Benefiting from statistical modeling in the analysis of current health expenditure to gross domestic product

1.   Introduction

The importance of the study lies in the fact that the decline in government support for the health sectors leads to a decline in social well-being indicators, which requires work to increase social expenditure on members of society, which is reflected in economic growth. Therefore, the importance of the study by explaining the nature and extent of the relationship between aspects of social government expenditure, especially in the field of health and GDP. Many countries in the world generally suffer from a lack of social programs and policies that affect the health of individuals and society, leading to a decline in social welfare and, at the same time, a decline in economic growth rates.

In this article, we address the statistical modeling of health protection expenditures relative to health system GDP for four different countries in terms of social policies, social welfare levels, credentials, and economic growth rates. The countries are Saudi Arabia, Egypt, Malaysia, and the United States, which provides a future vision for policy makers.

The researchers examined the issue of spending on health protection in relation to gross domestic product from different angles, including political, social, and statistical. Sardar et al. ^[1] examined the issue of creating a measure of social welfare based on unadjusted GDP, as the researchers relied on the benefit-cost analysis method for economic growth. The researchers found that there is a significant difference between GDP per capita and adjusted GDP per capita, so GDP can be used as a measure of social welfare. Sardar and Matthew ^[2] used a modern methodology to analyse the relationships between economic growth, health outcomes for individuals, and social well-being in both developing and developed countries. The researchers concluded that economic growth can improve health outcomes and expand the health sector and social well-being, but that its impact is limited due to biological laws. In addition, achieving economic growth may have a negative external effect that reduces health outcomes. Therefore, researchers used the health-to-GDP indicator to show the relationship between economic growth, health outcomes, and social well-being of individuals from the perspective of social choice. Therefore, the importance of improving health to social well-being is significant. Brent ^[3] has developed a number of alternative applications for measuring well-being that have emerged in recent decades. These applications are linked to specific indicators to assess the application of well-being when economic progress is achieved. Romina et al. ^[4] has concluded that there are several indicators to measure the specific social indicators related to well-being. In the countries of the Organisation for Economic Cooperation, the values of most of these social indicators are significantly related to average GDP per capita, but the changes in these indicators over time are not significant.

Malin ^[5] examined the relationship between economic growth and happiness from intellectual, political, and economic perspectives with what is happening in the Western world today. Gabriel ^[6] analyzed firstly, the expenditure on social protection spending in the Greek social system and second estimated the relationship between this expenditure on social protection spending and economic growth. The researcher found that the relationship between social protection spending and GDP is low compared to other European Union countries, such as Spain and Portugal. Hong ^[7] examined standard tests of the impact of economic growth on public social spending using its main components: Income Support, Pensions, and Other Health Benefits. The researcher used the mutual effect model of cross-sectional data for countries of the Organisation for Economic Cooperation and Development, which showed a strong and negative correlation between the rate of social spending. Partha and Edward ^[8] usd two classes of nonlinear statistical models to described the health care utilisation and spending. Malehi et al. found that the gamma regression model behaved well in estimating the population mean of health care costs. The approximate results are consistent when the sample size is increased. Cuckler and Sisko ^[10] described the methods underlying the econometric model. Guemmegne ^[11] et al. examined the dynamics of national health care spending in the United States from 1960 to 2011.

Statistical methods play a crucial role in analysis of medical data ^[12,13], environmental data ^[14,15], engineering data ^[16,17], social data ^[18], actuarial data ^[19], test data ^[20], reliability data ^[21], sports data ^[22], educational data ^[23], measurement system errors ^[24], risk assessment ^[25], robust analysis ^[26].

Let X be a random variable (R.V.) follows the three-parameters Weibull$(\beta, \varphi, \phi)$, and $x\ge \phi; \beta, \varphi, \phi > 0$, then its CDF (cumulative distribution function) is:

The probability density function (PDF) is

In general, actuarial are skewed positively^[27,28], unimodally shaped ^[29] and have heavy tails ^[30]. Therefore, some right-skewed and unimodal models have been utilized to modelling such data ^{[31,32,33,34]}.

