Confidence Intervals for the Coefficient of Variation in Delta Inverse Gaussian Distributions

Abstract

The inverse Gaussian distribution is characterized by its asymmetry and right-skewed shape, indicating a longer tail on the right side. This distribution represents extreme values in one direction, such as waiting times, stochastic processes, and accident counts. Moreover, depending on if the accident counts data can occur or not and may have zero value, the Delta Inverse Gaussian (Delta-IG) distribution is more suitable. The confidence interval (CI) for the coefficient of variation (CV) of the Delta-IG distribution in accident counts is essential for risk assessment, resource allocation, and the creation of transportation safety policies. Our objective is to establish CIs of CV for the Delta-IG population using various methods. We considered seven CI construction methods, namely Generalized Confidence Interval (GCI), Adjusted Generalized Confidence Interval (AGCI), Parametric Bootstrap Percentile Confidence Interval (PBPCI), Fiducial Confidence Interval (FCI), Fiducial Highest Posterior Density Confidence Interval (F-HPDCI), Bayesian Credible Interval (BCI), and Bayesian Highest Posterior Density Credible Interval (B-HPDCI). We utilized Monte Carlo simulations to assess the proposed CI technique for average widths (AWs) and coverage probability (CP). Our findings revealed that F-HPDCI and AGCI exhibited the most effective coverage probability and average widths. We applied these methods to generate CIs of CV for accident counts in India.

1. Introduction

Observations that are non-negative with an excess of zeros are frequently seen in areas such as economics, biomedical sciences, life sciences, and meteorology. These data often exhibit asymmetry, with a distribution tilted to the right (positively skewed) and heteroscedastic, meaning that the spread of the data varies. In such cases, many observations cluster at zero, while the positive values follow a right-skewed pattern. For example, in a clinical experiment, the duration of episodes such as low blood pressure, excessive perspiration, and rapid heartbeat is measured minutes after administering a particular medicine. Some patients may show a measurement value of zero, while those with non-zero values tend to follow an asymmetric, right-skewed distribution.

The data model that was appropriate for incorporating zero-inflated non-negative observations into a continuous distribution was examined by Liu et al. [1]. The data that were utilized encompassed information derived from the microbiota, medical bills, and the amount of alcohol consumed. Later, several researchers proposed utilizing the Inverse Gaussian (IG) distribution to analyze these kinds of data. To illustrate this, Heller et al. [2] investigated insurance data with zero-inflated information and discovered that the Inverse Gaussian distribution was better for analysis than the Gamma distribution. Ma and Yan [3] found that the Inverse Gaussian distribution was suitable for analyzing data above zero values through examining the frequency of traffic accidents in Colorado, USA. Regression models were employed to predict accident frequencies and estimate parameters using the maximum likelihood (ML) method. Punzo [4] also employed the Inverse Gaussian distribution to analyze insurance data about bodily damage claims and economic data on family wages in Italy.

Researchers have examined several distributions to develop confidence intervals (CIs) for zero-inflated data, considering the unique peculiarities of this type of data. Zhou and Tu [5] examined CIs for the ratio of means in lognormal distributions using maximum likelihood (ML)-based approaches and Bootstrap CI. Several researchers have examined techniques for determining CIs for various lognormal distribution parameters and proposed a range of approaches.

Nevertheless, further study by Chhikara and Folks [6] has shown that the Inverse Gaussian Distribution offers a variety of preferences and is less complex than other distributions like the lognormal distribution. This suggests that the Inverse Gaussian Distribution is more adaptable. Chatzidiamantis et al. [7] further suggested that the lognormal distribution is deficient in some features due to the absence of a distinct Moment Generating Function (MGF). Moreover, the mean of the lognormal distribution has a distinct distribution pattern that deviates from the overall population distribution. This characteristic might provide challenges when analyzing the lognormal distribution compared to other distributions. On the other hand, the Inverse Gaussian Distribution is more suited for wind energy data and facilitates the estimate of parameters. The distribution appears comparable to the three-parameter Weibull distribution but has less complexity [8].

There are several circumstances in which the Delta Inverse Gaussian (Delta-IG) distribution can be applied for estimating. This encompasses approximating the mean expense of insurance replacement, the daily frequency of accidents, the volume of precipitation in a certain region, and the mortality rate due to COVID-19 during a specified timeframe in each nation. Various research studies have been conducted on regression models for data that follows a Delta-IG distribution. Some of these investigations have focused on establishing CIs for the mean, differences in means, and ratios of means for the zero-adjusted inverse Gaussian distribution [9]. However, there is still a lack of studies examining the calculation of CIs for the coefficient of variation (CV) of the Delta-IG distribution. The CV measures the ratio between the standard deviation and the mean, which indicates data variability with various units.

This research aims to concentrate on researching the estimate of CIs for the CV of the Delta-IG distribution using different approaches. The seven approaches under consideration are: Generalized Confidence Interval (GCI), Adjusted-Generalized Confidence Interval (AGCI), Parametric Bootstrap Percentile Confidence Interval (PBPCI), Fiducial Confidence Interval (FCI), Fiducial Highest Posterior Density Confidence Interval (F-HPDCI), Bayesian Credible Interval (BCI), and Bayesian Highest Posterior Density Credible Interval (B-HPDCI). The CIs values produced are subsequently compared under various scenarios, considering the constraints and peculiarities of each estimating method.

