Measurement of dispersion of PM 2.5 in Thailand using confidence intervals for the coefficient of variation of an inverse Gaussian distribution

The GCI method

Since the GCI approach was first introduced by Weerahandi (1993), several researchers have used it to provide statistical inferences (Tsui & Weerahandi, 1989; Weerahandi, 1993; Weerahandi, 1995; Ye & Wang, 2007; Krishnamoorthy & Tian, 2008). After that, Ye, Ma & Wang (2010) presented the generalized pivot quantity (GPQ) criterion for the parameters and the constructed confidence intervals for the common mean of several inverse Gaussian populations. Furthermore, Chankham, Niwitpong & Niwitpong (2019) recommended GCI for constructing confidence intervals for the coefficient of variation of an IG distribution; they found that GCI provide coverage probabilities greater than or equal to the nominal confidence level at 0.95. Therefore, GCI was selected as the baseline for comparison with the proposed methods of this study.

The confidence interval for the CV is calculated using the concept of the GPQ. Let X = (X₁, X₂, …, X_n) be random variables from a distribution defined by probability density function f_X(x; θ, δ), with θ and δ being the sought after and nuisance parameters, respectively. Meanwhile, GPQ R(X; x, θ, δ) satisfies the following two conditions.

1. The probability distribution of function R(X; x, θ, δ) is independent of the unknown parameters.

2. The observed value of R(X; x, θ, δ), X = x does not depend on the nuisance parameters.

If R(X; x, θ, δ) satisfies both conditions, then the GCI for the parameter of interest is calculated using the percentiles of the GPQ. Let [R_α/2, R_1−α/2] be a 100(1 − α)% two-sided GCI for the parameter of interest, where R_α and R_1−α are denoted as 100(α/2)% and 100(1 − α/2)% of R(X; x, θ, δ), respectively.

Ye, Ma & Wang (2010) proposed the respective GPQs for λ_i and μ_i as follows: (6)${\displaystyle R_{{\lambda }_i}=\frac{n_i{\lambda }_i{V}_i}{n_i{\upsilon }_i}\sim \frac{{\chi }_{n_{i-1}}^2}{n_i{\upsilon }_i},i=1,2,\dots ,k,}$

where ${\chi }_{n_{i-1}}^2$ denotes a chi-squared distribution with n_i − 1 degrees of freedom. Thus, the GPQ for μ_i is given by (7)${\displaystyle R_{{\mu }_i}=\frac{{\overline{x}}_i}{|1+\frac{\sqrt{n_i{\lambda }_i}\left({\overline{x}}_i-\mu \right)}{\mu \sqrt{{\overline{x}}_i}}\sqrt{\frac{{\overline{x}}_i}{n_i{R}_{{\lambda }_i}}}|}\stackrel{d}{\sim }\frac{\overline{x_i}}{\left|1+Z_i\sqrt{\frac{{\overline{x}}_i}{n_i{R}_{{\lambda }_i}}}\right|},}$ where $\stackrel{d}{\sim }$ is approximately distributed and Z_i ∼ N(0, 1). The approximate distribution is derives from the moment matching method of Chhikara & Folks (1989). Note that $\sqrt{n_i{\lambda }_i}\left(\overline{X}-{\mu }_i\right)/\left({\mu }_i\sqrt{{\overline{X}}_i}\right)$ is a limiting distribution of N(0, 1). Consequently, the observed value of R_μi is μ_i. Therefore, the GCI for the CV of an IG distribution is given by (8)${\displaystyle R_{\theta }=\sqrt{\frac{R_{\mu }}{R_{\lambda }}}.}$

The 100(1 − α)% two-sided confidence interval for the CV of an IG distribution based on the GCI method is given by (9)${\displaystyle C{I}_{GCI}=\left[L_{GCI},U_{GCI}\right]=\left[R_{\theta }\left(\alpha /2\right),R_{\theta }\left(1-\alpha /2\right)\right],}$

where R_θ(α/2) and R_θ(1 − α/2) are the 100(α/2)-th and 100(1 − α/2)-th percentiles of the distribution of R_θ, respectively.

The following algorithm was used to construct GCI:

Algorithm 1

(1) Generate X₁, X₂, …, X_n from an IG distribution.

(2) Compute $\hat{\mu }$ and $\hat{\lambda }$.

(3) Generate ${\chi }_{n-1}^2$ from a Chi-square distribution with n − 1 degrees of freedom and Z from a standard normal distribution.

(4) Use Eqs. (6), (7) and (8) to calculate R_λ, R_μ,  and R_θ, respectively.

(5) Repeat Steps 3-4, 5,000 times and obtain an array of R_θ.

(6) Calculate the 95% confidence intervals for θ by using Eq. (9). If L ≤ θ ≤ U, then set cp = 1 ; else , set cp = 0.

