A Seasonal Transmuted Geometric INAR Process: Modeling and Applications in Count Time Series

Abstract

In this paper, the authors introduce the transmuted geometric integer-valued autoregressive model with periodicity, designed specifically to analyze epidemiological and public health time series data. The model uses a transmuted geometric distribution as a marginal distribution of the process. It also captures varying tail behaviors seen in disease case counts and health data. Key statistical properties of the process, such as conditional mean, conditional variance, etc., are derived, along with estimation techniques like conditional least squares and conditional maximum likelihood. The ability to provide k-step-ahead forecasts makes this approach valuable for identifying disease trends and planning interventions. Monte Carlo simulation studies confirm the accuracy and reliability of the estimation methods. The effectiveness of the proposed model is analyzed using three real-world public health datasets: weekly reported cases of Legionnaires’ disease, syphilis, and dengue fever.

1. Introduction

Statistical time series models are widely used to understand and forecast disease incidence, supporting early detection of unusual trends and aiding in public health planning. Their ability to predict future counts is especially useful in surveillance systems.

Count time series models are essential in epidemiology, particularly for monitoring disease incidence, hospital admissions, etc. These models are well-suited for non-negative integer values that appear frequently in public health and related fields. The integer-valued autoregressive (INAR) model is commonly used to describe such data. Steutel and van Harn [1] introduced the binomial thinning operator, which forms the basis of the classical INAR(1) model proposed by McKenzie [2] and Al-Osh and Alzaid [3]. Extensions such as autoregressive moving average models with negative binomial and geometric marginals were also developed by McKenzie [4], along with binomial ARMA models by Al-Osh and Alzaid [5]. Applications in medicine were discussed by Cardinal et al. [6].

Over time, several variants of the INAR model have been proposed. Freeland and McCabe [7,8] and Park and Kim [9] focused on low-count and diagnostic issues. Periodic structures were incorporated by Monteiro et al. [10] and later by Bourguignon et al. [11]. Kim and Park [12], Maiti and Biswas [13,14], and Maiti et al. [15,16] studied coherent forecasting for zero-inflated and overdispersed data. Spatial and seasonal modeling in infectious disease data were explored by Held and Paul [17], while compound Poisson INAR models were discussed by Schweer and Weiss [18].

Additional developments include models with threshold-based random coefficient INAR models by Li et al. [19] and seasonal extensions like those by Tian et al. [20] and Awale et al. [21]. Almuhayfith et al. [22] applied these ideas to influenza data.

Other studies have proposed models using negative binomial [23], geometric [24], shifted geometric [25], model with deflation or inflation of zeros [26]. Manaa and Souakri [27] proposed a zero-inflated Poisson INAR(1) model with periodic structure to handle seasonality and excess zeros in count data. The seasonal count time series model which can take care of negative dependence is developed by Kong and Lund [28]. Coherent forecasting for the overdispersed novel geometric AR(1) (NoGeAR(1)) model is discussed in [29]. Models handling underdispersion and equidispersion are covered in Bourguignon and Weiß [30], and a simple INAR(1) model with zero-distorted generalized geometric innovations was proposed by Kang et al. [31]. A comprehensive overview of modeling approaches for count time series is provided in Davis et al. [32], including models with fixed marginal distributions, state-space methods, multivariate extensions, and Bayesian techniques.

Although many distributional forms have been explored, traditional count models may still fall short in capturing asymmetry or heavy tails. The quadratic rank transmutation (QRT) technique [33] enhances flexibility by modifying standard distributions. The transmuted geometric (TGD) distribution introduced by Chakraborty and Bhati [34] adds a transmutation parameter $\alpha$ to adjust skewness and covers both underdispersion and overdispersion. This paper proposes a seasonal transmuted geometric INAR model that builds on these ideas to accurately model seasonal variation in count data time series.

The Transmuted Geometric Distribution

Let X be a non-negative random variable having transmuted geometric distribution (TGD) [34] with parameters $\alpha$ and q, then its probability mass function (pmf) is given as

$$P(X=x)=(1-\alpha )q^x(1-q)+\alpha (1-q^2)q^{2x},x=0,1,2\dots ,0<q<1,-1\le \alpha \le 1.$$

(1)

This model is a finite mixture of two geometric distributions: The first component follows a geometric distribution with parameter q ($GD\left(q\right)$) and mixing proportion $(1-\alpha )$. The second component follows a geometric distribution with the parameter $q^2$ ($GD\left(q^2\right)$) and the mixing proportion $\alpha$.

Also, we can see this distribution as

$$P(X=x)={\displaystyle \frac{w\left(X\right)\times GD\left(q\right)}{E\left[w\right(X\left)\right]}},$$

which represents a weighted geometric distribution with a weight function given by

$$w\left(X\right)=(1-\alpha )+\alpha (1+q)q^x.$$

The expectation of $w\left(X\right)$ is computed as

$$E\left[w\left(X\right)\right]=\sum _{x=0}^{\infty }\left[(1-\alpha )+\alpha (1+q)q^x\right](1-q)q^x=1.$$

Note that in (1), when $\alpha =0$, the transmuted geometric distribution reduces exactly to the ordinary geometric distribution. The extra parameter $\alpha$ effectively mixes the geometric distribution with a second weighted component, so by adjusting $\alpha$ we can widen or narrow the tail of the distribution to capture under/overdispersion in the data. It can model a variety of count data patterns by changing $\alpha$.

The probability generating function (pgf) of X is given by

$${\mathrm{\Phi }}_X\left(\gamma \right)={\displaystyle \frac{(1-q)(1-\alpha q(-1+\gamma )-q^2\gamma )}{(1-q\gamma )(1-q^2\gamma )}},\left|q\gamma \right|<1.$$

Using the above pgf, we can derive the mean and variance of X as follows:

$${\mathrm{\Phi }}_X^{\prime }\left(1\right)={\displaystyle \frac{q(1-\alpha +q)}{(1-q^2)}},{\mathrm{\Phi }}_X^{″}\left(1\right)={\displaystyle \frac{2q^2[{(1+q)}^2-\alpha (1+2q)]}{{(1-q^2)}^2}}.$$

Here, ${\mathrm{\Phi }}_X^{\prime }\left(1\right)$ and ${\mathrm{\Phi }}_X^{″}\left(1\right)$ are the first and second derivatives of the pgf of X evaluated at $\gamma =1$.

The variance of X is given as

$$\mathrm{Var}\left(X\right)={\mathrm{\Phi }}_X^{″}\left(1\right)+{\mathrm{\Phi }}_X^{\prime }\left(1\right)-{\left({\mathrm{\Phi }}_X^{\prime }\left(1\right)\right)}^2.$$

Using this, the mean and variance of X are

$${\mu }_x=E\left(X\right)={\displaystyle \frac{q(1-\alpha +q)}{(1-q^2)}}\mathrm{and}{\sigma }_x^2=Var\left(X\right)={\displaystyle \frac{q[-{\alpha }^{2}q+{(1+q)}^2-\alpha (1+q^2)]}{{(1-q^2)}^2}},$$

respectively.

