing Statistical Modelling: A Comparative Study of  Zero-Truncated Distributions 
in Economic Analysis
 B. E. Omokaro1, C. O. Aronu2*

Volume 4 Issue 1, Year 2025
ISSN: 2992-927X (Online)

DOI: https://doi.org/10.54536/ajase.v4i1.4235
https://journals.e-palli.com/home/index.php/ajase

Article Information ABSTRACT

Received: December 19, 2024

Accepted: January 21, 2025

Published: July 03, 2025

This study explores the application of  two zero-truncated distributions, Geometric-Zero 
Truncated Poisson (GZTP) and Zero-Truncated Poisson Pareto (ZTPP), in modelling eco-
nomic datasets, with a particular focus on Nigeria’s key economic indicators. Secondary data 
from the Central Bank of  Nigeria’s Statistical Bulletin (2021), spanning from 1989 to 2020, 
was used, covering variables such as Real Gross Domestic Product (RGDP), export and 
import goods, money supply, and Brent crude oil prices. The objectives were to: Introduce 
and describe the mathematical properties of  the GZTP and ZTPP distributions; compare 
the performance of  these distributions across datasets using metrics such as AIC, BIC, and 
MSE; and recommend the most efficient distribution for modelling economic variables and 
predicting trends. The study employs the Maximum Likelihood Estimation (MLE) method 
for parameter estimation, implemented in R programming. Model performance was evaluat-
ed using the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and 
Mean Squared Error (MSE). The results indicate that the ZTPP distribution outperforms 
the GZTP distribution across all datasets, with significantly lower AIC, BIC, and MSE val-
ues. Specifically, the ZTPP achieved an average AIC of  -1422.54, BIC of  -1422.55, and MSE 
of  30,161.94, compared to the GZTP’s -766.39, -762.82, and 31,163.2, respectively. These 
findings highlight the superior model fit and predictive accuracy of  the ZTPP distribution, 
making it a more robust tool for economic analysis, particularly in cases where zero occur-
rences are impossible. This comparative study underscores the importance of  choosing the 
right distribution for economic modelling to achieve high accuracy and reliable results.

Keywords

AIC, Economic Analysis, 
Geometric-Zero Truncated Poisson, 
Maximum Likelihood Estimation, 
Model Fit, MSE, Zero-Truncated 
Poisson Pareto

1 Department of  Statistics, Delta State Polytechnic, Otefe, Oghara, Delta State, Nigeria
2  Department of  Statistics, Chukwuemeka Odumegwu Ojukwu University, Uli, Anambra State, Nigeria
* Corresponding author’s e-mail: amaro4baya@yahoo.com

INTRODUCTION 
Considering the growing significance of  zero-truncated 
distributions in econometrics and economic forecasting. 
The Zero-truncated distributions, including the Zero-
Truncated Poisson (ZTP), Gamma Zero-Truncated 
Poisson (GZTP), and Zero-Truncated Poisson Pareto 
(ZTPP) distributions, have garnered attention for 
their ability to model count data that exclude zero 
observations, a common occurrence in economic 
datasets. Recent advancements in these models, 
such as the work by Niyomdecha and Srisuradetchai 
(2023), who introduced the complementary gamma 
zero-truncated Poisson (CGZTP) distribution, have 
demonstrated the effectiveness of  these distributions 
in modelling lifetime and economic data. The research 
by Niyomdecha et al. (2023) proposed the gamma zero-
truncated Poisson (GZTP) distribution, combining 
gamma and zero-truncated Poisson distributions 
using the minimum function. Their study explored the 
distribution’s characteristics, including hazard function 
and MLE estimation, with simulation tests confirming its 
adaptability in modelling lifetime data. Ngamkham and 
Panta (2023) addressed the estimation challenges of  the 
Zero-Truncated Poisson (ZTP) distribution, proposing 
a delta method for parameter estimation, which was 
applied to a real dataset on unrest events in southern 
Thailand. Similarly, the development of  the ZTPP 
distribution by Badr et al. (2023) has provided superior fit 
metrics, including Akaike Information Criteria (AIC) and 

