Introduction

Current estimation of the prevalence of disabilities at a global level is about 15%, which reaches more than a billion people worldwide1. These are expected to increase further, considering factors such as an ageing population, a rise in chronic health conditions, and higher survival rates for life-threatening illnesses2,3. To solve this problem, the World Health Organization (WHO), with Resolution WHA74.8, adopted by the 77th World Health Assembly in 2021, initiated a disability inclusion strategy to incorporate people with disabilities into health systems4. The resolution emphasizes enhancing the access of people with disabilities to effective health service delivery in all health care systems4. Such services are critical to addressing the peculiar needs of people with disabilities to effectively live, work, and learn while fully being assimilated into society1,5,6. The disability rate of 7.1% has been reported for Saudi Arabia with a significant rate of physical disabilities in the population7.

According to WHO statistics, the disability rate is lower in Saudi Arabia compared to the rest of the world8. However, this rate could be higher or lower as there are variations in definitions and data collection mechanisms in the countries9. Disabilities in Saudi Arabia have multiple causes, such as genetic diseases, congenital disorders, accidents, chronic diseases, societal discrimination, economic problems, domestic conflicts, mental disorders, among others9,10. In recent years, government and non-profit organizations have played a pivotal role in expanding access to essential healthcare services in Saudi Arabia11,12. As part of the Vision 2030, Saudi Arabia introduced the Health Sector Transformation Program, which seeks to transform the quality and access to health care for every citizen13. The key elements of these programs include intellectual and physical strategies, policies and legal regulations, funding for disability programs, services to detect and address disabilities, a steady supply of medications and rehabilitation services, and training of affected individuals, healthcare providers, policymakers, and other stakeholders10,14. Furthermore, the Saudi government is working to establish social measures that encourage community participation and develop income-support programs for individuals with various disabilities. This process includes efforts that ensure the continued establishment of a political, social, cultural, and economic setting open to all and conducive for the development of proper policies15.

Statistical modeling seeks to construct probability distributions that can accurately describe the features of survey-generated, observational study-derived, and experiment-generated data sets. The last decades have also seen significant development in probability theory with the advent of novel, more sophisticated distributions like beta and Kumaraswamy families. Several authors have proposed different classes of generated beta distributions. The first work on the introduction of generated distributions is one by16. The authors introduced a new family of models with two extra parameters by using a random variable following beta distribution in order to get more flexibility. However, this family brings some mathematical complexities since the CDF is not in a closed form. To overcome the deficiencies of beta distribution17 proposed the Kumaraswamy distribution. It has become a good substitute to the beta distribution in recent decades due to its simplicity and traceability. Cordeiro 18 developed another G-class so called the Kumaraswamy generalised family of distributions to extend the standard distributions. Some other examples of such models are the inverted Kumaraswamy (IKw) distribution (see e.g.19) with two extra parameters using the transformation \(Y = 1/X - 1\), when X has a Kumaraswamy distribution. Iqbal 20 generalized the IKw distribution by utilizing the power transformation T = \(X^{\gamma }\) and introduced a more flexible model known as generalized inverted Kumaraswamy (GIKum) distribution. In like manner21, proposed the exponentiated Ailamujia distribution. These models have widely been applied in diverse disciplines including biological sciences, engineering, environmental studies, and econometrics.

However, no single probability distribution is suitable for different data sets, and hence the need to improve the existing distributions or generate new and so more sophisticated and flexible models are needed. Hence, different techniques to develop new distributions have been proposed. A distribution could be generalized, by adding extra parameter(s) from the family of distributions used. The flexibility can also be improved by combining two or more distributions such as student t-distribution (see e.g.22). Besides, the study by23 in the European Journal of Operational Research presents a two-phase degradation model utilizing a reparameterized inverse Gaussian process to enhance the prediction of remaining useful life (RUL). This approach effectively captures degradation trends and provides accurate lifetime estimates, improving reliability assessment in engineering applications. Similarly24, introduce a multivariate Student-t process model in IISE Transactions to analyze dependent tail-weighted degradation data, addressing the challenge of modeling heavy-tailed dependencies in reliability studies. Their model demonstrates superior performance in capturing uncertainty and correlation structures in degradation processes, making it particularly useful for complex failure prediction.