2.   The statistical model

Cordeiro et al. ^[35] propose the generalized Weibull distribution (GWD-X) family. The CDF and PDF can be, respectively, written as

and

where $A(x) = -\log\left(1-{G}(x)\right)$. The special sub-models of the GWD-X family provide symmetric, asymmetric, density, unimodal, and bimodal shapes. By replacing $g(x)$ and $G(x)$ in Eqs (2.1) and (2.2) by $w(x; \beta, \varphi, \phi)$ and $W(x; \beta, \varphi, \phi)$ in Eq (1.2), give the CDF of the GWD-W model as

where $\beta' = \delta \beta$. The corresponding PDF (f), hazard function (h), survival function (S), cumulative hazard function (H), reverse hazard (r), and the quantile ($X_p$) function are

The GWD-W model are motivated by: (ⅰ) a convenient way to mutate the Weibull model, as well as have simple and closed forms; (ⅱ) improve the flexibility of Weibull model, can provide right-skewed, left-skewed form, unimodal, increasing, decreasing, symmetric, asymmetric curved densities; (ⅲ) improve the statistical properties of Weibull model; (ⅳ) includes the Weibull distribution as a special case and it can provide adequate fit for positively skewed actuarial data; (ⅴ) offers also the heavy-tailed behavior and the regular variational property.

3.   Distributional properties

Here, we derive some distribution properties of the GWD-W model. These distribution properties include linear representation, Renyi entropy, skewness, kurtosis, heavy-tailed characteristic, regular variational property, VAR (value at risk), TVAR (tail value at risk), and the identifiability property (I-P).

3.1.   Linear representation

Using the real exponential function ${\colon \mathbb {R} \to \mathbb {R} }$ that commonly defined by the following power series:

The corresponding GWD-W density can be written as

Also the corresponding GWD-W CDF can be expressed as

Based on Eq (3.2), if X is R.V. has the GWD-W model, then the transformed R.V. Z $( = \frac{X-\phi}{\varphi }\ge 0)$ has CDF $F(z; \kappa)$, and is defined by

Hence, the distribution function of the GWD-W model can be expressed as a linear combination of the transformed R.V. Z, which $Z = \frac{X-\phi}{\varphi }$. Using the same procedures, all the GWD-W mode functions Eq (2.3)–(2.9), can be derived in the form of a linear combination in terms of Z.

3.2.   Renyi entropy

Let $X$ is R.V. has the PDF $f(x)$, the Renyi entropy ($R_{\varrho}\left(X\right)$) of $X$ is a measure of variation of uncertainty. The $R_{\varrho}\left(X\right)$ for any $\varrho > 0, \varrho\neq1$ (see, Golomb ^[36]), is given by

Now, let the R.V. $X \sim$ GWD-W model, $(\kappa, \beta', \varphi, \phi)$. By substituting (2.3) in (3.5), $R_{\varrho}\left(X\right)$ become

By solving the integration and using the transformation $y = \kappa \left(\frac{x-\phi}{\varphi }\right)^{\beta' }$, we have

3.3.   The skewness and kurtosis functions

The Skewness ($S_{k}$) (see, Bowley ^[37]) and Kurtosis (${K}$) (see, Moor ^[38]) formulas are given by, respectively

And by using Eq (2.9), we have

where $\beta'' = \frac{1}{\beta' }$. Table 1 represents some numerical values of the $X_p$ at the median ($p = 0.50$) when $\kappa = 0.2, \varphi = 0.2, \phi = 0.2, )$, $S_k$ and $K$ for $\beta' = 0.1, 0.3, 0.5, 0.7, 0.9, 1, 2, 5, 10$. Figure 1 plots for the $S_{k}$ and ${K}$ and for different values of $\beta'$.

3.4.   The heavy-tailed behavior

The GWD-W model offers also the heavy-tailed behavior. A probability model is called a heavy tailed distribution, if it satisfies

Theorem 1. Let, $p, \kappa, \varphi, \phi > 0$ and $0 < \beta' < 1$, the PDF $f(y; \kappa, \beta', \varphi, \phi)$ that given in Eq (2.3) is heavy tailed distribution as $y \to \infty$.

Proof. Based on Eq (2.3), we can write

According to Seneta's ^[39] theorem, the GWD-W model in terms of CDF $S(y; \kappa, \beta', \varphi, \phi)$ is regularly varying, if it satisfies

deqnarray where $\Delta$ represents an index of regular variation.