The variances of geometric and transmuted geometric distributions are given as

$$Var\left(X_{Geom}\right)={\displaystyle \frac{q}{{(1-q)}^2}}\mathrm{and}Var\left(X_{TGD}\right)={\displaystyle \frac{q[-{\alpha }^{2}q+{(1+q)}^2-\alpha (1+q^2)]}{{(1-q^2)}^2}}.$$

(2)

Skewness (Sk) and kurtosis (Kurt) of the geometric and transmuted geometric distributions are given as

$$\mathrm{Sk}\left(X_{Geom}\right)={\displaystyle \frac{1+q}{\sqrt{q}}},\mathrm{and}\mathrm{Kurt}\left(X_{Geom}\right)=7+{\displaystyle \frac{1}{q}}+q$$

(3)

$$\mathrm{Sk}\left(X_{TGD}\right)={\displaystyle \frac{q\left(2{\alpha }^3{q}^2-{(1+q)}^4+3{\alpha }^2(q+q^3)+\alpha (1+4q^2+q^4)\right)}{{\left[q{(1+q)}^2-\alpha q(1+q^2)-{\alpha }^2{q}^2\right]}^{\frac{3}{2}}}}.$$

(4)

$${\displaystyle \mathrm{Kurt}\left(X_{TGD}\right)=-3-\frac{12q{(1+q)}^2}{{\left(\alpha -1+q[q(\alpha -1)+{\alpha }^2-2)\right]}^2}-\frac{(1+q(4+q\left)\right)(1+q(8+q\left)\right)}{q(\alpha -1+q[q(\alpha -1)+{\alpha }^2-2])}.}$$

(5)

In Figure 1, we compare the variance, skewness, and kurtosis of the transmuted geometric distribution (red) with the baseline geometric distribution (blue) for different values of $\alpha$ ($-0.99$, $0.50$, $0.99$). The TGD exhibits a higher variance than the geometric distribution for positive $\alpha$ ($0.50$, $0.99$). For negative $\alpha$ ($-0.99$), it aligns closely with the geometric distribution. Skewness decreases as q increases, indicating a reduced asymmetry. Skewness remains higher for the TGD at positive $\alpha$, especially for small q, and remains close to the geometric distribution for negative $\alpha$. Kurtosis also decreases as q increases, with the TGD showing a higher peakedness for positive $\alpha$ and a closer resemblance to the geometric distribution for negative $\alpha$. These results show that $\alpha$ controls the spread (variability) and skewness (asymmetry).

We examine the quantiles of the geometric distribution $GD\left(q\right)$ and the transmuted geometric distribution $TGD(q,\alpha )$ with $q=0.80$:

• For $\alpha =-0.98$, the $99.99$th quantile is 41 for $GD$ and 44 for $TGD$, indicating a heavier tail for the transmuted geometric distribution.
• For $\alpha =0.98$, the $99.99$th quantile is 41 for $GD$ and 24 for $TGD$, showing a lighter tail for the transmuted geometric distribution.

Thus, the transmutation parameter $\alpha$ controls the tail behavior of the distribution, with negative $\alpha$ leading to heavier tails and positive $\alpha$ to lighter tails.

Figure 2 presents simulated data for a transmuted geometric distribution (TGD) and its geometric counterpart ($\alpha =0$), using a fixed value of $q=0.6$. The figure includes plots for three different values of the transmutation parameter $\alpha$: −0.99, 0.50, and 0.99. Each line highlights the distinct behavior of the geometric and transmuted geometric distributions under varying $\alpha$. The plots clearly capture how the parameter $\alpha$ influences the shape and spread of the pmf. For negative values of $\alpha$ (plots (a) and (b)), the distribution becomes more dispersed, with higher probabilities for larger values of x. In contrast, positive values of $\alpha$ (plots (c) and (d)) result in a more concentrated distribution, with a sharp peak near smaller values of x. The neutral case ($\alpha =0$, plot (e)) represents the geometric distribution (baseline). This shows that $\alpha$ controls the concentration and dispersion of probabilities in the distribution.

The paper is organized as follows. Section 2 introduces the TGDINAR(1) model and discusses its basic properties. The estimation of parameters is presented in Section 3. The analysis of real-world datasets is given in Section 4. Section 5 concludes the paper.

2. Construction and Properties of the Process

In this section, we introduce a transmuted geometric INAR process with the period ‘s’ to handle stationary non-negative seasonal integer-valued time series. This model is based on the binomial thinning operator ‘∗’, which is defined as

$$\lambda \ast X=\sum _{i=1}^X{W}_i,$$

where $W_i$’s are i.i.d. Bernoulli random variables with parameter $\lambda$. The pgf of W is ${\mathrm{\Phi }}_W\left(\gamma \right)=(1-\lambda +\lambda \gamma ),\left|\gamma \right|\le 1.$ The random variables $\left\{W_i\right\}$ are mutually independent of X. The transmuted geometric INAR model with period ‘s’ is defined as follows.

Definition 1.

A discrete-time non-negative integer-valued stochastic process $\left\{X_t\right\}$ is said to be a transmuted geometric INAR process with period s (TGDINAR(1)s) based on a binomial thinning operator if the following conditions are satisfied:

$$X_t=\lambda \ast X_{t-s}+{ϵ}_t,0\le \lambda <1;t\ge s.$$

(6)

1. 

$\left\{X_t\right\}$ is a sequence of non-negative integer-valued random variables, each marginally following a transmuted geometric distribution with parameters $(q,\alpha )$.

Remark: When $s=1$ and $\alpha =0$, this model reduces to the first-order geometric INAR model (GINAR(1)) [4].

2. 

$s\in \mathbb{N}$ denotes the seasonal period (with $s\ge 1$), where $\mathbb{N}$ is the set of positive integers.

3. 

$\left\{{ϵ}_t\right\}$ is a sequence of innovation random variables (i.i.d.) with non-negative integer values, independent of past values of $\left\{X_t\right\}$ and the thinning variables.

4. 

The thinning variables involved in $\lambda \ast X_{t-s}$ are mutually independent and are also independent of the random variables $X_{t-s},X_{t-2s},\dots$ as well as the innovation sequence $\left\{{ϵ}_t\right\}$.

Using the pgf of $X_t$ and W, the pgf of $\lambda \ast X_{t-s}$ is given as

$${\mathrm{\Phi }}_{\lambda \ast X_{t-s}}\left(\gamma \right)={\mathrm{\Phi }}_{X_{t-s}}\left({\mathrm{\Phi }}_W\left(\gamma \right)\right)={\displaystyle \frac{(1-q)[1-\alpha q\lambda (\gamma -1)-q^2(1-\lambda +\lambda \gamma )]}{[1-q(1-\lambda +\lambda \gamma )][1-q^2(1-\lambda +\lambda \gamma )]}}.$$

(7)

In addition, the pgf of the random variable ${ϵ}_t$ is given as

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{{ϵ}_t}\left(\gamma \right)& ={\displaystyle \frac{{\mathrm{\Phi }}_{X_t}\left(\gamma \right)}{{\mathrm{\Phi }}_{\lambda \ast X_{t-s}}\left(\gamma \right)}}\hfill \\ \hfill & ={\displaystyle \frac{(1-\alpha q(\gamma -1)-q^2\gamma )(1-q(1-\lambda (1-\gamma )))[1-q^2(1-\lambda (1-\gamma ))]}{(1-q\gamma )(1-q^2\gamma )[1-\alpha q\lambda (\gamma -1)-q^2(1-\lambda (1-\gamma ))]}},\left|q\gamma \right|<1.\hfill \end{array}$$

The pgf of the innovation term ${ϵ}_t$ can also be written as

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{{ϵ}_t}\left(\gamma \right)& =\lambda +(1-\lambda )[{\displaystyle \frac{(1-q)(1-\alpha )(1+q(1-\lambda \left)\right)}{(1-q\gamma )(1+q(1-\lambda )-\alpha \lambda )}}+{\displaystyle \frac{\alpha (1-q^2)((1-\lambda )q-\lambda )}{(1-\gamma q^2)((1-\lambda )q-\alpha \lambda )}}\hfill \\ \hfill & +{\displaystyle \frac{(1-\alpha )\alpha \lambda (1-q^2)}{(1+q(1-\lambda )-\alpha \lambda )((1-\lambda )q-\alpha \lambda )(1-q^2+q(q+\alpha )(1-\gamma )\lambda )}}].\hfill \end{array}$$