Building on this foundation, this study proposes the Extended Generalized Inverted Kumaraswamy Standard Exponential (EGIKwS-Exp) distribution. This study will rely on disability statistics from Saudi Arabia. The data in this case focuses on communication disabilities in terms of percent and degree of severity-which was classified as either severe or unable-to respond-across 13 administrative areas. The data came from the Disability Survey 2023, implemented by GASTAT, the General Authority for Statistics. The administrative areas put in the analysis are Al-Riyadh, Makkah Al-Mokarramah, Al-Madinah Al-Monawarah, Al-Qaseem, the Eastern Region, Aseer, Tabouk, Hail, Northern Borders, Jazan, Najran, Al-Baha, and Al-Jouf. The data, obtained from the table titled Percentage Distribution of Individuals with One Disability and Degree of Severity (Severe or Unable) in the Age Group (Two Years and Over) by Type of Disability and Administrative Area, indicates the following: the level of communication disability distribution and severity that has regional disparities and possible underlying factors. The data is accessible to the public on the website of the General Authority for Statistics. For more information, see the Disability Survey 2023. In addition, the methodological approach combines descriptive and advanced statistical analysis to explore the properties of the EGIKwS-Exp distribution, such as its survival function (SF), hazard rate function (HRF), reverse hazard rate functions (RHRF), quantile function(QF), and maximum likelihood estimators (MLE). This research contributes to two main areas. First, it introduces the EGIKwS-Exp distribution, detailing its properties and statistical measures. Second, it applies this model to analyze the percentage data of Saudi individuals with communication disabilities, highlighting the distribution’s practical importance and comparing its performance to existing models. The findings show that the EGIKwS-Exp distribution outperforms competing models in terms of goodness-of-fit measures, making it a valuable tool for understanding disability prevalence and guiding future interventions.

The study uses Mathematica and statistical software R version 4.4.2 to make all the computations, and this ensures methodological precision.

The research provides essential information on the prevalence and management of communication disabilities with an all-inclusive analysis of the disability data in Saudi Arabia. The findings from the study highlight the crucial role of advanced statistical models in addressing public health issues by providing a robust framework for resource allocation, policy formulation, and social inclusion. The EGIKwS-Exp model is developed on established methods, which enhances the flexibility and applicability in modeling data under consideration. With an application of advanced statistical methods, the research is to probe into the reliability and robustness of health interventions and policies implemented in Saudi Arabia. It is also important to establish a framework that can be used for projecting disability trends and evaluating the long-term effects of governmental and nongovernmental intervention programs. Integrating statistical and reliability analyses, the findings are expected to provide actionable insights to guide policymakers, healthcare providers, and other stakeholders in enhancing the quality of life for individuals with disabilities. The research also emphasizes the global relevance of disability management, encouraging stakeholders worldwide to adopt evidence-based strategies to support individuals with disabilities effectively.

The structure of the paper is as follows: Section “Proposed probability distribution” describes the EGIKwS-Exponential distribution, which makes emphasis on some distributional and reliability properties. Section “Maximum likelihood estimation” deals with parameter estimation using the MLE method. Section “Applications to communication disability data” shows the practical application of the proposed distribution by analyzing the percentage data of Saudi individuals with communication disabilities and varying degrees of severity (Severe or Unable) in the age group of two years and above across 13 administrative areas, and compares its performance with other well-known distributions. Finally, Section “Conclusions and policy implications” concludes the paper with final remarks and recommendations for future research.

Proposed probability distribution

This section of the study is dedicated to introducing a new extension of the Exponential distribution. A random variable X is characterized as following the standard Exponential distribution if its CDF and probability density function (PDF) for \(x \>\) 0 are given by:

$$\begin{aligned} \varLambda (x)= & 1-\exp (-x ) \end{aligned}$$
(1)
$$\begin{aligned} \lambda (x)= & \exp (-x ) \end{aligned}$$
(2)

respectively.

Due to the widespread study and various applications of lifetime distributions, there is always a further need to search for newer generalizations. Many generalized forms of these distributions have been proposed in the literature. For example, Kumaraswamy generalized power Exponential distribution is studied by25 and exponentiated power generalized Exponential distribution by26. In addition, a new three-parameter lifetime model, the Truncated Exponential Lomax (TWL) distribution, has been presented. Among the above mentioned, the generalized inverted Kumaraswamy (GIKum) distribution, which was introduced by20, has attracted special attention. This distribution has parameters that make it flexible enough to model different types of data and is expressed as:

$$\begin{aligned} F(x)= & \left[ 1-\left( 1+x^{\vartheta }\right) ^{-\psi }\right] ^{\xi } \end{aligned}$$
(3)
$$\begin{aligned} f(x)= & \, \psi \xi \vartheta x^{\vartheta -1}\left( 1+x\right) ^{-\psi -1}\left[ 1-\left( 1+x^{\vartheta }\right) ^{-\psi }\right] ^{\xi -1} \end{aligned}$$
(4)

where \(\psi>0,\xi>0,\vartheta >0\) and \(x>0\). Utilizing the GIKum distribution in conjunction with an arbitrary baseline CDF and the generator \(\frac{G^{\eta }\left( x,\theta \right) }{1-G^{\eta }\left( x,\theta \right) }\)27, developed the T-X family of distributions, which serve as the basis for further generalizations. Building on this foundation, a new class of distributions, the EGIKw-G class, was proposed by28, where the CDF integrates the GIKum generator into the T-X framework as:

$$\begin{aligned} F_{EGIKw-G}(x,\varTheta )& = \psi \xi \vartheta \int _{0}^{\frac{G^{\eta }\left( x,\theta \right) }{1-G^{\eta }\left( x,\theta \right) } } t^{\vartheta -1}\left( 1+t\right) ^{-\psi -1}\left[ 1-\left( 1+t^{\vartheta }\right) ^{-\psi }\right] ^{\xi -1} dt, \nonumber \\ & = \left\{ 1-\left[ 1+\left( \frac{G^{\eta }\left( x,\theta \right) }{ 1-G^{\eta }\left( x,\theta \right) } \right) ^{\vartheta } \right] ^{-\psi } \right\} ^{\xi }, \end{aligned}$$
(5)

with parametric vector \(\varTheta =\psi ,\xi ,\vartheta ,\eta ,\). Here \(\psi>0,\xi>0,\eta>0 \; \text {and} \; \vartheta >0\) are additional parameters which introduce skewness and \(\theta\) is the parametric space of the baseline model. In this work, we focus on the development of a specific generalization within this class, termed the EGIKwS-Exp distribution. This novel distribution demonstrates greater flexibility compared to other commonly used standard distributions. By selecting the Standard Exponential distribution as the baseline CDF, the newly proposed EGIKwS-Exp distribution exhibits higher adaptability in modeling data, making it a valuable addition to the family of generalized distributions. By taking \(G\left( x,\theta \right)\) as the CDF of the Standard Exponential distribution in Eq. (5), the CDF and the pdf of the newly proposed EGIKwS-Exp distribution are given by:

$$\begin{aligned} F_{EGIKwS-Exp}(x, \varTheta )= & \, \psi \xi \vartheta \int _{0}^{\frac{ \varLambda (x) ^{\eta }}{ 1- \varLambda (x) ^{\eta }} } t^{\vartheta -1}\left( 1+t\right) ^{-\psi -1}\left[ 1-\left( 1+t^{\vartheta }\right) ^{-\psi }\right] ^{\xi -1} dt.\nonumber \\= & \, \left[ 1-\left\{ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta }\right\} ^{-\psi } \right] ^{\xi } \end{aligned}$$
(6)
$$\begin{aligned} f_{EGIKwS-Exp}(x, \varTheta )= & \, \psi \xi \vartheta \eta \lambda (x) \varLambda (x) ^{\eta \vartheta -1} \left[ 1- \varLambda (x) ^{\eta }\right] ^{-\vartheta -1}\nonumber \\ & \left\{ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta }\right\} ^{-\psi -1} \left[ 1-\left\{ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta }\right\} ^{-\psi } \right] ^{\xi -1} \end{aligned}$$
(7)

where \(x,\psi ,\xi ,\vartheta , \eta > 0\). Besides, \(\zeta \left( \varLambda (x),\eta \right) =\frac{ \varLambda (x) ^{\eta }}{ 1- \varLambda (x) ^{\eta }}\) and \(\varLambda (x)\) & \(\lambda (x)\) are respectively given by Eqs. (1) and (2). The main motivation for this extension is that the new EGIKwS-Exp distribution is a highly flexible life model which admits different degrees of kurtosis and asymmetry. The plot in Fig. 1a gives an indication that the shape of the EGIKwS-Exp distribution is highly flexible and offers various interesting curve shapes. It can also be said that the distribution is positively skewed. The graphical representation of the CDF of the EGIKwS-Exp distribution is as shown in Fig. 1b. The plot at other parameter values produces a similar shape. Furthermore, this study explores several essential functions associated with the proposed distribution, including the HRF, RHRF, CHRF, and SF, which are fundamental tools in reliability theory. The HRF, denoted by h(x), represents the instantaneous rate of occurrence of events, given no previous events have occurred. The HRF, RHRF, and CHRF of the EGIKwS-Exp distribution are provided, highlighting their mathematical structure.

Fig. 1
figure 1

The graphs for EGIKwS-Exp distribution with selected parameters.

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The HRF, RHRF and CHRF for EGIKwS-Exp are respectively given as follows:

$$\begin{aligned} h_{EGIKwS-Exp}(x, \varTheta )= & \, \psi \xi \vartheta \eta \lambda (x) \varLambda (x) ^{\eta \vartheta -1} \left[ 1- \varLambda (x) ^{\eta }\right] ^{-\vartheta -1} \left\{ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta }\right\} ^{-\psi -1} \nonumber \\ & \left[ 1-\left\{ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta }\right\} ^{-\psi } \right] ^{\xi -1}\nonumber \\ & \left\{ 1-\left[ 1-\left\{ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta }\right\} ^{-\psi } \right] ^{\xi } \right\} ^{-1}.\end{aligned}$$
(8)
$$\begin{aligned} H_{EGIKwS-Exp}(x, \varTheta )= & \, \psi \xi \vartheta \eta \;\lambda (x)\; \varLambda (x)^{\eta \vartheta -1} \left[ 1- \varLambda (x)^\eta \right] ^{-\vartheta -1} \left[ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta } \right] ^{-\psi -1} \nonumber \\ & \left\{ 1-\left[ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta } \right] ^{-\psi } \right\} ^{-1},\nonumber \\ \Omega _{EGIKw-G}(x, \varTheta )= & -\log \left[ 1-F(x, \varTheta ) \right] \nonumber \\= & -\log \left[ 1-\left\{ 1-\left[ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta } \right] ^{-\psi } \right\} ^{\xi } \right] . \end{aligned}$$
(9)

where \(x>0\) and \(\varTheta =\psi ,\xi ,\vartheta ,\eta .\)