Theorem 2. Let, $p, \kappa, \varphi > 0$, $\phi = 0$ and $0 < \beta' < 1$, the PDF $f(y; \kappa, \beta', \varphi, \phi)$ that given in Eq (2.3), the PDF $f(y; \kappa, \beta', \varphi, \phi)$ that given in Eq (2.3) is regularly varying model.

Proof. Using Eq (2.3), we have

Using Eq (3.13), the PDF $f(y; \kappa, \beta', \varphi, \phi)$ that given in Eq (2.3), the PDF $f(y; \kappa, \beta', \varphi, \phi)$ that given in Eq (2.3) with index of regular variation $\Delta = \frac{ \kappa }{\log\left(p\right)} {\left(1-p^{\beta' }\right)\left(\frac{y}{\varphi }\right)^{\beta' }}$ is regularly varying model.

3.5.   The VAR and TVAR

Based on the Monte Carlo simulation, the actuarial measures VAR and TVAR are the empirical approaches to find heavy-tailed models. Mathematically, VAR $(VAR _{q}(X))$ can be specified with a certain confidence level (C.L.) $q\in(0, 1)$, which is typically 99%, 95%, or 90%, see, ^[40]. Explicit expressions of the VAR and TVAR can be produced, as

deqnarray Let X follow the GWD-W model, based on Eq (2.9), then

The TVAR is given by

Let $\phi = 0$, and using Eqs (3.14) and (3.16), we have

and by using the transformation $z = \left(\frac{x}{\varphi }\right)^{\beta' }$, we have

On solving, we get

where $\Gamma\left(., .\right)$ is incomplete gamma constant.

Using Eqs (3.16) and (3.19), a numerical simulation study of VAR and TVAR to show empirically the heaviness of the GWD-W model tail is provided. The simulation algorithm:

(1) Using Eq (2.9), we generating random samples of size n = 100 from both GWD-W and Weibull models.

(2) Using the maximum likelihood estimation (MLE) for estimating the parameters of both models.

(3) Based on Eqs (3.16) and (3.19), we calculate the VAR and TVAR of both models.

(4) Table 2, report the numerical simulation results of VAR and TVAR for GWD-W $(\kappa = 3.5, \beta' = 2.5, \varphi = 2.5, \phi = 0)$ and Weibull $(\beta = 2.5, \varphi = 2.5, \phi = 0)$.

(5) Tables 3, report the numerical simulation results of VAR and TVAR for GWD-W $(\kappa = 0.5, \beta' = 2.5, \varphi = 0.5, \phi = 0)$ and Weibull $(\beta = 0.5, \varphi = 0.5, \phi = 0)$.

(6) For visual comparison and based on the numerical simulation results of Tables 2 and 3, Figures 2 and 3, respectively, show the shapes of the proposed two risk measures of both models.

The numerical simulation results of Figures 2 and 3 illustrate that, the VAR and TVAR results for the GWD-W models are higher than those of the Weibull models, indicating that the GWD-W models can better capture extreme events. The results also show that the GWD-W models have a higher risk measure than the Weibull models, which is beneficial for actuarial applications.

3.6.   The I-P

This subsection offers proof of the I-P of the GWD-W model for the parameters $\kappa, \beta', \varphi,$ and $\phi$.

3.6.1.   The I-P using $\kappa$

Let, $F(x; \kappa_1, \beta', \varphi, \phi) = F(x; \kappa_2, \beta', \varphi, \phi)$, the parameter $\kappa$ of the GWD-W model is called identifiable, if $\kappa_1 = \kappa_2$. To prove the I-P property of the GWD-W model for $\kappa$, we start with

3.6.2.   The I-P using $\beta'$

Let, $F(x; \kappa, \beta'_1, \varphi, \phi) = F(x; \kappa, \beta'_2, \varphi, \phi)$, the parameter $\beta'$ of the GWD-W model is identifiable, such that

3.6.3.   The I-P using $\varphi$

Let, $F(x; \kappa, \beta', \varphi_1, \phi) = F(x; \kappa, \beta', \varphi_2, \phi)$, the parameter $\varphi$ of the GWD-W model is identifiable, such that