2. The Confidence Interval for the Coefficient of Variation in a Delta-IG Distribution

First, it is assumed that all non-zero observations in a population can be represented by a suitable continuous distribution. In this population, there exist both zero and non-zero positive observations, with a probability of containing zero observations denoted as $p$, where $0<p<1$, which is represented as $\delta$ (delta) in the Delta-IG distributions. The sample consists of $n$ observations, which may be divided into two categories: zero observations ($n_0$) and nonzero observations ($n_1$), while the total number of observations in the sample is denoted as $n$ where $n=n_0+n_1$. We employ an Inverse Gaussian distribution as a suitable model for the non-zero data. The population’s random variable $X=\left(X_1, X_2,\dots X_n\right),$ which include only non-negative observations, is said to conform to the Delta-IG distribution, characterized by a mean ($\mu$) and shape parameter ($\lambda$), where both $\mu$ and $\lambda$ are greater than zero ($\mu , \lambda >0$) with $p$ representing the probability of containing zero observations. This distribution is indicated as $X~ \mathrm{D}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{a}$-$\mathrm{I}\mathrm{G}(p,\mu ,\lambda )$ and the density function is given by

$$f\left(x;p,\mu ,\lambda \right)=\left\{\begin{array}{c}p ;x=0\\ \left(1-p\right)\sqrt{{\displaystyle \frac{\lambda }{2\pi x^3}}}exp\left[{\displaystyle \frac{-\lambda {\left(x-\mu \right)}^2}{2{\mu }^{2}x}}\right] ;x>0\end{array}\right., \mu ,\lambda >0, \mathrm{where} P\left(X=0\right)=p.$$

(1)

The maximum likelihood estimates (MLEs) of the parameters $p, \mu ,$ and $\lambda$ of the Delta-IG distribution are as follows: $\widehat{p}=n_0/n$, $\widehat{\mu }=\overline{X}={\displaystyle \frac{1}{n_1}}{\sum }_{i=1}^{n_1}{X}_i,$ and $n_1{\widehat{\lambda }}^{-1}=V={\sum }_{i=1}^{n_1}(X_i^{-1}-{\overline{X}}^{-1})$, respectively.

The expected value of the Delta-IG distribution is given by $E\left(X\right)=q\mu$ while the variance is given by $Var\left(X\right)={\mu }^{2}q\left({\displaystyle \frac{\mu }{\lambda }}+p\right)$. The CV of $X$ may be calculated using the formula $\theta =\sqrt{(\mu +\lambda p)/q\lambda }$. Suppose that a random sample, denoted as $n$, is selected from the Delta-IG population, resulting in a set of values $X=(X_1, X_2,\dots , X_n)$. Here, $n_0$ is the number of successful outcomes in $n$ Bernoulli trials and $n_0$ represents the occurrence of zero observations as a success. We may denote this as $n_0~Bin(n,p).$

2.1. Generalized Confidence Interval (GCI)

Weerahandi [10] introduced the GCI technique, which relies on the Generalized Pivotal Quantity (GPQ) methodology. The Pivotal Quantity (PQ) and its generalization, the GPQ approach, are different in that the former may be seen from the viewpoint of the sample group and the parameters of interest, while the latter is dependent on nuisance parameters. The nuisance parameter has no bearing on the observed value of the GPQ technique, and the parameters of the method have unknown values that are independent of any distribution.

Ye, Ma, and Wang [11] proposed a technique for creating CIs for the mean value of the inverse Gaussian distribution over many populations using the GPQ approach. Chankham, Niwitpong, and Niwitpong [12] proposed that the estimation of CIs for the CV of the inverse Gaussian distribution should have a coverage probability of around 0.95 and the smallest average width. This may be carried out by utilizing the GCI technique to estimate the intervals. Thus, the GCI technique was selected as the foundational approach for evaluating the accuracy of CI estimation compared to other approaches.

The Generalized Pivotal Quantities (GPQs) for the parameters $\lambda$ and $\mu$, as proposed by Ye, Ma, and Wang, are

$$R_{\lambda }={\displaystyle \frac{n_1\lambda V_i}{n_1{\upsilon }_i}} ~{\displaystyle \frac{{\chi }_{n_1-1}^2}{n_1{\upsilon }_i}} ,i=\mathrm{1,2},\dots ,k,$$

(2)

where ${\chi }_{n_1-1}^2$ is a Chi-square distribution with $n_1-1$ degrees of freedom and

$$R_{\mu }={\displaystyle \frac{{\overline{X}}_i}{\left|1+\frac{\sqrt{n_1\lambda }({\overline{X}}_i-\mu )}{\mu \sqrt{{\overline{X}}_i}}\sqrt{\frac{{\overline{X}}_i}{n_1{R}_{{\lambda }_i}}}\right|}}~{\displaystyle \frac{{\overline{X}}_i}{\left|1+Z_i\sqrt{\frac{{\overline{X}}_i}{n_1{R}_{{\lambda }_i}}}\right|}},$$

(3)

where $Z_i~N\left(\mathrm{0,1}\right)$. And the notation $~$ means approximately distributed.