The HRF plots for different parameter values, as shown in Fig. 1c, reveal diverse shapes, making this distribution suitable for modeling hazard behaviors encountered in real-world applications, such as human mortality and biomedical studies. Similarly, the CHRF and RHRF plots in Figs. 1d,e, respectively, further demonstrate the versatility of the proposed distribution in capturing complex patterns. Moreover, in healthcare and biomedical contexts, the SF is often employed to describe the probability of survival beyond a certain time x. For a random variable X representing survival time, the SF is defined as the complement of the CDF. The SF of the EGIKwS-Exp distribution is presented as:

$$\begin{aligned} S_{x}(x)=\mathbb {P}(X>x)=1-F_{x}(x). \end{aligned}$$

That is, the SF is the probability of survival beyond time x. The SF of \(X \sim\) EGIKwS\(-Exp\) is given by

$$\begin{aligned} S_{EGIKwS-Exp}(x, \varTheta )=1-\left[ 1-\left\{ 1+\zeta \left( \varLambda (x),\eta \right) ^{\vartheta }\right\} ^{-\psi } \right] ^{\xi } , \end{aligned}$$
(10)

For brevity purpose, graphical representation of SF at selected parameter values is shown in Fig. 1f. These features collectively demonstrate the suitability of the EGIKwS-Exp distribution for analyzing diverse types of lifetime data and for use in practical scenarios requiring advanced modeling flexibility. Let u be a standard uniform variable, then the quantile function or inverse CDF of the proposed distribution is obtained by solving \(F_{EGIKw-G}(x_{u}) = u\), thus yielding

$$\begin{aligned} Q_{EGIKwS-Exp}(u, \varTheta )= & \, F^{-1}_{EGIKwS-Exp}(u, \varTheta ) = x_{u},\nonumber \\= & \left[ - \log \left( 1 - \left( 1 + \left[ \left( 1 - u^{\frac{1}{\xi }} \right) ^{-\frac{1}{\psi }} - 1 \right] ^{-\frac{1}{\vartheta }} \right) ^{-\frac{1}{\eta }} \right) \right] . \end{aligned}$$
(11)

The random numbers from EGIKwS-Exp distribution can be simulated using the expression in Eq. (11) where \(U \sim Uniform(0,1).\) In particular, the median of the EGIKwS-Exp distribution can be derived by substituting \(u = 0.5\) in Eq. (11) as follows:

$$\begin{aligned} Median= & \left[ - \log \left( 1 - \left( 1 + \left[ \left( 1 - 0.5^{\frac{1}{\xi }} \right) ^{-\frac{1}{\psi }} - 1 \right] ^{-\frac{1}{\vartheta }} \right) ^{-\frac{1}{\eta }} \right) \right] . \end{aligned}$$
(12)

Likewise, the three quartiles and seven octiles can be respectively obtained by \(Q_{i} = Q(i/4),\; i \in (1,2,3)\; and\; O_{j} = Q(j/8),\; j \in (1,2,3,4,5,6,7)\). Furthermore, the Bowley skewness, denoted by ,\(S_{kb}\) is defined by

$$\begin{aligned} S_{kb} = \frac{Q_{3} + Q_{1} -2Q_{2}}{ Q_{3} -Q_{1}}. \end{aligned}$$
(13)

Also, a measure of the kurtosis, the Moors kurtosis, denoted by \(K_{um}\), can be defined as follows

$$\begin{aligned} K_{um} = \frac{O_{3} -O_{1} + O_{7} -O_{5}}{ O_{6} -O_{2}}. \end{aligned}$$
(14)

See29 for more details on this topic.

Maximum likelihood estimation

In this section we employ MLE to estimate the unknown parameters of EGIKwS-Exp distribution. We consider independent random variables \(X_{1},X_{2}...,X_{n},\) from an EGIKwS-Exp distribution with parameter vector \(\varTheta .\) The likelihood and log-likelihood \(l(\varTheta )\) = \(\log L(\varTheta )\) for the model parameters obtained from (7) is,