Bayesian Information Criterion (BIC), when applied to 
economic datasets. The work by Panichkitkosolkul (2023) 
proposed the zero-truncated Poisson-Ishita distribution 
and evaluated various bootstrap methods for estimating 
confidence intervals, concluding that the simple bootstrap 
approach was most efficient for larger sample sizes. Irshad 
et al. (2023) developed the Lagrangian Intervened Poisson 
Distribution (LIPD), a generalized approach for over-
dispersed and under-dispersed datasets, and showcased 
its application using MLE and simulations. Akdogan et al. 
(2019) introduced the geometric-zero truncated Poisson 
(GZTP) distribution, demonstrating its usefulness for 
discrete systems with increasing hazard rates. Shukla et al. 
(2020) adapted the Poisson-Ishita distribution to the zero-
truncated Poisson-Ishita distribution (ZTPID), showing 
its superior fit in count data without zero values. Agarwal 
and Pandey (2024) introduced the inflated zero-truncated 
Poisson Ailamujia distribution (IMZTPAD), which 
demonstrated a better fit in modelling child mortality 
and genetic count data. Panichkitkosolkul (2024) further 
investigated the zero-truncated Poisson-Lindley (ZTPL) 
distribution and found that non-parametric bootstrap 
methods performed best for larger sample sizes. Pankaj 
et al. (2023) examined system reliability using a dual repair 
technique and regenerative methodologies, providing 
insights into improving system reliability. Ghosh et al. 
(2023) introduced a bivariate geometric distribution 
for negatively correlated count data, and Abbas (2023) 
proposed a bivariate generalized geometric distribution 



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(BGGD) for correlated count data, utilizing a Bayesian 
approach for analysis. Shang et al. (2023) presented a 
novel method for predicting aero-engine coaxiality using 
geometric distribution error modelling and deep learning, 
achieving high prediction accuracy.
These advancements underline the potential of  zero-
truncated models to enhance the precision and efficiency of  
economic analysis, particularly when dealing with data that 
exhibits truncation at zero, such as income, expenditure, 
or event counts. Despite these advancements, a notable 
gap remains in the comprehensive comparison of  these 
zero-truncated distributions in the context of  economic 
analysis. While individual studies have demonstrated the 
utility of  these distributions, there is a lack of  systematic 
evaluation across various economic datasets, with limited 
attention to performance metrics like AIC, BIC, and 
Mean Squared Error (MSE) in this domain. This study 
aims to fill this gap by introducing and comparing the 
mathematical properties of  the GZTP and ZTPP 
distributions, alongside their performance in economic 
modelling. By applying these distributions to real-world 
economic datasets, the study seeks to identify the most 
efficient distribution for modelling economic variables 
and predicting trends. Through this comparison, the study 
will provide valuable insights into the suitability of  these 
distributions for addressing specific economic modelling 
challenges, thereby contributing to the advancement of  
statistical methods in economic analysis. Hence, the aim 

of  this study was to advance the understanding of  zero-
truncated statistical distributions and their application 
in economic data modelling. The specific objectives are 
to: Introduce and describe the mathematical properties 
of  the GZTP and ZTPP distributions; compare the 
performance of  these distributions across datasets using 
metrics such as AIC, BIC, and MSE; and recommend 
the most efficient distribution for modelling economic 
variables and predicting trends.

MATERIALS AND METHODS
Source of  Data collection for the study 
The study utilized secondary data, sourced from reliable 
publications such as the Central Bank of  Nigeria’s 
Statistical Bulletin for 2021, included key economic 
indicators from 1989 to 2020. These indicators were Real 
Gross Domestic Product (RGDP), export and import 
goods, money supply, and Brent crude oil prices. This 
data set provided crucial insights into Nigeria’s economic 
performance, trade, monetary policy, and the impact of  
oil prices, forming the basis for detailed econometric 
analysis.

MATERIALS AND METHODS
Table 1 presents the Probability Mass Functions (PMFs) 
of  two advanced distributions the Geometric-Zero 
Truncated Poisson (GZTP) and Zero-Truncated Poisson 
Pareto (ZTPP)

Table 1: The PMF of  the GZTP and ZTPP distribution
S/No. Distribution PMF Source 
1. GZTP distribution e-λ((1-p)x-(1-p)x-1),/(1-e-λ),λ>0,0≤p≤1,x∈{1,2,3,…} Niyomdecha et al. (2023)

2 ZTPP distribution (λne-λ)/n!(1-e-λ) ,λ>0,n∈{0,1,2,3,…} Badr et al. (2023)

The Geometric-Zero Truncated Poisson (GZTP) and 
Zero-Truncated Poisson Pareto (ZTPP) distributions in 
Table 1 exhibit notable flexibility and unique properties, 
making them suitable for modelling diverse real-world 
phenomena. The GZTP distribution incorporates a 
geometric component through the parameter p, allowing 
it to capture overdispersion and varying probabilities for 
successive events, which is crucial in applications with 
decaying probabilities. Its range of  p between 0 and 1 
further enhances its adaptability to different datasets. On 
the other hand, the ZTPP distribution, characterized by its 
Poisson-based structure truncated at zero, is particularly 
effective in handling count data where zero occurrences 
are impossible. Its dependency solely on the rate parameter 
λ simplifies its application while maintaining robustness 
in capturing event frequencies. Both distributions are 
zero-truncated, addressing scenarios where non-zero 
occurrences are mandatory, and their closed-form 
PMFs enable straightforward parameter estimation and 
interpretation. These properties underscore their utility 
in fields such as ecology, reliability analysis, and actuarial 
science.