The Generalized Pivotal Quantity (GPQ) of parameter $p$ proposed by Wu and Hsieh [13] is

$$R_p={\mathit{sin}}^2\left[arcsin\sqrt{\widehat{p}}-{\displaystyle \frac{T}{2\sqrt{n}}}\right],$$

(4)

where $T=2\sqrt{n}(arcsin\sqrt{\widehat{p}}-arcsin\sqrt{p})~N\left(\mathrm{0,1}\right)$.

From the GPQs of the three parameters, the GPQ for the CV of a Delta-IG distribution using the GCI method is given by

$$R_{\theta }=\sqrt{\left(R_{\mu }+R_{\lambda }{R}_p\right)/(1-R_p)R_{\lambda }}.$$

(5)

Then, the 100(1 − $\alpha$)% two-sided CI of CV for Delta-IG distribution based on the GCI method is given by

$${CI}_{\mathrm{G}\mathrm{C}\mathrm{I}}=\left[L_{\mathrm{G}\mathrm{C}\mathrm{I}},U_{\mathrm{G}\mathrm{C}\mathrm{I}} \right]=\left[R_{\theta }\left(\alpha /2\right),R_{\theta }(1-\alpha /2)\right],$$

(6)

where $R_{\theta }\left(\alpha /2\right)$ and $R_{\theta }(1-\alpha /2)$ are the ${100}_{{\displaystyle \frac{\alpha }{2}}}$-th and ${100}_{\left(1-{\displaystyle \frac{\alpha }{2}}\right)}$-th percentiles of the distribution of $R_{\theta }$, respectively, defined in Algorithm 1.

Algorithm 1: The GCI method
1.	$\mathrm{Generate} X_1, X_2,\dots , X_n$$\mathrm{from} \mathrm{Delta}-\mathrm{IG} \mathrm{distribution} \mathrm{and} \mathrm{compute} \mathrm{for} {\widehat{\mu }}_i$$\mathrm{and} {\widehat{\lambda }}_i$.
2.	$\mathrm{Generate} {\chi }_{n_1-1}^2$$\mathrm{and} Z_i$ from Chi-square and standard normal distributions, respectively.
3.	$\mathrm{Compute} R_{{\lambda }_i}$$\mathrm{and} R_{{\mu }_i}$ using (2) and (3), respectively, and compute $R_{p_i}$ by using (4).
4.	$\mathrm{Calculate} R_{{\theta }_i}$ using (5).
5.	$\mathrm{Repeat} \mathrm{steps} 2–4 \mathrm{for} i$ = 1, 2, …, 5000 times.
6.	$\mathrm{Obtain} {CI}_{\mathrm{G}\mathrm{C}\mathrm{I}}$$\mathrm{from} R_{\theta }\left(\alpha /2\right)$$\mathrm{and} R_{\theta }(1-\alpha /2)$.

2.2. Adjusted-Generalized Confidence Interval (AGCI)

After the GCI approach was applied to estimate the CIs of CV for the Delta-IG distribution, there was another generalized pivotal quantity (GPQ) of parameter $\mu$, as recommended by Krishnamoorthy and Tian [14]. This AGCI method will use this GPQ of parameter $\mu$ instead of GPQ for the parameter $\mu$ in (3) of the GCI method. The GPQ of parameter $\mu$ suggested by Krishnamoorthy and Tian is given by

$$R_{\tilde{\mu }}={\displaystyle \frac{{\overline{X}}_i}{max\left\{\mathrm{0,1}+t_{n_1-1}\sqrt{\frac{{\overline{X}}_i{v}_i}{n_1-1}}\right\}}},$$

(7)

where $t_{n_1-1}$ is Student’s t distribution with $n_1-1$ degrees of freedom. The GPQs of parameters $\lambda$ and $p$ are like the GCI method.

Thus, the GPQ for the CV of a Delta-IG distribution using the AGCI method is given by

$$R_{\tilde{\theta }}=\sqrt{\left(R_{\tilde{\mu }}+R_{\lambda }{R}_p\right)/(1-R_p)R_{\lambda }}.$$

(8)

The 100(1 − $\alpha$)% two-sided CI of CV for Delta-IG distribution estimated by the AGCI method is given by

$${CI}_{\mathrm{A}\mathrm{G}\mathrm{C}\mathrm{I}}=\left[L_{\mathrm{A}\mathrm{G}\mathrm{C}\mathrm{I}},U_{\mathrm{A}\mathrm{G}\mathrm{C}\mathrm{I}}\mathrm{}\right]=\left[R_{\tilde{\theta }}\left(\alpha /2\right),R_{\tilde{\theta }}(1-\alpha /2)\right].$$

(9)

The algorithm of the AGCI method is like the GCI method but uses (7) instead of (3) to compute the GPQ of $\mu$.

2.3. Parametric Bootstrap Percentile Confidence Interval (PBPCI)

Efron and Tibshirani [15] introduced the PBPCI approach by utilizing resampling procedures, wherein a resampled subgroup is selected from the original sample group. Each distribution is analyzed to determine the particular properties of the estimated statistics. The PBPCI approach is employed in this study to estimate CIs, utilizing 1000 resamples.

Chankham, Niwitpong, and Niwitpong [11] found that the GCI approach is more accurate than the PBPCI method in calculating CIs. However, to demonstrate the range of estimating approaches, this method will be contrasted with others.