(8)

The innovation term is distributed as a degenerated 0, $\mathrm{geometric}\left(q\right),\mathrm{geometric}\left(q^2\right),\mathrm{geometric}\left({\displaystyle \frac{\lambda q(\alpha +q)}{q^2-\lambda q(\alpha +q)-1}}\right)$ with probabilities $\lambda ,{\displaystyle \frac{(1-\alpha )(1-\lambda )\left(\right(1-\lambda )q+1)}{1+\alpha \lambda +(1-\lambda )q}},$${\displaystyle \frac{(1-\alpha )\alpha (1-\lambda )}{1+\alpha \lambda +(1-\lambda )q}},{\displaystyle \frac{\lambda q(\alpha +q)}{1-q^2+\lambda q(\alpha +q)}}$, respectively.

Hence, the pmf of the innovations $\left\{{ϵ}_t\right\}$ is given as follows:

$$\begin{array}{cc}\hfill P({ϵ}_t=ϵ)& =\lambda {\mathbb{I}}_{\left\{0\right\}}+{\displaystyle \frac{(1-q)q^{ϵ}\left[(1-\alpha )(1-\lambda )((1-\lambda )q+1)\right]}{1-\alpha \lambda +(1-\lambda )q}}\hfill \\ \hfill & +{\displaystyle \frac{(1-\alpha )\alpha (1-\lambda )\lambda }{(-\alpha \lambda +(1-\lambda )q+1)\left(\right(1-\lambda )q-\alpha \lambda )}}\left(1-{\displaystyle \frac{\lambda q(\alpha +q)}{1-q^2+\lambda q(\alpha +q)}}\right)\times \hfill \\ \hfill & {\left({\displaystyle \frac{\lambda q(\alpha +q)}{1-q^2+\lambda q(\alpha +q)}}\right)}^{ϵ}+{\displaystyle \frac{(1-q^2)q^{2ϵ}\left(\alpha (1-\lambda )((1-\lambda )q-\lambda )\right)}{(1-\lambda )q-\alpha \lambda }}.\hfill \end{array}$$

(9)

Alternatively, the pmf of ${ϵ}_t$ can be written as

$$P({ϵ}_t=ϵ)=\left\{\begin{array}{cc}{\displaystyle \frac{(\alpha q+1)(1-(1-\lambda )q)(1-(1-\lambda )q^2)}{1-(1-\lambda )q^2+\alpha \lambda q}},\hfill & ϵ=0,\hfill \\ {\displaystyle \frac{(1-q)q^{ϵ}\left[(1-\alpha )(1-\lambda )((1-\lambda )q+1)\right]}{1-\alpha \lambda +(1-\lambda )q}}+{\displaystyle \frac{(1-\alpha )\alpha (1-\lambda )\lambda }{(1-\alpha \lambda +(1-\lambda \left)q\right)\left(\right(1-\lambda )q-\alpha \lambda )}}\hfill \\ \times \left(1-{\displaystyle \frac{\lambda q(\alpha +q)}{1-q^2+\lambda q(\alpha +q)}}\right){\left({\displaystyle \frac{\lambda q(\alpha +q)}{1-q^2+\lambda q(\alpha +q)}}\right)}^{ϵ}\hfill \\ +{\displaystyle \frac{(1-q^2)q^{2ϵ}\left(\alpha (1-\lambda )((1-\lambda )q-\lambda )\right)}{(1-\lambda )q-\alpha \lambda }},\hfill & ϵ=1,2,3,\dots \hfill \end{array}\right.$$

(10)

Thus, the mean and variance of the innovation sequence $\left\{{ϵ}_t\right\}$, respectively, are given as

$$\begin{array}{c}\hfill {\mu }_{ϵ}=E\left({ϵ}_t\right)={\displaystyle \frac{q(1-\lambda )(1-\alpha +q)}{(1-q^2)}},\end{array}$$

and

$$\begin{array}{c}\hfill {\sigma }_{ϵ}^2=V\left({ϵ}_t\right)={\displaystyle \frac{q(\lambda -1)}{{(1-q^2)}^2}}\left[\alpha +q{\alpha }^2(1+\lambda )+q^2\alpha (1+2\lambda )-{(1+q)}^2(1+\lambda q)\right].\end{array}$$

Theorem 1.

The TGDINAR(1) process has a unique stationary solution given by

$$\begin{array}{c}\hfill X_t\stackrel{d}{=}\sum _{j=0}^{\infty }{\lambda }^{\left(j\right)}\ast {ϵ}_{t-js}.\end{array}$$

The symbol $\stackrel{d}{=}$ stands for equality in distribution, when $j\ge 1$, ${\lambda }^{\left(j\right)}\triangleq \underset{\underset{j\mathrm{times}}{⏟}}{\lambda \ast \lambda \ast \cdots \ast \lambda }$, and ${\lambda }^{\left(0\right)}\ast {\epsilon }_t\triangleq {\epsilon }_t$. For all $t\in {\mathbb{N}}_0$, the infinite sum is interpreted as the limit in probability of its corresponding finite sums.

The proof of this theorem is deferred to the Appendix A.1.

Theorem 2.

One-step-ahead conditional distribution of the process $\left\{X_t\right\}$ defined in (6) is given by