Parameter Estimation 
Parameter estimates for each distribution were derived 
using the Maximum Likelihood Estimation (MLE) 
method, which maximizes the likelihood function L(θ|x) 
given by:

where f(xi;θ)  represents the Probability Mass Function 
(PMF) of  the distribution, θ is the vector of  parameters, 
and xi are the observed data points (Casella & Berger, 
2002). MLE implementation was performed in the R 
programming language (R Core Team, 2023).

Model Performance Measures of  the distributions 
The model performance was evaluated using the 
following criteria:

i. Akaike Information Criterion (AIC):
AIC=-2ln(L)+2k                                                                                    (2)
where L Where likelihood of  the model, and k is the 
number of  estimated parameters (Akaike, 1974).

ii. Bayesian Information Criterion (BIC):
BIC=-2ln(L)+kln(n)                                                                           (3)



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Am. J. Appl. Stat. Econ. 4(1) 65-69, 2025

where n is the sample size (Schwarz, 1978).
iii. Mean Squared Error (MSE):

where y_i represents the observed values, and y ̂_i are the 
predicted values.
These metrics were computed for both the GZTP and 
ZTPP distributions across datasets, as shown in Table 3. 
To ensure a comprehensive evaluation, the average values 
of  AIC, BIC, and MSE for each distribution were also 
computed and summarized in Table 4.
Comparative Analysis

The comparative analysis focused on the ability of  the 
GZTP and ZTPP distributions to minimize AIC, BIC, 
and MSE. The ZTPP distribution demonstrated superior 
performance across the datasets, as evidenced by its 
consistently lower average AIC, BIC, and MSE values 
compared to the GZTP distribution. This suggests 
that the ZTPP distribution provides a better fit for the 
economic datasets under consideration.

RESULTS AND DISCUSSIONS
Table 2 provides an overview of  the descriptive statistics 
for key economic variables employed in the study, 
highlighting their central tendencies, variability, and 

Table 2: Descriptive Statistics of  dataset
Variable Mean St. Dev Minimum Median Maximum Skewness Kurtosis
RGDP 365 99 237 330 569 0.72 -0.72
Export_Goods 40 30 10 31 99 0.76 -0.78
Import_Goods 26 19 6 20 66 0.77 -0.78
Money_Supply(M2) 59694188 20216127 32262332 56782642 109951956 0.57 -0.36
Brent Crude (BRT) 47 32 11 35 133 0.89 -0.32

Table 3: Performance Comparison of  GZTP and ZTPP Distributions across Economic Datasets
Dataset Distributions Parameter estimates AIC BIC MSE
RGDP GZTP λ=0.1000,p=1.0000 -2258.61900 -2255.8700 148809.8000

ZTPP  λ=1.0000 -2150.6200 -2154.2200 143807.4000
Export_Goods GZTP λ=1.0000,p=0.1000 -158.61900 -155.8170 2488.2640

ZTPP  λ=33.5437 -791.1342 -791.733 2487.7510
Import_Goods GZTP λ=1.0000,p=0.1000 -158.6300 -155.8170 1009.2800

ZTPP  λ=25.4986 -541.2217 -543.8205 1009.6800
L o g ( M o n e y _
Supply)

GZTP λ=1.0000,p=0.1000 -167.8410 -164.273 318.6976
ZTPP  λ=17.8489 -202.3526 -198.5684 315.3769

BRT GZTP λ=1.0000,p=0.1000 -1088.2200 -1082.3400 3189.9470
ZTPP  λ=38.1595 -3427.3550 -3424.4130 3189.4930

distributional characteristics. The Real Gross Domestic 
Product (RGDP) exhibits a mean of  365 units with a 
standard deviation of  99, indicating moderate variability 
around the average, and ranges from 237 to 569 units, 
with a median of  330. The skewness of  0.72 suggests a 
slight rightward skew, while the negative kurtosis (-0.72) 
indicates a relatively flat distribution compared to the 
normal curve. Exported goods have a mean value of  40 
units and a standard deviation of  30, with values spanning 
from 10 to 99 units. Its skewness (0.76) and kurtosis (-0.78) 
reflect a similar pattern to RGDP, indicating asymmetry 
and platykurtic tendencies. Import goods average 26 
units with a standard deviation of  19, ranging from 6 to 
66 units, showing a slightly higher skewness (0.77) and 
comparable kurtosis (-0.78). The Money Supply (M2) 