Therefore, the 100(1 − $\alpha$)% two-sided CI of CV for Delta-IG distribution from the PBPCI method is given by

$${CI}_{\mathrm{P}\mathrm{B}\mathrm{P}\mathrm{C}\mathrm{I}}=\left[L_{\mathrm{P}\mathrm{B}\mathrm{P}\mathrm{C}\mathrm{I}},U_{\mathrm{P}\mathrm{B}\mathrm{P}\mathrm{C}\mathrm{I}} \right]=\left[{\theta }^{*}\left(\alpha /2\right),{\theta }^{*}(1-\alpha /2)\right],$$

(10)

where ${\theta }^{*}\left(\alpha /2\right)$ and ${\theta }^{*}(1-\alpha /2)$ are the ${100}_{{\displaystyle \frac{\alpha }{2}}}$-th and ${100}_{\left(1-{\displaystyle \frac{\alpha }{2}}\right)}$-th percentiles of the distribution of ${\theta }^{*}$, defined in Algorithm 2.

Algorithm 2: The PBPCI method
1.	$\mathrm{Generate} X_1, X_2,\dots , X_n$ from Delta-IG distribution.
2.	Obtain bootstrap sample $X_1^{*}, X_2^{*},\dots , X_n^{*}$ $\mathrm{from} X_1, X_2,\dots , X_n$ obtained in step 1.
3.	$\mathrm{Compute} {\theta }^{*}$.
4.	Repeat steps 2–3 for 1000 times.
5.	$\mathrm{Obtain} {CI}_{\mathrm{P}\mathrm{B}\mathrm{P}\mathrm{C}\mathrm{I}}$ using (10).

2.4. Fiducial Confidence Interval (FCI)

In the context of estimating, the idea of fiducial CIs includes the process of deriving a probability distribution for an unknown quantity directly from observed data. This is accomplished without the utilization of previous distributions or the reliance on repeated sampling. In this method, which was first presented by Fisher (1930) [16], a fiducial distribution is generated. This distribution represents the uncertainty of the parameter, and it is based only on the data and the model. The fiducial CIs that were produced consequently provide an objective estimate of the parameter. This interval is comparable to Bayesian credible intervals, but it does not need either subjective or objective priors. However, even though it provides a fresh viewpoint on interval estimation, the fiducial technique is frequently criticized for its unorthodox handling of the parameters’ uncertainty.

The Gibbs sampling method by Geman and Geman (1984) [17] can be used in fiducial inference. The sample distributions of the MLEs of parameters $p, \mu , \mathrm{a}\mathrm{n}\mathrm{d} \lambda$ are used in the Delta-IG distribution, and the Fiducial distribution of parameters $\mu \mathrm{a}\mathrm{n}\mathrm{d} \lambda$ can be obtained by substituting the parameters with the MLEs. When both parameters appear in the sample distribution, they are treated as follows $\mu ~ IG\left(\widehat{\mu },n_1\widehat{\lambda }\right)$ and $\lambda ~ \left(\widehat{\lambda }/n_1\right){\chi }_{n_1-1}^2$, where $\widehat{\mu } \mathrm{a}\mathrm{n}\mathrm{d} \widehat{\lambda }$ are the MLEs of $\mu$ and $\lambda$, respectively.

The 100(1 − $\alpha$)% two-sided CI of CV for Delta-IG distribution from the FCI method is given by Algorithm 3 and

$${CI}_{\mathrm{F}\mathrm{C}\mathrm{I}}=\left[L_{\mathrm{F}\mathrm{C}\mathrm{I}},U_{\mathrm{F}\mathrm{C}\mathrm{I}} \right]=\left[{\theta }^t\left(\alpha /2\right),{\theta }^t(1-\alpha /2)\right].$$

(11)

Algorithm 3: The FCI method
1.	$\mathrm{Given} \mathrm{the} \mathrm{initial} \mathrm{values} \left(\mathrm{MLEs}\right) \mathrm{of} \mathrm{parameters} \left({\mu }^{\left(0\right)},{\lambda }^{\left(0\right)}\right)$
2.	$\mathrm{Generate} {\mu }^{\left(t\right)}~IG({\widehat{\mu }}^{(t-1)},n{\widehat{\lambda }}^{(t-1)})$.
3.	$\mathrm{Generate} {\lambda }^{\left(t\right)}~({\widehat{\lambda }}^{\left(t-1\right)}/n){\chi }_{n-1}^2$.
4.	$\mathrm{Repeat} \mathrm{steps} 2–3 \mathrm{for} t$ = 1, 2, …, 20,000 times, which is the number of replications of MCMC.
5.	Burn-in 1000 samples and calculate for the interest parameter.
6.	$\mathrm{Obtain} {CI}_{\mathrm{F}\mathrm{C}\mathrm{I}}$ using (11).

2.5. Fiducial Highest Posterior Density Confidence Interval (F-HPDCI)

The MCMC method may be utilized to calculate the CIs derived from the parameter with the highest posterior density function [18], provided that all of the values contained within the interval have higher posterior densities than the values considered to be outside of the interval [19]. To determine the $100(1-\alpha )\%$ HPD intervals, we used the HDInterval package in the R programming language to compute the $L_{F-HPDCI}$ and $U_{F-HPDCI}$ statistics.