$$\begin{array}{c}P(X_t=y\mid X_{t-s}=x)=\hfill \\ \left\{\begin{array}{c}{\displaystyle \frac{{(1-\lambda )}^x(-1-\alpha q)AC}{R},ifx\ge 0,y=0,}\hfill \\ \\ \\ {\displaystyle \frac{{(1-\lambda )}^x}{R}[-\lambda q(1+\alpha q)(Aq+C)-AC(1+\alpha q)(q+q^2)+AC(\alpha q+q^2)}\hfill \\ {\displaystyle -AC(1+\alpha q)x{\lambda }_1+\frac{AC(1+\alpha q)(\alpha \lambda q+\lambda q^2)}{R}],ifx\ge 0,y=1,}\hfill \\ \\ {\displaystyle \frac{{(1-\lambda )}^x{q}^{2}AC}{R}\left[(\alpha +q)(1+q)-(1+\alpha q)(1+q+q^2)\right]+\frac{{(1-\lambda )}^{x}q}{R}}\hfill \\ {\displaystyle \left[\alpha +q+(-1-\alpha q)(1+q)\right]\times \left[\lambda q(Aq+C)+ACx{\lambda }_1-\frac{AC(\alpha \lambda q+\lambda q^2)}{R}\right]}\hfill \\ {\displaystyle +\frac{{(1-\lambda )}^x(-1-\alpha q)}{R}\{{\lambda }^2{q}^3+\lambda q(Aq+C)\times \left(\frac{-(\alpha \lambda q+\lambda q^2)}{R}+{\lambda }_{1}x\right)}\hfill \\ {\displaystyle +AC\left[\frac{{(\alpha \lambda q+\lambda q^2)}^2}{R^2}-\frac{{\lambda }_1(\alpha \lambda q+\lambda q^2)x}{R}+{\lambda }_1^2\left(\genfrac{}{}{0pt}{}{x}{2}\right)\right]\},ifx\ge 0,y=2,}\hfill \\ \\ \\ {\displaystyle \frac{{(1-\lambda )}^x{q}^{y}AC}{R}\left[(\alpha +q)\sum _{j=0}^{y-1}{q}^j+(-1-\alpha q)\sum _{j=0}^y{q}^j\right]+\frac{{(1-\lambda )}^x{q}^{y-1}}{R}[(\alpha +q)\sum _{j=0}^{y-2}{q}^j}\hfill \\ {\displaystyle -(1+\alpha q)\sum _{j=0}^{y-1}{q}^j]\times \left[\lambda q(Aq+C)+AC\left(\frac{-(\alpha \lambda q+\lambda q^2)}{R}+{\lambda }_{1}x\right)\right]+\sum _{i=2}^{y-1}\frac{{(1-\lambda )}^x{q}^{y-i}}{R}\times }\hfill \\ {\displaystyle \left((\alpha +q)\sum _{j=0}^{y-i-1}{q}^j+(-1-\alpha q)\sum _{j=0}^{y-i}{q}^j\right)\times [{\lambda }^2{q}^3\sum _{j=0}^{i-2}{(-1)}^{i-j-2}{\lambda }_1^j{\left(\frac{\alpha \lambda q+\lambda q^2}{R}\right)}^{i-j-2}\left(\genfrac{}{}{0pt}{}{x}{j}\right)}\hfill \\ {\displaystyle +\left(\lambda q^{2}A+\lambda qC\right)\times \sum _{j=0}^{i-1}{(-1)}^{i-j-1}{\lambda }_1^j{\left(\frac{\alpha \lambda q+\lambda q^2}{R}\right)}^{i-j-1}\left(\genfrac{}{}{0pt}{}{x}{j}\right)}\hfill \\ {\displaystyle +AC\sum _{j=0}^i{\lambda }_1^j{\left(\frac{\alpha \lambda q+\lambda q^2}{R}\right)}^{i-j}\left(\genfrac{}{}{0pt}{}{x}{j}\right){(-1)}^{i-j}]-\frac{{(1-\lambda )}^x(1+\alpha q)}{R}\times }\hfill \\ {\displaystyle \{{\lambda }^2{q}^3\sum _{j=0}^{y-2}{(-1)}^{y-j-2}{\lambda }_1^j{\left(\frac{\alpha \lambda q+\lambda q^2}{R}\right)}^{y-j-2}\left(\genfrac{}{}{0pt}{}{x}{j}\right)+\left(\lambda q^{2}A+\lambda qC\right)\times }\hfill \\ {\displaystyle \sum _{j=0}^{y-1}{(-1)}^{y-j-1}{\lambda }_1^j{\left(\frac{\alpha \lambda q+\lambda q^2}{R}\right)}^{y-j-1}\left(\genfrac{}{}{0pt}{}{x}{j}\right)+AC\sum _{j=0}^y{(-1)}^{y-j}{\lambda }_1^j{\left(\frac{\alpha \lambda q+\lambda q^2}{R}\right)}^{y-j}\left(\genfrac{}{}{0pt}{}{x}{j}\right)\},}\hfill \\ ifx\ge 0,y\ge 3,\hfill \end{array}\right.\hfill \end{array}$$

where $A=-1+q-\lambda q$, $C=-1+q^2-\lambda q^2$, $R=-1-\alpha \lambda q-(\lambda -1)q^2$, ${\lambda }_1={\displaystyle \frac{\lambda }{1-\lambda }}$.

One-step-ahead conditional mean and variance of the process $\left\{X_t\right\}$ are given as

$${\displaystyle E\left(X_t\right|X_{t-s})=\lambda X_{t-s}+\frac{q(1-\lambda )(1-\alpha +q)}{(1-q^2)},}$$

and

$${\displaystyle Var\left(X_t\right|X_{t-s})=\lambda (1-\lambda )X_{t-s}+\frac{q(\lambda -1)}{{(1-q^2)}^2}\left[\alpha +q{\alpha }^2(1+\lambda )+q^2\alpha (1+2\lambda )-{(1+q)}^2(1+\lambda q)\right].}$$

Using recursion in Equation (6), we can write

$$X_{t+k}={\lambda }^{\left(m\right)}\ast X_{t-r}+\sum _{j=0}^{m-1}{\lambda }^{\left(j\right)}\ast {ϵ}_{t-js}$$

(11)

where $m=⌈{\displaystyle \frac{k}{s}}⌉$ and $r=ms-k$, where $\lceil ·\rceil$ denotes the ceiling function (i.e., the smallest integer greater than or equal to ‘.’.

The proof of this theorem is deferred to the Appendix A.2.

Theorem 3.

Let $\left\{X_t\right\}$ be a $TGDINAR{\left(1\right)}_s$ process defined as in (6). Then, the k-step-ahead conditional pmf of $X_{t+k}$ given $X_{t-r}$ is given by

$$\begin{array}{c}P(X_{t+k}=y\mid X_{t-r}=x)=\hfill \\ \left\{\begin{array}{c}{\displaystyle \frac{{(1-{\lambda }^m)}^x(-1-\alpha q)AC}{R},\mathrm{if}x\ge 0,y=0,}\hfill \\ \\ \\ {\displaystyle \frac{{(1-{\lambda }^m)}^x}{R}[-{\lambda }^{m}q(1+\alpha q)(Aq+C)-AC(1+\alpha q)(q+q^2)+AC(\alpha q+q^2)}\hfill \\ {\displaystyle -AC(1+\alpha q)x{\lambda }_1+\frac{AC(1+\alpha q)(\alpha {\lambda }^{m}q+{\lambda }^m{q}^2)}{R}],\mathrm{if}x\ge 0,y=1,}\hfill \\ \\ {\displaystyle \frac{{(1-{\lambda }^m)}^x{q}^{2}AC}{R}\left[(\alpha +q)(1+q)-(1+\alpha q)(1+q+q^2)\right]+\frac{{(1-{\lambda }^m)}^{x}q}{R}\left[\alpha +q+(-1-\alpha q)(1+q)\right]\times }\hfill \\ {\displaystyle \left[{\lambda }^{m}q(Aq+C)+ACx{\lambda }_1-\frac{AC{\lambda }^{m}q(\alpha +q)}{R}\right]-\frac{{(1-{\lambda }^m)}^x(1+\alpha q)}{R}\{{\lambda }^{2m}{q}^3+{\lambda }^{m}q(Aq+C)\times }\hfill \\ {\displaystyle \left(\frac{-{\lambda }^{m}q(\alpha +q)}{R}+{\lambda }_{1}x\right)+AC\left[\frac{{\lambda }^{2m}{q}^2{(\alpha +q)}^2}{R^2}-\frac{{\lambda }_1{\lambda }^{m}q(\alpha +q)x}{R}+{\lambda }_1^2\left(\genfrac{}{}{0pt}{}{x}{2}\right)\right]\},\mathrm{if}x\ge 0,y=2,}\hfill \\ \\ {\displaystyle \frac{{(1-{\lambda }^m)}^x{q}^{y}AC}{R}\left[(\alpha +q)\sum _{j=0}^{y-1}{q}^j-(1+\alpha q)\sum _{j=0}^y{q}^j\right]+\frac{{(1-{\lambda }^m)}^x{q}^{y-1}}{R}\left[(\alpha +q)\sum _{j=0}^{y-2}{q}^j-(1+\alpha q)\sum _{j=0}^{y-1}{q}^j\right]\times }\hfill \\ {\displaystyle \left[{\lambda }^{m}q(Aq+C)+AC\left(\frac{-{\lambda }^{m}q(\alpha +q)}{R}+{\lambda }_{1}x\right)\right]+\sum _{i=2}^{y-1}\frac{{(1-{\lambda }^m)}^x{q}^{y-i}}{R}\left((\alpha +q)\sum _{j=0}^{y-i-1}{q}^j-(1+\alpha q)\sum _{j=0}^{y-i}{q}^j\right)}\hfill \\ {\displaystyle \times [{\lambda }^{2m}{q}^3\sum _{j=0}^{i-2}{(-1)}^{i-j-2}{\lambda }_1^j{\left(\frac{{\lambda }^{m}q(\alpha +q)}{R}\right)}^{i-j-2}\left(\genfrac{}{}{0pt}{}{x}{j}\right)+{\lambda }^{m}q(qA+C)\times \sum _{j=0}^{i-1}{(-1)}^{i-j-1}{\lambda }_1^j\times }\hfill \\ {\displaystyle {\left(\frac{{\lambda }^{m}q(\alpha +q)}{R}\right)}^{i-j-1}\left(\genfrac{}{}{0pt}{}{x}{j}\right)+AC\sum _{j=0}^i{\lambda }_1^j{\left(\frac{{\lambda }^{m}q(\alpha +q)}{R}\right)}^{i-j}\left(\genfrac{}{}{0pt}{}{x}{j}\right){(-1)}^{i-j}]-\frac{{(1-{\lambda }^m)}^x(1+\alpha q)}{R}\times }\hfill \\ {\displaystyle \{{\lambda }^{2m}{q}^3\sum _{j=0}^{y-2}{(-1)}^{y-j-2}{\lambda }_1^j{\left(\frac{{\lambda }^{m}q(\alpha +q)}{R}\right)}^{y-j-2}\left(\genfrac{}{}{0pt}{}{x}{j}\right)+{\lambda }^{m}q(qA+C)\times }\hfill \\ {\displaystyle \sum _{j=0}^{y-1}{(-1)}^{y-j-1}{\lambda }_1^j{\left(\frac{{\lambda }^{m}q(\alpha +q)}{R}\right)}^{y-j-1}\left(\genfrac{}{}{0pt}{}{x}{j}\right)+AC\sum _{j=0}^y{(-1)}^{y-j}{\lambda }_1^j{\left(\frac{{\lambda }^{m}q(\alpha +q)}{R}\right)}^{y-j}\left(\genfrac{}{}{0pt}{}{x}{j}\right)\},}\hfill \\ ifx\ge 0,y\ge 3,\hfill \end{array}\right.\hfill \end{array}$$