$$\begin{aligned} L(\varTheta )= & \, ( \psi \xi \vartheta \eta ) ^{n} \prod _{i=1}^{n}\Bigg [\lambda (x_{i}) \varLambda (x_{i}) ^{\eta \vartheta -1} \left[ 1- \varLambda (x_{i}) ^{\eta }\right] ^{-\vartheta -1}\nonumber \\ & \left\{ 1+\zeta \left( \varLambda (x_{i}),\eta \right) ^{\vartheta }\right\} ^{-\psi -1} \left[ 1-\left\{ 1+\zeta \left( \varLambda (x_{i}),\eta \right) ^{\vartheta }\right\} ^{-\psi } \right] ^{\xi -1} \Bigg ], \end{aligned}$$
(15)
$$\begin{aligned} l(\varTheta )= & \, n\log (\psi \xi \vartheta \eta ) - \sum _{i=1}^{n} x -\left( \eta +1 \right) \sum _{i=1}^{n}\log \varLambda (x) - \left( \vartheta +1\right) \sum _{i=1}^{n}\log \left[ \varLambda (x) ^{-\eta }-1 \right] \nonumber \\ & - \left( \psi +1\right) \sum _{i=1}^{n}\log \left[ 1+\left( \varLambda (x)^{-\eta }-1 \right) ^{-\vartheta } \right] \nonumber \\ & +\left( \xi -1\right) \sum _{i=1}^{n}\log \left[ 1-\left[ 1+\left( \varLambda (x)^{-\eta }-1\right) ^{-\vartheta } \right] ^{-\psi } \right] . \end{aligned}$$
(16)

The components of score vector U = \((U_{\psi },U_{\xi },U_{\vartheta }, U_{\eta },U_{\vartheta })^{'}\) are given by Eq. (17–20). The MLE against Eq. 16 can be obtained by solving the following nonlinear likelihood equations:

$$\begin{aligned} U_{\psi }= & \frac{n}{\psi }-\sum _{i=1}^{n}\log \left[ 1+\left( \varLambda (x)^{-\eta }-1\right) ^{-\vartheta } \right] \nonumber \\ & + \left( \xi -1\right) \sum _{i=1}^{n} \frac{\log \left[ 1+\left( \varLambda (x)^{-\eta }-1\right) ^{-\vartheta } \right] }{\left[ 1+\left( \varLambda (x)^{-\eta }-1\right) ^{-\vartheta } \right] ^{\psi }-1}, \end{aligned}$$
(17)
$$\begin{aligned} U_{\xi }= & \frac{n}{\xi }+\sum _{i=1}^{n}\log \left[ 1-\left[ 1+\left( \varLambda (x)^ {-\eta }-1\right) ^{-\vartheta } \right] ^{-\psi } \right] , \end{aligned}$$
(18)
$$\begin{aligned} U_{\vartheta }= & \frac{n}{\vartheta } - \sum _{i=1}^{n}\log \left( \varLambda (x)^{-\eta }-1\right) + \left( \psi +1\right) \sum _{i=1}^{n}\frac{\log \left( \varLambda (x)^{-\eta }-1\right) }{ 1+\left( \varLambda (x)^{-\eta } -1\right) ^{\vartheta }} \nonumber \\ & -\left( \xi -1 \right) \psi \sum _{i=1}^{n}\frac{\log \left( \varLambda (x)^{-\eta }-1\right) \left[ 1+\left( \varLambda (x)^{-\eta }-1\right) ^{\vartheta } \right] ^{-1} }{ \left[ \left( 1+\left( \varLambda (x) ^{-\eta }-1\right) ^{-\vartheta }\right) ^{\psi } -1\right] },\end{aligned}$$
(19)
$$\begin{aligned} U_{\eta }= & \frac{n}{\eta }- \sum _{i=1}^{n}\log \varLambda (x)-\left( \vartheta +1 \right) \sum _{i=1}^{n}\frac{\log \varLambda (x)}{\left( \varLambda (x)^{\eta }-1\right) }\nonumber \\ & - \left( \psi +1\right) \vartheta \sum _{i=1}^{n}\frac{\log \varLambda (x) }{\left( 1- \varLambda (x)^{\eta } \right) \left[ 1+\left( \varLambda (x)^{-\eta }-1\right) ^{\vartheta }\right] }\nonumber \\ & + \sum _{i=1}^{n} \frac{\log \varLambda (x) }{\left[ 1+\left( \varLambda (x)^{-\eta } -1\right) ^{-\vartheta } \right] ^{\psi }-1 } \frac{\psi \vartheta \left( \xi -1\right) }{\left( 1- \varLambda (x)^{\eta } \right) \left[ 1+\left( \varLambda (x) ^{-\eta }-1\right) ^{\vartheta }\right] }. \end{aligned}$$
(20)

Asymptotic confidence interval

In30, Neyman introduced confidence intervals based on the inversion of a family of hypothesis tests. With the assumption of a large sample size, the ACI provides a set of estimators whose density function approaches normality. For more details, consult31,32. To obtain the ACI of the unknown parameters the ML estimates \(\hat{\varTheta }\) are assumed that they conform to a bivariate normal distribution with a given mean \(\varTheta\) and covariance matrix I(\(\varTheta\)), the negative expectation of the inverse of the observed information matrix

$$\begin{aligned} I(\varTheta )=-E \left( \begin{array}{cccc} \hat{I}_{11}& \quad \hat{I}_{12}& \quad \hat{I}_{13}& \quad \hat{I}_{14}\\ \hat{I}_{21}& \quad \hat{I}_{22}& \quad \hat{I}_{23}& \quad \hat{I}_{24}\\ \hat{I}_{31}& \quad \hat{I}_{32}& \quad \hat{I}_{33}& \quad \hat{I}_{34}\\ \hat{I}_{41}& \quad \hat{I}_{42}& \quad \hat{I}_{43}& \quad \hat{I}_{44}\\ \end{array}\right) ^{-1}. \end{aligned}$$

The 100(1-\(\alpha\))% ACI for parameters is defined as \(\hat{\delta }\pm z_{\alpha /2}\sqrt{I_{\delta \delta }(\Theta )}\) where \(I_{\delta \delta }(\Theta )\) is \((\delta \delta )^{th}\) element of \(I(\Theta )\).