The 100(1 − $\alpha$)% two-sided CI of CV for Delta-IG distribution based on the F-HPDCI method is given by

$${CI}_{\mathrm{F}-\mathrm{H}\mathrm{P}\mathrm{D}\mathrm{C}\mathrm{I}}=\left[L_{\mathrm{F}-\mathrm{H}\mathrm{P}\mathrm{D}\mathrm{C}\mathrm{I}},U_{\mathrm{F}-\mathrm{H}\mathrm{P}\mathrm{D}\mathrm{C}\mathrm{I}}\right].$$

(12)

2.6. Bayesian Credible Interval (BCI)

The Bayesian method, also known as Bayesian parameter estimation, is a technique that is used to discover a parameter estimate. This estimator is created by combining a prior distribution with a likelihood function derived from random sample data. We will obtain the posterior probability density function of the random sample dataset provided to us. To find the posterior probability density function of the CV of the Delta-IG distribution, we need to combine the likelihood function for the Delta-IG distribution and the prior density function. The likelihood function for the Delta-IG distribution is given by

$$L(\theta |\underset{\_}{x})\propto p^{n_0}{\left(1-p\right)}^{n_1}{\lambda }^{n_1/2}\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{\lambda }{2{\mu }^2}}\left(t_1-2n_1\mu +t_2{\mu }^2\right)\right),$$

(13)

where $t_1={\sum }_{i=1}^{n_1}{X}_i$, and $t_2={\sum }_{i=1}^{n_1}{X}_i^{-1}$.

The posterior density of $\theta$ is given by

$$f\left(\theta \right|\underset{\_}{y})\propto f\left(\theta \right)L(\theta \left|\underset{\_}{x}\right).$$

(14)

However, the posterior distributions derived from the Delta-IG distribution cannot be computed. As a result, we employed the Markov Chain Monte Carlo simulation (MCMC) with Gibbs sampling to supply them with the parameter. We used Markov Chain Monte Carlo simulation to calculate the single CV, define the prior distribution of the parameter $\mu$ to a uniform distribution, a gamma distribution for parameter $\lambda$ and a beta distribution for parameter $p$. We computed Bayesian parameters with the package R2OpenBUGS in the R programming language.

The 100(1 − $\alpha$)% two-sided CI of CV for Delta-IG distribution based on the BCI method is given by Algorithm 4 and

$${CI}_{\mathrm{B}\mathrm{C}\mathrm{I}}=\left[L_{\mathrm{B}\mathrm{C}\mathrm{I}},U_{\mathrm{B}\mathrm{C}\mathrm{I}}\right].$$

(15)

Algorithm 4: The BCI method
1.	$\mathrm{Generate} X_1, X_2,\dots , X_n$ from Delta-IG distribution.
2.	Given prior distribution $\mu ~\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(\mathrm{a},\mathrm{b})$, $\lambda ~\mathrm{g}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}({\alpha }_1,{\beta }_1)$, and $p~\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}({\alpha }_2,{\beta }_2)$ by trial hyperparameters.
3.	Using Gibbs sampling in R programming, generate posterior distribution for three parameters $(p,\mu ,\lambda )$.
4.	$\mathrm{Compute} \widehat{\theta }=\sqrt{(\widehat{\mu }+\widehat{\lambda }\widehat{p})/\widehat{q}\widehat{\lambda }}$
5.	$\mathrm{Repeat} \mathrm{steps} 3–4 \mathrm{for} t$ = 1, 2, …, 20,000 times, which is the number of replications of MCMC.
6.	Burn-in 5000 samples.
7.	$\mathrm{Obtain} {CI}_{\mathrm{B}\mathrm{C}\mathrm{I}}$ using (15).

2.7. Bayesian Highest Posterior Density Credible Interval (B-HPDCI)

The Bayesian-highest posterior density CIs refers to a range where every point inside it holds greater significance than any point outside of it. Additionally, it holds the distinction of being the briefest duration [19]. This work employs the package HDInterval in R programming to compute the Bayesian-highest posterior density CIs.

The 100(1 − α)% two-sided CI of CV for Delta-IG distribution based on the B-HPDCI method is given by

$${CI}_{\mathrm{B}-\mathrm{H}\mathrm{P}\mathrm{D}\mathrm{C}\mathrm{I}}=\left[L_{\mathrm{B}-\mathrm{H}\mathrm{P}\mathrm{D}\mathrm{C}\mathrm{I}},U_{\mathrm{B}-\mathrm{H}\mathrm{P}\mathrm{D}\mathrm{C}\mathrm{I}}\right].$$

(16)

3. Simulation Studies

The study examined seven approaches (GCI, AGCI, PBPCI, FCI, F-HPDCI, BCI, and B-HPDCI) for constructing CIs for the CV within the Delta-IG distribution. The research used a Monte Carlo simulation implemented in the R programming language. These seven approaches were evaluated by analyzing their coverage probabilities (CPs) and average widths (AWs). Since there are no asymptotic proofs for the coverage probabilities and expected widths of these confidence intervals. Thus, this research solely compares these confidence intervals utilizing the simulation approach provided in Algorithms 1–4. Choosing an optimal strategy relies on two crucial factors: the coverage probabilities of the method should be greater or close to the nominal confidence level at 0.95, then comparing the average widths of each method should give the shortest width. The simulation was conducted with a total of 5000 replications. To produce sample data for the CV of the Delta-IG distribution, data were generated for $X$ following the $\mathrm{D}\mathrm{e}\mathrm{l}\mathrm{t}\mathrm{a}$-$\mathrm{I}\mathrm{G}(p,\mu ,\lambda )$ distribution. The sample size ($n$) was 50, 100, and 200. The mean ($\mu$) was set to either 0.5 or 1.0. The shape parameter ($\lambda$) was set to 1, 5, and 10. The probability ($p$) was set to 0.3 and 0.5.