where $A=-1+q-{\lambda }^{m}q$, $C=-1+q^2-{\lambda }^m{q}^2$, $R=-1-\alpha {\lambda }^{m}q-({\lambda }^m-1)q^2$, ${\lambda }_1={\displaystyle \frac{{\lambda }^m}{(1-{\lambda }^m)}}$.

The proof of this theorem is deferred to the Appendix A.3.

Corollary 1.

Suppose $X_{t+k}$ and $X_{t-r}$ satisfy the process given in (6), and $k\in \mathbb{N}$. Then we have the following results:

1. 

The conditional expectation of $X_{t+k}$ given $X_{t-r}$ is

$$E\left(X_{t+k}\right|X_{t-r})={\lambda }^m{X}_{t-r}+{\displaystyle \frac{q(1-{\lambda }^m)(1-\alpha +q)}{(1-q^2)}}.$$

(12)

when k$\to \infty ,$ implies $E\left(X_{t+k}\right|X_{t-r})$$\to {\mu }_x,$ which is the unconditional mean.

2. 

The conditional variance of $X_{t+k}$ given $X_{t-r}$ is

$$\begin{array}{cc}\hfill Var\left(X_{t+k}\right|X_{t-r})& ={\displaystyle \frac{(1-{\lambda }^m)}{{(1-q^2)}^2}}[(1-\alpha )q-(\alpha -1)(1+2{\lambda }^m)q^3+{\lambda }^m{X}_{t-r}\hfill \\ \hfill & +{\lambda }^m{q}^4(1+X_{t-r})-q^2\left(-2+{\alpha }^2(1+{\lambda }^m)+{\lambda }^m(2X_{t-r}-1)\right)].\hfill \end{array}$$

(13)

when k$\to \infty ,$ implies $Var\left(X_{t+k}\right|X_{t-r})$$\to {\sigma }_x^2,$ which is the unconditional variance.

3. 

The autocovariance function of the process is given by

$$\begin{array}{cc}\hfill \mathit{Cov}(X_t,X_{t-ks})& ={\displaystyle \frac{{\lambda }^{k}q\left[-{\alpha }^{2}q+{(1+q)}^2-\alpha (1+q^2)\right]}{{(1-q^2)}^2}}\hfill \\ \hfill & ={\lambda }^k{\sigma }_x^2,k=0,1,2,\dots \hfill \end{array}$$

(14)

Proof. 

The proof of this corollary is deferred to the Appendix B. □

3. Estimation of Parameters

In this section, two methods for estimating the parameters of the TGDINAR(1)s process with seasonal period s are discussed: the conditional least squares (CLS) method and the conditional maximum likelihood (CML) method.

3.1. Conditional Least Squares (CLS) Estimation

To obtain the conditional least squares (CLS) estimators of the parameters $\theta ={(\alpha ,q,\lambda )}^{\prime }$, we differentiate the objective function $Q_n\left(\theta \right)$ with respect to each parameter ($\alpha$, q, and $\lambda$) and equate the resulting expressions to zero. The objective is to determine the values of $\alpha$, q, and $\lambda$ that minimize $Q_n\left(\theta \right)$. The objective function is given by

$$Q_n\left(\theta \right)=\sum _{t=s+1}^n{\left[X_t-E\left(X_t\right|X_{t-s})\right]}^2=\sum _{t=s+1}^n{\left(X_t-\lambda X_{t-s}-{\displaystyle \frac{q(1-\lambda )(1-\alpha +q)}{1-q^2}}\right)}^2.$$

(15)

The CLS estimators are obtained by solving the system of equations

$${\displaystyle \frac{\partial Q_n\left(\theta \right)}{\partial \lambda }}=0,{\displaystyle \frac{\partial Q_n\left(\theta \right)}{\partial q}}=0,{\displaystyle \frac{\partial Q_n\left(\theta \right)}{\partial \alpha }}=0.$$

The estimator ${\widehat{\lambda }}_{CLS}$ is derived by differentiating $Q_n\left(\theta \right)$ with respect to $\lambda$ and solving for $\lambda$. The closed-form solution for ${\widehat{\lambda }}_{CLS}$ is

$${\widehat{\lambda }}_{CLS}={\displaystyle \frac{(n-s){\sum }_{t=s+1}^n{X}_t{X}_{t-s}-{\sum }_{t=s+1}^n{X}_t{\sum }_{t=s+1}^n{X}_{t-s}}{(n-s){\sum }_{t=s+1}^n{X}_{t-s}^2-{\left({\sum }_{t=s+1}^n{X}_{t-s}\right)}^2}}.$$

(16)

Substituting ${\widehat{\lambda }}_{CLS}$ into $Q_n\left(\theta \right)$ yields the function $Q_n(\alpha ,q)$, which depends only on $\alpha$ and q. Since closed-form solutions for ${\widehat{\alpha }}_{CLS}$ and ${\widehat{q}}_{CLS}$ are not available, numerical optimization techniques are used to obtain the estimates.