3.6.4.   The I-P using $\phi$

Let, $F(x; \kappa, \beta', \varphi, \phi_1) = F(x; \kappa, \beta', \varphi, \phi_2)$, the parameter $\phi$ of the GWD-W model is identifiable, such that

4.   Estimations

This section assigns the MLEs to estimate the parameters of the GWD-W model. Suppose $X_{1:n}, X_{2:n}, \dots X_{r:n}$ is R.V.s from the GWD-W$(\kappa, \beta', \varphi, \phi)$ model that given in (2.3). The GWD-W$(\kappa, \beta', \varphi, \phi)$ likelihood is

and the log-likelihood function $log(l)$ is

The first partial derivatives of (4.2) w.r.t $\kappa, \beta', \varphi, \phi$ are

The MLEs $\hat{\kappa}_{ML}, \hat{\beta'}_{ML}, \hat{\varphi}_{ML},$ and $\hat{\phi}_{ML}$ of the GWD-W$(\kappa, \beta', \varphi, \phi)$ parameters are the solutions of the Eqs (4.3)–(4.6) after equating each of them by zero. One can solve them numerically to obtain the MLEs. By using any statistical software, these nonlinear system of equations can be solved.

5.   Simulation study

The numerical simulation findings is executed for the GWD-W model by R software (version 4.1.3) with the optim() function, see, Appendix A, and the argument method = "L-BFGS-B". The simulation algorithm is:

(1) We generate sample of sizes n = 25, 50, ..., 1000 from the GWD-W for two different sets of initial parameters.

set 1: $\kappa^{(0)} = 1.7, \beta'^{(0)} = 2.4, \varphi^{(0)} = 0.01,$ and $\phi^{(0)} = 3.4$,

set 2: $\kappa^{(0)} = 0.8, \beta'^{(0)} = 1.5, \varphi^{(0)} = 0.02,$ and $\phi^{(0)} = 1.2$.

(2) Use sets 1 and 2 for conducting the MLEs of each parameter $\kappa, \beta'_1, \varphi,$ and $\phi$.

(3) Repeated the steps (1) and (2) M = 1000 times.

(4) Obtain the estimates, then calculate the Bias and the Estimated Risk (ER).

(5) The Biases and ERs for the parameter $\lambda( = \kappa, \beta', \varphi, \phi)$ are given by, respectively

and

(6) Table 4 presents the estimates, Bias and ER, respectively, for set 1.

(7) Table 5 presents the estimates, Bias and ER, respectively, for set 2.

(8) Figure 4 shows graphically the simulation results of Table 4.

(9) Figure 5 shows graphically the simulation results of Table 5.

The simulation results in Tables 4 and 5 and Figures 4 and 5 can be explained though the following steps:

(1) The MLE estimates of ${\kappa}$ and $\varphi$ are overestimated for both sets 1 and 2.

(2) The MLE estimates of ${\phi}$ and $\beta'$ are underestimated for both set 1, while the MLE estimates of ${\phi}$ and $\beta'$ are overestimated set 2.

(3) The performance of MLE was good even when the small sample sizes.

(4) Increasing the sample size n leads to a decrease in the estimated biases and ERs, which approach zero as n increases. These results demonstrate both the efficiency and consistency properties of the MLEs.

6.   Application of the GWD-W model

Probability distributions are widely used in various real data modeling applications. Heavy tailed distributions play an important role in data analysis and modeling in various application areas of life such as economics, financial mathematics, actuarial science, risk management, banking, etc. The quality of statistical procedures mainly depends on the probability distributions given for the application data in question. Based on the data were collected by the World Bank (accessed on 15 September 2021), we address the statistical modeling of current health expenditure (% of GDP) for four different countries in terms of social policies, social welfare levels, credentials, and economic growth rates. The countries are United States, Malaysia, Egypt, and Saudi Arabia. The data are analysed and the maximum likelihood estimates of the model parameters are obtained, see, Appendix B. The data sets for the last fifty years are provided in Table 6.