4. Results

The findings from

Table 1. The coverage probabilities of the 95% confidence intervals for the coefficient of variation in a Delta-IG distribution.

n	m	l	p	Coverage Probability
				GCI	AGCI	PBPCI	FCI	F-HPDCI	BCI	B-HPDCI
50	0.5	1	0.3	0.9822	0.9454	0.8712	0.9540	0.9484	0.9670	0.9620
			0.5	0.9792	0.9466	0.8422	0.9528	0.9414	0.9560	0.9370
		5	0.3	0.9570	0.9472	0.8796	0.9488	0.9480	0.9600	0.9100
			0.5	0.9592	0.9470	0.8464	0.9488	0.9444	0.9560	0.9350
		10	0.3	0.9540	0.9484	0.8794	0.9492	0.9494	0.9590	0.9350
			0.5	0.9544	0.9478	0.8476	0.9482	0.9456	0.9650	0.9150
	1	1	0.3	0.9940	0.9462	0.8628	0.9526	0.9398	0.9410	0.9320
			0.5	0.9906	0.9462	0.8392	0.9542	0.9364	0.9570	0.9400
		5	0.3	0.9654	0.9482	0.8758	0.9506	0.9486	0.9580	0.9380
			0.5	0.9670	0.9474	0.8456	0.9514	0.9438	0.9610	0.9420
		10	0.3	0.9590	0.9468	0.7724	0.9530	0.9450	0.9650	0.9360
			0.5	0.9592	0.9470	0.8464	0.9488	0.9444	0.9410	0.9210
100	0.5	1	0.3	0.9828	0.9514	0.9086	0.9538	0.9500	0.9570	0.9480
			0.5	0.9846	0.9554	0.9024	0.9590	0.9518	0.9440	0.9280
		5	0.3	0.9632	0.9522	0.9132	0.9504	0.9496	0.9610	0.9310
			0.5	0.9656	0.9540	0.9008	0.9556	0.9508	0.9560	0.9390
		10	0.3	0.9580	0.9536	0.9166	0.9514	0.9500	0.9570	0.9360
			0.5	0.9590	0.9538	0.8980	0.9540	0.9508	0.9590	0.9320
	1	1	0.3	0.9912	0.9534	0.9042	0.9546	0.9490	0.9420	0.9290
			0.5	0.9926	0.9550	0.8948	0.9538	0.9450	0.9410	0.9140
		5	0.3	0.9692	0.9530	0.9120	0.9500	0.9480	0.9590	0.9400
			0.5	0.9724	0.9540	0.8998	0.9546	0.9500	0.9560	0.9450
		10	0.3	0.9632	0.9522	0.9132	0.9504	0.9496	0.9390	0.9160
			0.5	0.9656	0.9540	0.9008	0.9556	0.9508	0.9410	0.9200
200	0.5	1	0.3	0.9814	0.9476	0.9302	0.9536	0.9518	0.9620	0.9380
			0.5	0.9826	0.9528	0.9246	0.9536	0.9494	0.9440	0.9330
		5	0.3	0.9542	0.9464	0.9350	0.9510	0.9502	0.9610	0.9340
			0.5	0.9614	0.9516	0.9278	0.9502	0.9496	0.9450	0.9290
		10	0.3	0.9506	0.9466	0.9356	0.9482	0.9500	0.9410	0.9260
			0.5	0.9560	0.9516	0.9256	0.9498	0.9488	0.9590	0.9400
	1	1	0.3	0.9942	0.9470	0.9272	0.9532	0.9476	0.9560	0.9350
			0.5	0.9934	0.9538	0.9214	0.9516	0.9478	0.9620	0.9420
		5	0.3	0.9630	0.9464	0.9336	0.9518	0.9504	0.9540	0.9410
			0.5	0.9690	0.9530	0.9284	0.9530	0.9490	0.9550	0.9420
		10	0.3	0.9542	0.9464	0.9350	0.9510	0.9502	0.9430	0.9360
			0.5	0.9614	0.9516	0.9278	0.9502	0.9496	0.9410	0.9300

Note: Bold values mean the value closest to the nominal level of 0.95 in each case.

and

Table 2. The average widths of the 95% confidence intervals for the coefficient of variation in a Delta-IG distributions.