variable, measured in millions, has a mean of  59,694,188 
and exhibits substantial variability (standard deviation of  
20,216,127), ranging from 32,262,332 to 109,951,956. 
Its skewness (0.57) and kurtosis (-0.36) indicate a mild 
rightward skew and a distribution closer to normal. Lastly, 
Brent Crude (BRT) prices show the highest variability, 
with a mean of  47, a standard deviation of  32, and a range 
from 11 to 133. Its skewness (0.89) highlights a more 
pronounced rightward skew, while the kurtosis (-0.32) 
remains slightly platykurtic. These statistics collectively 
reveal distinct patterns and variability across the variables, 
providing insights into their economic implications and 
informing further analyses. 
The result presented in Table 3 summarizes the performance 

metrics of  the GZTP and ZTPP distributions across 
various economic datasets, providing insights into their 
parameter estimates, model fit, and predictive accuracy. 
For RGDP, the GZTP distribution achieved a lower AIC 

(-2258.62) and BIC (-2255.87) compared to the ZTPP 
(-2150.62 and -2154.22, respectively), but the ZTPP 
showed a marginally lower Mean Squared Error (MSE) 
of  143,807.4 versus 148,809.8 for the GZTP. Similarly, 



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for Export_Goods, ZTPP demonstrated superior fit 
metrics (AIC = -791.13, BIC = -791.73) and a slightly 
better MSE (2487.75) than GZTP (AIC = -158.62, BIC = 
-155.82, MSE = 2488.26). In the Import_Goods dataset, 
GZTP slightly outperformed ZTPP in MSE (1009.28 
vs. 1009.68), but ZTPP had better AIC and BIC values 
(-541.22 and -543.82, respectively). For Log(Money_
Supply), ZTPP consistently outperformed GZTP with a 
lower MSE (315.38 vs. 318.70), AIC (-202.35 vs. -167.84), 
and BIC (-198.57 vs. -164.27). Lastly, in the Brent Crude 
(BRT) dataset, ZTPP exhibited a significant advantage 
in AIC (-3427.36) and BIC (-3424.41) while maintaining 
a marginally lower MSE (3189.49 vs. 3189.95). These 
results underscore the flexibility and robustness of  ZTPP 
across datasets, particularly in achieving better model fit 
and predictive accuracy. 
Table 4 presents the average performance metrics of  

Table 4: Comparative Analysis of  Average Performance 
Metrics for GZTP and ZTPP Distributions
Distributions Average 

AIC
Average 
BIC

Average 
MSE

GZTP -766.386 -762.823 31163.2
ZTPP -1422.54 -1422.55 30161.94

the Geometric-Zero Truncated Poisson (GZTP) and 
Zero-Truncated Poisson Pareto (ZTPP) distributions 
across multiple datasets, highlighting their comparative 
effectiveness. On average, the ZTPP distribution 
significantly outperformed the GZTP in terms of  model 
fit, as indicated by its lower Average AIC (-1422.54) 
and Average BIC (-1422.55), compared to the GZTP’s 
Average AIC (-766.39) and Average BIC (-762.82). 
Furthermore, the ZTPP demonstrated superior predictive 
accuracy, with a lower Mean Squared Error (MSE) 
of  30,161.94, as opposed to 31,163.2 for the GZTP. 
These results suggest that the ZTPP distribution offers 
a more robust modelling framework, with consistently 
better performance metrics across datasets, making it a 
preferable choice for applications requiring high accuracy 
and reliable fit.

CONCLUSION
This study highlights the flexibility and robustness of  the 
GZTP and ZTPP distributions in modelling real-world 
phenomena, particularly in economic datasets. The GZTP 
distribution, with its geometric component, is well-suited 
for capturing overdispersion and varying probabilities, 
making it adaptable to datasets with decaying probabilities. 
Conversely, the ZTPP distribution, relying solely on the 
rate parameter λ, excels in handling count data where 
zero occurrences are impossible, providing a simpler 
yet effective approach. The descriptive statistics of  key 
economic variables, such as Real GDP (RGDP), Exported 
Goods, Import Goods, Money Supply (M2), and Brent 
Crude (BRT), reveal distinct patterns and variability, which 
inform further analyses. The comparative performance 
of  the two distributions indicates that, on average, the 

ZTPP outperforms the GZTP in terms of  model fit, 
with significantly lower AIC, BIC, and Mean Squared 
Error (MSE) values across multiple datasets. Specifically, 
the ZTPP achieved lower AIC and BIC values, such as 
-1422.54 and -1422.55, respectively, compared to the 
GZTP’s -766.39 and -762.82, and demonstrated superior 
predictive accuracy with an average MSE of  30,161.94 
versus 31,163.2 for the GZTP. These findings underscore 
the ZTPP distribution’s superior capability in providing 
reliable model fit and predictive performance, making it 
a more suitable choice for applications that demand high 
accuracy in economic modelling.

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