Bootstrap CIs

In this section of the article, we introduce two parametric Bootstrap techniques: the percentile Bootstrap method (Boot-P) pioneered by33, and the Bootstrap-T method (Boot-T) developed by34. These methods are utilized to construct confidence intervals for parametric functions \(w \left( \Theta \right) =\nu (\varTheta )\). The algorithm to generate Boot-P CI, is:

figure a

Algorithm 1

The computational algorithm to generate Boot-T CI, is given as under:

figure b

Algorithm 2

Bayes estimation

This section includes the BEs of \(\varTheta\). We utilize MCMC approache for parameter estimation of the proposed distribution. Assuming independence, gamma prior distributions are assigned to the \(\varTheta =\psi ,\xi ,\vartheta ,\eta\), with hyper parameters \(a_{i},b_{i}\ for i=1,2,3,4\). The joint prior distribution is

$$\begin{aligned} h (\varTheta ) \propto \psi ^{ a_{1} -1} \xi ^{a_{2} -1}\vartheta ^{a_{3} -1}\eta ^{a_{4} -1} e^{- b_{1}\psi - b_{2}\xi - b_{3}\vartheta - b_{4}\eta }. \end{aligned}$$

Combining the likelihood function \(L(\varTheta |{X})\) with prior \(h(\varTheta )\) through the application of Bayes’ theorem, the joint posterior density function of the unknown parameters is derived as

$$\begin{aligned} f(.|{X})\propto & \, \psi ^{n+ a_{1} -1} \xi ^{n+a_{2} -1}\vartheta ^{n+a_{3} -1}\eta ^{n+a_{4} -1} e^{- b_{1}\psi - b_{2}\xi - b_{3}\vartheta - b_{4}\eta } \prod _{i=1}^{n}\Bigg [\lambda (x_{i}) \varLambda (x_{i}) ^{\eta \vartheta -1} \nonumber \\ & \left[ 1- \varLambda (x_{i}) ^{\eta }\right] ^{-\vartheta -1}\left\{ 1+\zeta \left( \varLambda (x_{i}),\eta \right) ^{\vartheta }\right\} ^{-\psi -1} \nonumber \\ & \left[ 1-\left\{ 1+\zeta \left( \varLambda (x_{i}),\eta \right) ^{\vartheta }\right\} ^{-\psi } \right] ^{\xi -1} \Bigg ]. \end{aligned}$$
(21)

And the Bayes estimate of any parametric function \(w \left( \varTheta \right)\) is defined as

$$\begin{aligned} w\hat{\left( \varTheta \right) }_{B }= & E_{p}\left[ w \left( \varTheta \right) \right] . \end{aligned}$$
(22)

where \(E_{p}\) denotes posterior expectation. It’s worth noting that evaluating this expression in closed-form is not feasible. In this respect, we use MCMC technique to compute explicit BEs of \(w \left( l(\varTheta )\right)\).

MCMC Techniques

Since data generation directly from joint posterior density Eq. (21) is not possible, In this subsection, we employ MCMC techniques, specifically the Gibbs sampler and the Metropolis-Hastings (M-H) algorithm, to compute the BEs of unknowns. To achieve this goal, we propose the following sequence of steps to produce posterior samples using the MCMC technique for Bayesian estimation

figure c

Algorithm 3

Applications to communication disability data

In this section, the flexibility of the proposed EGIKwS-Exp distribution is demonstrated by analyzing the percentage data of Saudi individuals with communication disabilities and the degree of severity (Severe or Unable) within the age group of two years and above across 13 administrative areas. The data has been sourced from Table 1, titled Percentage Distribution of Individuals with One Disability and Degree of Severity (Severe or Unable) in the Age Group (Two Years and Over) by Type of Disability and Administrative Area, in the Disability Statistics Publication 2023 of the Disability Survey 2023. This dataset is available online, and additional details can be found in the Disability Survey 2023 The publication includes data on various types of disabilities, such as communication difficulties, memory and concentration issues, mobility challenges, personal care limitations, hearing and vision impairments (with and without aids), and the total disabled female population categorized by three degrees of difficulty: mild, severe, and extreme (cannot perform at all). These statistics are reported for 13 administrative regions of Saudi Arabia, including: Al-Riyadh, Makkah Al-Mokarramah, Al-Madinah Al-Monawarah, Al-Qaseem, Eastern Region, Aseer, Tabouk, Hail, Northern Borders, Jazan, Najran, Al-Baha and Al-Jouf. This study specifically focuses on the analysis of communication disability data at the severity level of Severe or Unable within the age group of two years and above for the aforementioned administrative areas. The analysis adopts a novel methodological and descriptive approach by utilizing the newly proposed EGIKwS-Exp distribution. This extension of the exponential distribution builds on the framework developed by28. To evaluate the performance of the EGIKwS-Exp model, a comparison is made with several well-established probability distributions. These distributions are as follows:

Fig. 2
figure 2

The fitted PDF plots for communication disabilities data.