n	m	l	p	Average Width
				GCI	AGCI	PBPCI	FCI	F-HPDCI	BCI	B-HPDCI
50	0.5	1	0.3	0.4260	0.3380	0.3129	0.3732	0.3616	0.5462	0.5264
			0.5	0.5717	0.4444	0.3737	0.4880	0.4642	0.5991	0.5794
		5	0.3	0.1037	0.0982	0.0815	0.1004	0.0961	0.4660	0.4517
			0.5	0.1266	0.1193	0.0892	0.1217	0.1140	0.4867	0.4804
		10	0.3	0.0545	0.0530	0.0431	0.0536	0.0511	0.6303	0.6173
			0.5	0.0650	0.0631	0.0461	0.0637	0.0594	0.4882	0.4817
	1	1	0.3	0.7912	0.5268	0.5303	0.6303	0.6128	0.4127	0.4013
			0.5	1.1264	0.7166	0.6603	0.8495	0.8129	0.4621	0.4413
		5	0.3	0.1923	0.1734	0.1490	0.1810	0.1742	0.5129	0.5003
			0.5	0.2432	0.2174	0.1685	0.2261	0.2131	0.5322	0.5180
		10	0.3	0.1037	0.0982	0.0815	0.1004	0.0961	0.3377	0.3316
			0.5	0.1266	0.1193	0.0892	0.1217	0.1140	0.3620	0.3592
100	0.5	1	0.3	0.2849	0.2297	0.2323	0.2552	0.2509	0.3584	0.3432
			0.5	0.3679	0.2944	0.2848	0.3262	0.3179	0.5864	0.5719
		5	0.3	0.0692	0.0658	0.0603	0.0673	0.0658	0.5968	0.5817
			0.5	0.0816	0.0774	0.0678	0.0792	0.0766	0.6467	0.6299
		10	0.3	0.0363	0.0354	0.0319	0.0358	0.0349	0.5176	0.5019
			0.5	0.0418	0.0407	0.0350	0.0411	0.0397	0.4638	0.4439
	1	1	0.3	0.5216	0.3594	0.3973	0.4353	0.4288	0.6388	0.6236
			0.5	0.7041	0.4777	0.5055	0.5789	0.5629	0.4639	0.4506
		5	0.3	0.1289	0.1170	0.1105	0.1225	0.1200	0.3420	0.3317
			0.5	0.1572	0.1423	0.1281	0.1486	0.1441	0.3663	0.3509
		10	0.3	0.0692	0.0658	0.0603	0.0673	0.0658	0.4280	0.4143
			0.5	0.0816	0.0774	0.0678	0.0792	0.0766	0.3117	0.2926
200	0.5	1	0.3	0.1968	0.1598	0.1697	0.1778	0.1761	0.5886	0.5706
			0.5	0.2498	0.2021	0.2103	0.2247	0.2217	0.6634	0.6491
		5	0.3	0.0478	0.0455	0.0442	0.0465	0.0460	0.4026	0.3855
			0.5	0.0554	0.0527	0.0499	0.0539	0.0529	0.3471	0.3309
		10	0.3	0.0250	0.0244	0.0233	0.0247	0.0243	0.4719	0.4543
			0.5	0.0283	0.0276	0.0258	0.0279	0.0274	0.5896	0.5765
	1	1	0.3	0.3582	0.2502	0.2901	0.3043	0.3017	0.5712	0.5590
			0.5	0.4739	0.3289	0.3735	0.3990	0.3941	0.6705	0.6562
		5	0.3	0.0891	0.0811	0.0808	0.0849	0.0839	0.3445	0.3271
			0.5	0.1069	0.0972	0.0946	0.1017	0.1000	0.5728	0.5565
		10	0.3	0.0478	0.0455	0.0442	0.0465	0.0460	0.3824	0.3717
			0.5	0.0554	0.0527	0.0499	0.0539	0.0529	0.3514	0.3382

Note: Bold values mean the minimum value of each case.

regarding CPs and AWs were visually represented as figures. Figure 1 shows the CPs and AWs of each method with various sample sizes. Figure 2 shows the CPs and AWs of each method with various mean parameters ($\mu$). And Figure 3 shows the CPs and AWs of each method with various shape parameters ($\lambda$). These representations demonstrate that in nearly all cases, the CPs obtained by AGCI, FCI, and F-HPDCI closely approached the nominal coverage level of 0.95. Among the methods, AGCI demonstrated outstanding results when dealing with small sample quantities. However, high sample numbers showed that FCI and F-HPDCI performed better in CPs than other methods. When comparing the average widths, PBPCI had the smallest width, with AGCI and F-HPDCI following closely behind. Comparing the coverage probabilities of the BCI and B-HPDCI methods, the CPs of BCI were found to be closer to the nominal coverage level than B-HPDCI.

Furthermore, the average widths of these two Bayesian procedures were larger than those of other approaches. While evaluating two criteria simultaneously, the approaches that demonstrated exceptional performance were AGCI and F-HPDCI. Next in sequence are FCI, BCI, B-HPDCI, GCI, and PBPCI. Moreover, AGCI used less time to calculate the CIs than F-HPDCI. Nevertheless, F-HPDCI proved to be less complicated in estimating CIs than AGCI due to the availability of the HDInterval package in R programming.

5. An Empirical Application

The statistical analysis of collision counts/rates is crucial to transportation safety planning and evaluating specific transportation infrastructure. Previous models’ accuracy still needs to be improved to account for a number of crash count difficulties, most notably the well-known issue of excessive zero counts of accidents.

The objective of estimating the CI for the CV of a Delta-IG distribution in the framework of accident counts is to evaluate the comparative dispersion of accident events. The CV is a statistical metric that quantifies the dispersion of a dataset or group of sets by calculating the ratio of the standard deviation to the mean. Analysis of the CI of the CV allows analysts to ascertain the range in which the actual CV is expected to be found. This provides valuable information on the stability and predictability of accident numbers. This study is highly significant for evaluating risks, allocating resources, and formulating policies in the field of transportation safety.