3.2. Conditional Maximum-Likelihood (CML) Estimation

In this subsection, we obtain the estimates using the conditional maximum-likelihood (CML) method. The CML estimator of $\theta ={(\alpha ,q,\lambda )}^{\prime }$ is the value ${\theta }_{CML}$ that maximizes the likelihood function. Since the likelihood is not tractable, the maximum-likelihood estimate of $\theta$ is computed using numerical methods. The log-likelihood function is defined as

$$logL(x_1,\dots ,x_n;\alpha ,q,\lambda )=\sum _{t=s+1}^{n}logP(X_t=x_t\mid X_{t-s}=x_{t-s}),$$

(17)

where $P(X_t=x_t\mid X_{t-s}=x_{t-s})$ is given by Theorem 2. The CML estimates of the parameters are obtained by maximizing the log-likelihood function defined in Equation (17).

3.3. Simulation Study

A simulation study has been carried out to compare the performance of the two estimation procedures using Monte Carlo experiments. We simulated 1000 data series from the TGDINAR(1)s process, each of sizes 50, 100, 300, 500, 1000, and 1500, for various combinations of parameter values and with a seasonal period of $s=12$. The $TGD(q,\alpha )$ distribution is unimodal with a non-zero mode when $-1<\alpha <-{\left(q(2+q)\right)}^{-1}$, provided that $q>0.414$ [34]. We selected the following parameter combinations: $\theta =(-0.70,0.75,0.50)$, $\theta =(-0.85,0.50,0.10)$, $\theta =(-0.80,0.85,0.75)$ and $\theta =(0.15,0.30,0.25)$.

The estimation results, including mean squared errors (MSEs), are presented in

Table 1. Parameter estimates and their mean squared errors.

n	${\widehat{\mathit{\alpha }}}_{\mathit{CLS}}$	${\widehat{\mathit{q}}}_{\mathit{CLS}}$	${\widehat{\mathit{\lambda }}}_{\mathit{CLS}}$	${\widehat{\mathit{\alpha }}}_{\mathit{CML}}$	${\widehat{\mathit{q}}}_{\mathit{CML}}$	${\widehat{\mathit{\lambda }}}_{\mathit{CML}}$
	$\alpha =-0.7$, $q=0.75$, $\lambda =0.5$
50	$-0.648$	$0.742$	$0.471$	$-0.638$	$0.748$	$0.488$
	($0.024$)	($0.012$)	($0.003$)	($0.009$)	($0.002$)	($0.003$)
100	$-0.654$	$0.750$	$0.476$	$-0.682$	$0.748$	$0.493$
	($0.002$)	($0.000$)	($0.017$)	($0.090$)	($0.004$)	($0.000$)
300	$-0.665$	$0.754$	$0.495$	$-0.696$	$0.749$	$0.498$
	($0.001$)	($0.000$)	($0.002$)	($0.000$)	($0.000$)	($0.000$)
500	$-0.665$	$0.751$	$0.495$	$-0.700$	$0.749$	$0.499$
	($0.000$)	($0.000$)	($0.000$)	($0.001$)	($0.000$)	($0.000$)
1000	$-0.671$	$0.752$	$0.498$	$-0.703$	$0.750$	$0.499$
	($0.000$)	($0.000$)	($0.001$)	($0.001$)	($0.000$)	($0.000$)
1500	$-0.673$	$0.752$	$0.498$	$-0.702$	$0.750$	$0.499$
	($0.001$)	($0.000$)	($0.000$)	($0.000$)	($0.000$)	($0.000$)
	$\alpha =-0.85$, $q=0.50$, $\lambda =0.10$
50	$-0.820$	$0.501$	$0.070$	$-0.748$	$0.507$	$0.116$
	($0.009$)	($0.005$)	($0.028$)	($0.023$)	($0.003$)	($0.012$)
100	$-0.822$	$0.501$	$0.086$	$-0.806$	$0.504$	$0.010$
	($0.010$)	($0.002$)	($0.002$)	($0.167$)	($0.004$)	($0.000$)
300	$-0.828$	$0.502$	$0.096$	$-0.842$	$0.500$	$0.099$
	($0.000$)	($0.001$)	($0.000$)	($0.010$)	($0.000$)	($0.004$)
500	$-0.830$	$0.502$	$0.097$	$-0.849$	$0.500$	$0.100$
	($0.001$)	($0.000$)	($0.001$)	($0.004$)	($0.001$)	($0.001$)
1000	$-0.832$	$0.502$	$0.099$	$-0.853$	$0.500$	$0.099$
	($0.003$)	($0.000$)	($0.000$)	($0.000$)	($0.000$)	($0.000$)
1500	$-0.830$	$0.502$	$0.099$	$-0.847$	$0.500$	$0.100$
	($0.000$)	($0.000$)	($0.001$)	($0.005$)	($0.000$)	($0.000$)
	$\alpha =-0.8$, $q=0.85$, $\lambda =0.75$
50	$-0.750$	$0.824$	$0.728$	$-0.666$	$0.846$	$0.744$
	($0.000$)	($0.000$)	($0.078$)	($0.003$)	($0.002$)	($0.000$)
100	$-0.771$	$0.845$	$0.725$	$-0.729$	$0.846$	$0.746$
	($0.000$)	($0.001$)	($0.000$)	($0.014$)	($0.000$)	($0.000$)
300	$-0.781$	$0.849$	$0.737$	$-0.780$	$0.851$	$0.750$
	($0.001$)	($0.001$)	($0.002$)	($0.001$)	($0.000$)	($0.000$)
500	$-0.785$	$0.850$	$0.743$	$-0.788$	$0.849$	$0.749$
	($0.000$)	($0.000$)	($0.002$)	($0.002$)	($0.000$)	($0.000$)
1000	$-0.788$	$0.850$	$0.747$	$-0.793$	$0.850$	$0.749$
	($0.000$)	($0.000$)	($0.000$)	($0.001$)	($0.000$)	($0.000$)
1500	$-0.790$	$0.850$	$0.748$	$-0.798$	$0.850$	$0.750$
	($0.001$)	($0.000$)	($0.000$)	($0.000$)	($0.000$)	($0.000$)
	$\alpha =0.15$, $q=0.30$, $\lambda =0.25$
50	$0.174$	$0.293$	$0.161$	$-0.140$	$0.266$	$0.239$
	($0.000$)	($0.000$)	($0.005$)	($1.322$)	($0.046$)	($0.049$)
100	$0.172$	$0.299$	$0.199$	$0.044$	$0.296$	$0.241$
	($0.000$)	($0.000$)	($0.025$)	($1.215$)	($0.026$)	($0.026$)
300	$0.171$	$0.302$	$0.228$	$0.138$	$0.303$	$0.246$
	($0.001$)	($0.000$)	($0.005$)	($0.008$)	($0.002$)	($0.005$)
500	$0.170$	$0.304$	$0.239$	$0.158$	$0.308$	$0.248$
	($0.002$)	($0.000$)	($0.002$)	($0.859$)	($0.013$)	($0.001$)
1000	$0.171$	$0.303$	$0.243$	$0.145$	$0.302$	$0.248$
	($0.001$)	($0.000$)	($0.001$)	($0.034$)	($0.002$)	($0.000$)
1500	$0.170$	$0.303$	$0.248$	$0.163$	$0.304$	$0.250$
	($0.001$)	($0.000$)	($0.000$)	($0.017$)	($0.001$)	($0.001$)

. Estimates are obtained using the two methods, namely CLS and CML. We have computed the mean of the estimates over 1000 simulations and their MSEs. It can be seen that the CML has the lowest MSE compared to the CLS method. We plotted the box plots for three parameters using the estimates obtained from the CML and CLS for 1000 simulations. From the plots given in Figure 3, Figure 4, Figure 5 and Figure 6, we can say that as the sample size increases, the estimated parameter values converge to the actual value. Also, we have plotted the Q-Q plots based on 1000 simulations for standardized estimated values of the parameters for two estimation methods and three-parameter combinations. Due to space constraints, only one Q-Q plot is presented (Figure 7). The alignment of the sample quantiles with the theoretical quantiles along the diagonal line shows that the normality assumption is satisfied for a large sample size. The Q-Q plots confirm normality for all estimation methods and parameter combinations.