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Fig. 3
figure 3

The fitted CDF plots for communication disabilities data.

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Fig. 4
figure 4

The fitted PP plots for communication disabilities data.

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Fig. 5
figure 5

The fitted QQ plots for communication disabilities data.

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  1. (a)

    The proposed EGIKwS-Exp distribution,

  2. (b)

    Extended generalized inverted Kumaraswamy Weibull (EGIKw-Weibull) distribution by35,

  3. (c)

    Generalized inverted Kumaraswamy Weibull (GIKw-Weibull) distribution by36,

  4. (d)

    The inverse Weibull Weibull (IW-weibuII) distribution by37,

  5. (e)

    Generalized inverse Weibull distribution (GIWD) by38,

  6. (f)

    TypeII Kumaraswamy Half Logistic Weibull (TIIKwHL-Weibull) model by39.

Fig. 6
figure 6

The fitted TTT plot for communication disabilities data.

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Fig. 7
figure 7

The fitted reliability measures of EGIKwS-Exp distribution for communication disabilities data.

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Fig. 8
figure 8

Trace plot of the posterior samples from the EGIKwS-Exp distribution for communication disabilities data.

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Table 1 MLE (SE in parentheses) and goodness of fit measures for the communication disabilities data.
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Findings in Table 1 presents the MLE of the parameters, along with their standard errors (SE) and goodness-of-fit statistics, including AIC, CAIC, BIC, and HQIC. These measures are used to assess the comparative performance of the considered models, with lower values indicating a better fit. Additionally, various plots, such as the estimated PDF and histogram (Fig. 2), CDF (Fig. 3), PP (Fig. 4) and QQ (Fig. 5) of various competitor models clrearily highlight the superiority of the EGIKwS-Exp other models, confirming the capability of the EGIKwS-Exp distribution to analyze and model the the disability data effectively. Besides, the TTT-transform curve (Fig. 6) suggests an increasing HRF, supporting the suitability of the EGIKwS-Exp distribution for fitting the data. The comparative analysis highlights the clear superiority of the EGIKwS-Exp distribution over its competitor models. It demonstrates that the proposed four-parameter probability distribution provides a better fit for the percentage data of Saudi individuals with communication disabilities. Its strong performance in empirical fitting and reliability analysis underscores its potential utility in applications related to reliability theory and beyond. In conclusion, the EGIKwS-Exp distribution effectively captures the underlying characteristics of the disability data, showcasing its flexibility and applicability in statistical modeling. Likewise, in this study, we present various reliability measures associated with the proposed EGIKwS-Exp distribution, which are crucial in reliability theory and play a central role in assessing the performance of systems and datasets. The PDF plot (Fig. 7a) illustrates the likelihood of different outcomes, highlighting the distribution’s flexibility in modeling the data. The CDF plot (Fig. 7b) captures the cumulative probability, providing insights into the overall distribution of values. The HRF plot (Fig. 7c) describes the instantaneous failure rate, offering valuable information on the reliability of the system at any given time. The RHRF plot (Fig. 7d) reveals the rate of survival over time, further aiding in understanding system longevity. The CHRF plot (Fig. 7e) tracks the accumulated risk of failure, which is vital for assessing long-term reliability. Finally, the SF plot (Fig. 7f) represents the probability of survival, providing an essential measure of reliability over time. These graphical representations once again underscore the ability of the EGIKwS-Exp distribution to effectively analyze the reliability measures of communication disability data, specifically focusing on the severity (Severe or Unable) in Saudi Arabia’s population across 13 administrative areas.

It is evident form Table 1 that the MLEs obtained for the parameters \(\psi\), \(\xi\), \(\vartheta\), and \(\eta\) were 2.7529, 8.8516, 0.1366, and 36.9708, with standard errors of 5.0107, 34.2424, 0.1962, and 31.0949, respectively. In contrast, the Bayesian approach yielded estimates of 0.2731, 1.6463, 0.6290, and 0.7493 for \(\psi\), \(\xi\), \(\vartheta\), and \(\eta\), accompanied by considerably lower standard errors of 0.5513, 0.8218, 0.1954, and 0.0751. The markedly reduced uncertainty in the Bayesian estimates indicates a superior precision in parameter estimation compared to the MLE method. Furthermore, Fig. 8 presents the trace plots for the parameters estimated via the MCMC Metropolis-Hastings algorithm. The plots clearly demonstrate good mixing of the chains, thereby confirming the convergence and reliability of the Bayesian estimation. Consequently, these results substantiate the efficacy of the Bayesian framework in fitting the novel EGIKwS-Exp distribution to the complex disability data, thereby enhancing the model’s reliability and practical applicability in capturing the underlying characteristics of communication disabilities.