The goodness-of-fit results of the GWD-W model are compared with some other models. The comparison of the GWD-W model is made with some important distributions including generalized Weibull two-parameter Weibull distribution (GW-OWD), exponentiated distribution (EXP-WD) and two-parameter Weibull distribution (TW-D). The CDF of the competing probability models are, respectively, given by

Table 7 shows the descriptive statistics of the proposed current health expenditure data sets. Table 8 shows the result of the estimates as well as the one-sample Kolmogorov-Smirnov test. Table 9 compare the GWD-W model via some recognition criterion, such as, Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC) and consistent Akaike information Criterion (CAIC). The results in Tables 7 and 8 suggest that the GWD-W model provides a better fit than other competing models and could be chosen as a suitable model for analyzing all data sets. The boxplot, the simulated PDF, simulated CDF, the Kaplan-Meier survival function, the PP plot and Q-Q plot of the proposed current health expenditure data of United States, Malaysia, Egypt, and Saudi Arabia are shown in Figures 6–9, respectively. These figures confirm the best fitting of the GWD-W model for the statistical modeling of current health expenditure for all four different countries. The proposed simulated PDF fits the histogram plot very well. The simulated CDF fits the empirical plot very well. The proposed model fits the Kaplan-Meier survival plot very well. The PP plot and Q-Q plot fits the symmetric line very well.

7.   Discussion and future works

We have presented a new application for modeling health expenditures as a percentage of GDP in a comparative study for four different countries, namely the United States, Malaysia, Egypt, and the Kingdom of Saudi Arabia. We have presented a model with flexible properties and proven efficiency in statistical modeling of a new application that represents health spending as a percentage of GDP for four different countries in terms of social policies, welfare levels, credentials, and economic growth rates.

The Weibull distribution is a versatile tool that can be used to model a variety of phenomena. It has been used to model downtime in reliability engineering, survival times in medical research, and other applications. The generalized Weibull family of distributions provides additional flexibility for modeling data with more complex shapes. The new distributions generated from the generalized Weibull family can be used to better describe data that does not fit the traditional Weibull distribution.

In this work, we used an extended, updated version of the Weibull distribution, called a generalized Weibull distribution model, which has good statistical properties in terms of flexibility and goodness of fit. Some distributional properties and statistical functions were derived in closed forms, including Renyi entropy, skewness, kurtosis, highly fluctuating behavior, regular variation, and identifiable property.

Based on Monte Carlo simulation, the empirical studies of the actuarial measures provide evidence of the severity of the GWD-W model tail. The results of VAR and TVAR for the GWD-W models in Tables 2 and 3 and Figures 2 and 3 are higher than those of the Weibull models, indicating that the GWD-W models better represent extreme events. The results also show that the GWD-W models have a higher risk measure than the Weibull models, which is beneficial for actuarial applications.

The results of the statistical analysis show that the GWD-W model is a better fit to the data than the traditional Weibull distribution and other competing models. Our study provides useful information for decision makers with regard to allocating resources for health protection expenditures. AIC, BIC, HQC, and CAIC are all criteria used to compare different models and select the best model for a given data set. These criteria measure how well a model fits the data and penalize models with more parameters to prevent overfitting. Both the goodness of fit of the model and its complexity are considered when calculating the criteria, with more complex models being penalized more heavily. From the results in Table 8, the GWD-W model could be selected as the best model among the fitted models.

8.   Conclusions

In this article, we address statistical analysis and modeling of World Bank data to compare health protection expenditure as a percentage of GDP in four countries. In statistically modeling the data, we used an extended, updated version of the Weibull distribution with heavy tails, the generalized Weibull distribution (Weibull model). Some of the properties and characteristics of the distribution were discussed, and some of the properties were proved. The model presented is very flexible and can be used effectively for modeling data with heavy tails. The empirical studies of the actuarial measures show the tail heaviness of the generalized Weibull distribution of the Weibull model and can better capture extreme events. The results also show the efficiency and consistency properties of the maximum likelihood approaches based on the proposed model. The results show that the proposed model fits better than other competing models and could be chosen for statistical analysis and modeling of health protection expenditures relative to gross domestic product in a comparative study of the United States, Malaysia, Egypt, and the Kingdom of Saudi Arabia.

Acknowledgments

The study was funded by Researchers Supporting Project number (RSPD2023R749), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.