We analyze the data from a statistics report on road accidents released by the Ministry of State for Road Transport and Highways, Government of India, available at https://morth.nic.in/road-accident-in-india (accessed on 14 September 2024). Presented here is the aggregate count of individuals who had minor injuries in road accidents during the rainy season in 42 prominent cities in India: 45, 38, 76, 41, 202, 68, 210, 646, 13, 13, 565, 35, 8, 48, 38, 8, 68, 7, 46, 6, 44, 75, 30, 5, 77, 14, 29, 71, 15, 28, 7, 47, 0 (10 times). We want to illustrate the efficacies of the CI for the CV of Delta-IG distributions derived by using GCI, AGCI, PBPCI, FCI, F-HPDCI, BCI, and B-HPDCI approaches.

The data’s positive values may be fitted to the Normal, Lognormal, Weibull, Gamma, Exponential, Logistic, Cauchy, Birnbaum-Saunders, and Inverse Gaussian distributions. We analyzed the distributions of datasets measuring positive accident counts using the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), defined as

$$\mathrm{A}\mathrm{I}\mathrm{C}=-2\ln L+2k,$$

(17)

and

$$\mathrm{B}\mathrm{I}\mathrm{C}=-2\ln L+2k\ln \left(n\right),$$

(18)

respectively, where $L$ is the likelihood function, $k$ is the number of parameters, and $n$ is the number of observations.

Table 3. AIC and BIC values for the fitting of nine asymmetric distributions.

Distributions	AIC	BIC
Normal	412.612	415.5434
Lognormal	336.2688	339.2002
Weibull	345.0267	347.9582
Gamma	347.7497	350.6812
Exponential	348.0056	349.4714
Logistic	393.4866	396.4181
Cauchy	353.6207	356.5521
Birnbaum-Saunders	336.8754	341.2726
Inverse Gaussian	334.8293	337.7607

Note: Bold values mean the minimum value.

presents the AIC and BIC values for the fitting of nine asymmetric distributions, which help to determine which dataset is appropriate for which distributions. Since the values of AIC and BIC for the IG distribution were the smallest among the others, this indicates that the accident count dataset in India conforms to an IG distribution.

Table 4. Descriptive statistics for the accident counts data.

Country	n	Min	Median	Mean	Max	Variance	CV
India	42	0	28.5	62.4524	646	17,297.72	1.7662

presents the computed fundamental statistics for the accident counts data in India. The CV of the data was 1.7662.

Table 5. The 95% confidence intervals for the coefficient of variation from accident counts in India datasets.

Methods	Accident Counts in India
	Interval	Width
GCI	1.3464–3.3212	1.9748
AGCI	1.5500–2.4595	0.9095
PBPCI	1.3269–2.7557	1.4288
FCI	1.4935–3.0918	1.5982
F-HPDCI	1.4067–2.9654	1.5588
BCI	1.4539–1.9226	0.4686
B-HPDCI	1.4383–1.8824	0.4441

reports the 95% CIs for CV based on the GCI, AGCI, PBPCI, FCI, F-HPDCI, BCI, and B-HPDCI approach. The simulation findings indicated that the average widths of B-HPDCI were the shortest, followed by BCI, AGCI, PBPCI, and others. Unlike the simulation studies, the BPCI, AGCI, and F-HPDCI methods perfromed better than the BCI and B-HPDCI in terms of average width. Considering both criteria between the coverage probability and average width, the most suitable methods for estimating the CI of CV for the Delta-IG distribution were the AGCI and F-HPDCI methods.

6. Conclusions and Discussion

This work examined seven approaches (GCI, AGCI, PBPCI, FCI, F-HPDCI, BCI, and B-HPDCI) used to identify the most effective approach for estimating the CIs for the CV of Delta-IG distributions. There is no asymptotic proof for the coverage probabilities and expected widths of these confidence intervals. Thus, this research solely compares these confidence intervals utilizing the simulation approach provided in Algorithms 1–4. The simulation analysis findings indicate that AGCI, F-HPDCI, and FCI consistently had a nominal confidence level of 0.95 for almost all scenarios regarding the CPs. After carefully considering the AWs, the PBPCI, AGCI, and F-HPDCI methods emerged as the better options. However, when we evaluated both CPs and AWs, the AGCI and F-HPDCI approaches demonstrated outstanding results. AGCI performed better in terms of average widths, giving us the shortest AWs, while the F-HPDCI performed better in terms of coverage probabilities, especially when the sample size was large. And the performance of these methods was not different for nearly all cases of parameter settings. In an empirical application on accident counts to showcase the efficacy of these techniques, the B-HPDCI and BCI exhibited the narrowest average widths (AWs) and outperformed the other approaches. Bayesian intervals, which adapt to prior information, can yield narrower intervals than others. Selection for prior distributions of prior information from the dataset potentially results in shorter credible intervals when the information is informative. Based on the simulation studies and an empirical application, AGCI and F-HPDCI were the most appropriate approaches. Given the extensive usage of Delta-IG in economics, medicinal sciences, life science, and meteorology, it is crucial to choose optimal techniques for precise estimates of CIs for CV. The present work expanded the estimate of CIs for the CV of the IG distribution to the Delta-IG distribution, which is better appropriate for real-world situations. Thus, the AGCI and F-HPDCI approaches were shown to be the most suitable for constructing CIs for the CV of Delta-IG, based on the simulation studies and an empirical application.