4. Real-Life Data Analysis

4.1. Exploratory Analysis of Legionnaires’ Data in Bavaria (2001–2023)

Legionnaires’ disease is a type of pneumonia caused by inhaling water droplets contaminated with Legionella bacteria. These bacteria flourish in warm water environments such as hot tubs, air conditioning systems, and large plumbing systems. The incubation period typically ranges from 2 to 10 days but can extend up to 20 days. In regions like Bavaria, Germany, the incidence increases from late spring to early fall due to warmer temperatures and greater use of water-based cooling systems.

We begin by examining the Legionnaires’ case data in Bavaria over the period 2001–2023, extracted from the Robert Koch Institute (https://survstat.rki.de (accessed on 12 August 2023)). This dataset consists of weekly counts of reported Legionnaires’ cases per year. Figure 8 shows the total number of cases of Legionnaires’ disease each year. The number of cases has been increasing over time, especially after 2017, with the highest in 2023. Figure 9 shows the weekly average of cases of Legionnaires’ disease in all years. The number of cases stays low from around week 10 to week 30, then starts to increase. The highest average number of cases occurs near weeks 48 to 50, which means that more cases tend to occur in the late part of the year. Figure 10 shows a heat map of Legionnaires’ cases by week and year. The bright yellow and orange areas, especially in recent years like 2018 and 2022, show that more cases usually occur near the end of the year. It also shows that, while the number of cases varies from year to year, the peak tends to occur around the same time each year.

These exploratory insights justify the use of a flexible and seasonally structured count model like the proposed TGDINAR(1)s, which is specifically designed to handle overdispersion, tail behavior, and periodicity in count time series.

Legionnaires’ Disease

We analyzed the weekly number of cases of Legionnaires’ disease in Bavaria, Germany, from 2001 to 2003, available at the Robert Koch Institute (https://survstat.rki.de (accessed on 12 August 2023)). The data consists of 159 weekly observations.

The bar graph in Figure 11 shows that the data set is positively skewed. The average value of the data is $1.4277$, and its spread is $2.0311$, which is relatively high compared to the average. This indicates that the data are more spread out than usual. The time series plot in Figure 12 shows repeating peaks every 3 weeks, visually confirming the seasonal cycle, while the PACF plot supports this with a strong spike at lag 3, statistically validating the seasonal period $s=3$. Together, they indicate a consistent 3-week pattern in cases of Legionnaires’ disease. Figure 13 presents the marginal calibration plot, showing the difference between the fitted cumulative distribution functions (CDFs) and the empirical CDF of the observed data. Among the competing models, the curve corresponding to the transmuted geometric distribution remains closest to the zero line throughout the support, indicating a minimal deviation from the empirical CDF. This suggests that the TGD has a better fit to the data compared to alternative models.

To compare the performance of various models, we used the Akaike information criterion (AIC), Bayesian information criterion (BIC), and root mean square error (RMSE). Both AIC and BIC evaluate how well the model fits the data while accounting for the model complexity. Smaller values of these two criteria suggest a model with greater accuracy. RMSE measures prediction accuracy, with smaller values reflecting better predictive performance. Taking into account the AIC, BIC, and RMSE together in

Table 2. AIC and BIC for the seasonal INAR(1) models for Legionnaires’ disease data.

Model	Estimated Values	AIC	BIC	RMSE
PoINAR(1)s	$\widehat{\varphi }=0.1324$, $\widehat{\lambda }=1.2283$	505.0183	511.1561	1.3887
GINAR(1)s	$\widehat{\varphi }=0.0987$, $\widehat{\theta }=0.5752$	514.1081	520.2459	1.3939
NGINAR(1)s	$\widehat{\alpha }=0.1648$, $\widehat{\mu }=1.3424$	513.2207	519.3585	1.3876
ZMGINAR(1)s	$\widehat{\alpha }=0.3750$, $\widehat{\pi }=-0.3158$, $\widehat{\mu }=1.1636$	506.1553	515.3620	1.4088
TGDINAR(1)s	$\widehat{\alpha }=0.9999$, $\widehat{q}=0.4565$, $\widehat{\lambda }=0.1184$	498.3401	507.5468	1.3902

, the objective is to identify the model that best balances accuracy, complexity, and predictive performance. These criteria suggest that the transmuted geometric distribution and TGDINAR(1)s model are the better choices.

To predict future model values, we split the dataset into two parts; the first 149 observations are used to build the model, while the remaining ten observations are used for forecasting. We checked the independence of Pearson’s residual using the ACF plot in Figure 14. The highest posterior probability (HPP) intervals were constructed as the shortest intervals that contain 90% of the forecast distribution, based on the conditional pmf in Theorem 3.

Table 3. Point forecast for Legionnaires’ disease data with $\gamma$ = 0.90.

k	${\mathit{X}}_{\mathit{t}+\mathit{k}}$	Mean	Mode	Median	HPP
1	1	1.72	1	1	[0, 4)
2	3	1.96	1	2	[0, 4)
3	1	1.60	1	1	[0, 4)
4	1	1.41	1	1	[0, 3)
5	1	1.44	1	1	[0, 3)
6	4	1.41	1	1	[0, 3)
7	2	1.42	1	1	[0, 3)
8	2	1.42	1	1	[0, 3)
9	3	1.42	1	1	[0, 3)
10	0	1.42	1	1	[0, 3)

shows the predicted mean, median, mode, and HPP intervals for all ten-step-ahead forecasts.

Comparative Models:

• PoINAR(1)s: The INAR(1) process with Poisson marginal distributions, based on binomial thinning ($\varphi$) [11].
• GINAR(1)s: The INAR(1) process with a geometric marginal distribution and seasonal period s, based on binomial thinning ($\varphi$) [21].
• NGINAR(1)s: The INAR(1) process with new geometric marginal distributions, based on negative binomial thinning ($\alpha$), with seasonal period s [20].
• ZMGINAR(1)s: The INAR(1) process with zero-modified geometric marginal distributions, based on negative binomial thinning ($\alpha$) [35].
• TGDINAR(1)s: The proposed model.

4.2. Exploratory Analysis of Syphilis Cases (2007–2010)

Syphilis is a sexually transmitted infection (STI) caused by the bacterium Treponema pallidum. It can be transmitted through sexual contact, blood transfusion, or from mother to child during pregnancy. If not treated, it can cause serious complications, including neurological and cardiovascular damage, and adverse birth outcomes, such as stillbirth and neonatal death.

The syphilis data from 2007 to 2010 show clear trends and patterns. Figure 15 presents the average weekly cases in all years, with fluctuations throughout the year and peaks around weeks 6, 17, and 41. This suggests possible seasonal patterns or repeated increases in cases during certain times. The heatmap of weekly counts per year (Figure 16) highlights specific weeks, especially in 2008 and 2009, with higher case counts (in yellow and red), which supports the observed peaks and indicates consistent periods of higher incidence. Figure 17 shows the total number of cases per year, with a steady increase from 2007 to a peak in 2009, followed by a decrease in 2010. This pattern may reflect an outbreak or improved case detection in 2009, followed by better control or changes in reporting. Overall, 2009 had the highest number of syphilis cases during this period.