Besides, we provide the CI estimation findings for the novel the EGIKwS-Exp distribution in Table 2. The 95% ACI derived via the normal approximation are considerably wider than the bootstrap-based intervals and HPD CIs. Specifically, the ACI for \(\psi ,\xi ,\vartheta , \eta\) are \((-7.0679,\,12.5737)\), \((-58.2623,\,75.9654)\), \((-0.2479,\,0.5211)\), and \((-23.9742,\,97.9158)\), respectively. In contrast, the bootstrap percentile confidence intervals are \((2.1521,\,5.9301)\) for \(\psi\), \((4.9665,\,40.5326)\) for \(\xi\), \((0.0624,\,0.2152)\) for \(\vartheta\), and (12.0267, 126.5041) for \(\eta\), while the bootstrap t (studentized) confidence intervals are \((2.1482,\,5.9742)\), (4.9221, 40.8253), \((0.0622,\,0.2176)\), and \((11.9674,\,126.6455)\) for \(\psi ,\xi ,\vartheta , \eta\), respectively. On the other hand the Bayesian HPD results are (0.2384, 1.262), (0.5636, 3.3969), (0.2034, 0.8308), and (4.6631, 65.3298) for \(\psi ,\xi ,\vartheta , \eta\), respectively. The asymptotic intervals not only span a much broader range but also include negative values for parameters that are theoretically positive, indicating that the normal approximation may be inadequate in this context. In contrast, the close agreement between the bootstrap percentile and bootstrap t intervals suggests that the bootstrap approach more reliably captures the uncertainty in the parameter estimates. Overall, these results reinforce the robustness of the proposed the EGIKwS-Exp distribution in modeling the communication disability data, as the Bayesian HPD confidence intervals provide a more precise and interpretable quantification of parameter uncertainty.

Table 2 CI estimation for the communication disabilities data.
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All of the aforementioned key statistical measures and various graphical representations of the EGIKwS-Exp, and the comparative analysis with its competitive models clearly highlight that the EGIKwS-Exp probability model with four parameters provides a better fit than other competitor models to the considered data of Saudi Individuals with Communication Disability.

The model offers valuable insights for data-driven decision-making by identifying high-risk regions and critical age groups, enabling targeted interventions like speech therapy and early screening programs. Our model’s performance make it a reliable tool for policymakers to allocate resources effectively. In summary, its applicability can be extended to broader disability analyses, ensuring evidence-based strategies for healthcare planning and policy formulation.

Conclusions and policy implications

This research introduces the EGIKwS-Exp distribution, a new four-parameter probability model designed to analyze the percentage data of Saudi individuals with severe or complete communication disabilities, aged two years and above. The dataset, sourced from the Disability Statistics Publication 2023 by the General Authority for Statistics, provides critical insights into the prevalence and regional disparities of communication disabilities across 13 administrative areas in Saudi Arabia. The EGIKwS-Exp distribution, developed as an extension of the exponential distribution, offers enhanced flexibility and adaptability for complex data modeling. Key distributional and reliability properties, including SF, HRF, RHRF, CHRF, QF, and median, were derived to facilitate a deeper understanding of disability patterns. Parameter estimation was conducted using MLE, while goodness-of-fit measures confirmed the model’s superiority over existing distributions. To strengthen inferential accuracy, we incorporated multiple estimation techniques. The ACI was constructed through the inversion of a family of hypothesis tests, ensuring reliable confidence intervals under large sample assumptions. Additionally, parametric Bootstrap methods, including Boot-P and Boot-T, were implemented to assess estimation variability. Bayesian estimation using the MCMC Metropolis-Hastings approach further refined parameter estimation, while Bayesian HPD interval estimation provided credible region assessments, reinforcing the robustness of our proposed model.

The model’s outputs offer valuable applications for data-driven policymaking. The HRF identifies high-risk regions for severe communication disabilities, guiding efficient resource allocation for speech therapy programs, assistive communication devices, and specialized education services. Furthermore, trends in the HRF pinpoint critical age groups where early interventions?such as childhood screening programs-could mitigate long-term communication impairments. Estimated regional parameters highlight substantial disparities in disability prevalence across administrative areas, supporting the formulation of region-specific policies that promote equitable healthcare and education accessibility.

While the model demonstrates strong applicability in disability analysis, its generalization to other disability types, such as mobility or cognitive disabilities, requires further validation. Expanding the analysis to multiple datasets across different countries could enhance its robustness and adaptability. Moreover, the model’s potential applications extend beyond disability studies, with promising utility in censored data analysis, medical follow-up studies, risk assessment, and reliability analysis. Overall, the EGIKwS-Exp distribution presents a novel statistical framework for understanding regional variations and severity in communication disabilities. By integrating classical and Bayesian estimation methods, this study strengthens statistical inference and enhances real-world decision-making. Future research could explore its broader applicability to various domains requiring flexible and adaptive statistical modeling.