Syphilis Patient Data

The dataset records weekly syphilis cases in Massachusetts, USA, from 2007 to 2010, available in R (https://cran.r-project.org/web/packages/ZIM/ZIM.pdf (accessed on 3 March 2024)) [36]. It has 197 observations, with a mean of 2.7259 and a variation of 4.4449, indicating overdispersion.

Figure 18 presents a bar plot with a positively skewed distribution, while Figure 19 presents the time series plot along with the sample ACF and PACF for the syphilis dataset. The PACF plot exhibits a prominent spike at lag 6, providing statistical support for a seasonal period of 6 weeks. These findings suggest a consistent 6-week cyclical pattern in the incidence of syphilis cases.

Model performance was assessed using the AIC, BIC, and RMSE.

Table 4. AIC and BIC for the seasonal INAR(1) models for syphilis cases.

Model	Estimated Values	AIC	BIC	RMSE
PoINAR(1)s	$\widehat{\varphi }=0.1444$, $\widehat{\lambda }=2.3552$	820.7281	827.2945	2.0338
GINAR(1)s	$\widehat{\varphi }=0.1342$, $\widehat{\theta }=0.7196$	833.4564	840.0229	2.0413
NGINAR(1)s	$\widehat{\alpha }=0.3297$, $\widehat{\mu }=2.4377$	828.0160	834.5824	2.0591
ZMGINAR(1)s	$\widehat{\alpha }=0.3750$, $\widehat{\pi }=-0.1254$, $\widehat{\mu }=2.2853$	825.2454	835.0950	2.0664
TGDINAR(1)s	$\widehat{\alpha }=-0.8055$, $\widehat{q}=0.6526$, $\widehat{\lambda }=0.0643$	806.4264	816.2760	2.0332

shows that the proposed TGDINAR(1)s model achieves consistently better results than the others. Figure 20 is a marginal calibration plot in which the TGD curve remains closer to zero, suggesting a better fit than the other distributions. To predict future values, we used 199 observations for model building and the last 10 for forecasting.

Table 5. Point forecast for syphilis cases with $\gamma$ = 0.90.

k	${\mathit{X}}_{\mathit{t}+\mathit{k}}$	Mean	Mode	Median	HPP
1	2	2.74	1	2	[0, 6)
2	1	2.74	1	2	[0, 6)
3	0	2.68	1	2	[0, 6)
4	0	2.61	1	2	[0, 6)
5	2	2.74	1	2	[0, 6)
6	0	2.61	1	2	[0, 6)
7	0	2.79	1	2	[0, 6)
8	0	2.79	1	2	[0, 6)
9	0	2.79	1	2	[0, 6)
10	1	2.78	1	2	[0, 6)

shows the predicted mean, median, mode, and HPP intervals for all ten-step-ahead forecasts. The independence of Pearson’s residuals was assessed using the ACF plot in Figure 21.

4.3. Exploratory Analysis of Dengue Cases (2001–2023)

Dengue is a mosquito-borne viral infection common in tropical and subtropical regions. In places like Bavaria, Germany, while not endemic, dengue cases are reported due to travel-related exposures, with incidence peaking during warmer months when mosquito activity is higher.

We analyze weekly dengue case data in Bavaria from 2001 to 2023, obtained from the Robert Koch Institute (https://survstat.rki.de (accessed on 24 April 2024)). Figure 22 shows a growing trend in annual dengue cases, with peaks in 2018 and 2023. Figure 23 presents the average weekly number of cases, which tends to increase during the mid-year and peaks between weeks 35 and 45. The heatmap in Figure 24 further confirms this seasonal pattern, highlighting increased activity in late summer and early fall, particularly in recent years.

Dengue Fever Cases

We analyzed the weekly number of cases of dengue disease in Berlin, Germany, from 2010 to 2015. The data consists of a total of 298 observations.

The bar plot in Figure 25 shows that the data is positively skewed. The average value of the data is $1.0772$, and its spread is $1.4587$, which is relatively high compared to the average. This indicates that the data are more spread out than usual. The PACF plot in Figure 26 indicates that the periodicity of this model is $s=1$. This figure contains the time series plot and its sample ACF and PACF plot. Figure 27 is a marginal calibration plot suggesting that the transmuted geometric distribution provides a better fit to the data compared to the other distributions.

To compare the performance of various models for such data, we used the AIC, BIC, and RMSE. Obtaining the AIC, BIC, and RMSE measures presented in

Table 6. AIC and BIC for the seasonal INAR(1) models for dengue disease data.

Model	Estimated Values	AIC	BIC	RMSE
PoINAR(1)s	$\widehat{\varphi }=0.1363$, $\widehat{\lambda }=0.9348$	850.0882	857.4824	1.1896
GINAR(1)s	$\widehat{\varphi }=0.1133$, $\widehat{\theta }=0.5109$	855.7436	863.1378	1.1909
NGINAR(1)s	$\widehat{\alpha }=0.2114$, $\widehat{\mu }=1.0409$	852.6276	860.0218	1.1916
ZMGINAR(1)s	$\widehat{\alpha }=0.3750$, $\widehat{\pi }=-0.2332$, $\widehat{\mu }=0.9602$	849.2594	860.3507	1.2194
TGDINAR(1)s	$\widehat{\alpha }=-0.8059$, $\widehat{q}=0.4077$, $\widehat{\lambda }=0.1496$	840.0040	851.0953	1.1893

, we aim to identify the model that best balances accuracy, complexity, and predictive performance. These criteria suggest that the transmuted geometric distribution and TGDINAR(1)s model are the best choices.

We split the dataset into two parts to predict future model values: the first 288 observations are used to build the model, while the remaining 10 observations are used for forecasting. We checked the independence of Pearson’s residual using the ACF plot in Figure 28. We calculated the 90% highest predictive probability interval (HPP) instead of standard prediction intervals.

Table 7. Point forecast for dengue disease data with $\gamma$ = 0.90.

k	${\mathit{X}}_{\mathit{t}+\mathit{k}}$	Mean	Mode	Median	HPP
1	1	1.22	1	1	[0, 3)
2	2	1.10	0	1	[0, 3)
3	0	1.09	0	1	[0, 3)
4	1	1.08	0	1	[0, 3)
5	0	1.08	0	1	[0, 3)
6	0	1.08	0	1	[0, 3)
7	0	1.08	0	1	[0, 3)
8	0	1.08	0	1	[0, 3)
9	1	1.08	0	1	[0, 3)
10	3	1.08	0	1	[0, 3)

shows the predicted mean, median, mode, and HPP intervals for all ten-step-ahead forecasts.

5. Concluding Remarks

This paper deals with modeling overdispersed count time series with long or short tails using a transmuted geometric marginal distribution. Coherent forecasts are obtained based on the conditional distribution. A detailed Monte Carlo simulation study is carried out to check the performance of the estimation methods. The proposed approach is illustrated using both simulated data and data from practical applications. Model adequacy is assessed using Pearson residuals. The results show that the TGDINAR(1)s model provides a better fit compared to the others overdispersed count time series models. A key limitation of the proposed model is its focus on univariate count time series, restricting its applicability to more complex real-world scenarios. Future work will explore extending the model to capture structural breaks and non-linear patterns in the data. In addition, future extensions could consider a bivariate TGDINAR model with a seasonal structure to capture interdependencies between two related count processes, or address non-stationary cases to model time-varying behavior.