(7) Repeat Steps 1-6, 15,000 times to compute the coverage probability and the average length.

The AGCI method

According to Ye, Ma & Wang (2010), an approach similar to the GCI method can be utilized for the single coefficient of variation. The GPQ of λ uses the same of GCI method. Subsequently, Krishnamoorthy & Tian (2008) established an estimated GPQ of ${\tilde{\mu }}_i$ as follows: (10)${\displaystyle R_{\widetilde{{\mu }_i}}=\frac{{\bar{X}}_i}{\text{max}\left\{0,1+t_{n_i-1}\sqrt{\frac{{\bar{X}}_i{v}_i}{n_i-1}}\right\}},}$

where t_ni−1 denotes a t-distribution with t_ni−1 degrees of freedom. However, the denominator can become zero when t_ni−1 takes a negative value, and thus $R_{{\tilde{\mu }}_i}$ is an approximate GPQ.

Therefore, the AGCI for the CV of an IG distribution is given by (11)${\displaystyle R_{\tilde{\theta }}=\sqrt{\frac{R_{\tilde{\mu }}}{R_{\lambda }}}.}$

Subsequently, the 100(1 − α)% two-sided confidence interval for the CV of an IG distribution based on the AGCI method is given by (12)${\displaystyle C{I}_{AGCI}=\left[L_{AGCI},U_{AGCI}\right]=\left[R_{\tilde{\theta }}\left(\alpha /2\right),R_{\tilde{\theta }}\left(1-\alpha /2\right)\right],}$ where $R_{\tilde{\theta }}\left(\alpha /2\right)$ and $R_{\tilde{\theta }}\left(1-\alpha /2\right)$ which are the 100(α/2)-th and 100(1 − α/2)-th percentiles of the distribution of $R_{\tilde{\theta }}$, respectively can be obtained from the notion of Algorithm 1.

The BPCI method

When applying this method, the distribution of the bootstrap sample statistic is a direct approximation of the data sample (Efron & Tibshirani, 1986). It is proceeded by re-sampling the data with replacement from the distribution. Chankham, Niwitpong & Niwitpong (2019) reported that BPCI performed more poorly than GCI. However, to provide context, this approach was still included in the comparative analysis. Suppose X = (X₁, X₂, …, X_n) is a random sample of size n from an IG distribution. Sampling is replaced by $X^{\ast }=\left(X_1^{\ast },X_2^{\ast },\dots ,X_n^{\ast }\right)$, which can be obtained by the bootstrapping the sample B times. Efron & Tibshirani (1986) claimed that a minimum of approximately B = 1, 000 bootstrap resamples is usually sufficient for obtaining reasonable accurate confidence interval estimates for CV of IG distribution.

The 100(1 − α)% two-sided confidence interval for the CV of an IG distribution based on BPCI is defined as (13)${\displaystyle C{I}_{BPCI}=\left[L_{BPCI},U_{BPCI}\right]=\left[{\theta }^{\ast }\left(\alpha /2\right),{\theta }^{\ast }\left(1-\alpha /2\right)\right],}$

where θ^∗(α/2) and θ^∗(1 − α/2) are percentiles of the distribution.

The following algorithm is used to construct BPCI:

Algorithm 2

(1) Generate X₁, X₂, …, X_n from an IG distribution.

(2) Obtain bootstrap sample $X_1^{\ast },X_2^{\ast },\dots ,X_n^{\ast }$ from Step 1.

(3) Compute θ^∗.

(4) Repeat Steps 2 and 3, 1,000 times.

(5) Compute 95% confidence interval based on BPCI according to Eq. (13).

The FCI method

Although fiducial inference proposed by Fisher (1973) is similar to the Bayesian framework, it does not require prior knowledge of the distribution for estimating the parameters involved. Fiducial inference is the only type of inference with frequentist interpretation that uses conditionality on the data. Hence, it allows the implementation of Gibbs sampling (Geman & Geman, 1984), which is an MCMC method commonly used to generate a sample from the posterior distribution for Bayesian inference by sweeping through a variable to sample from its conditional distribution while the remaining variables are fixed at their current values. The sampling distributions of the MLEs of both μ and λ are used for an IG distribution. The fiducial distributions of μ and λ are easily obtained simply by replacing them with their MLEs when they appear in their sample distributions as follows (14)${\displaystyle \mu \sim \mathit{IG}\left(\hat{\mu },n\hat{\lambda }\right),}$ and (15)${\displaystyle \lambda \sim \left(\hat{\lambda }/n\right){\chi }_{n-1}^2,}$ where $\hat{\mu }$ and $\hat{\lambda }$ are the MLEs of the μ and λ.

Subsequently, the 100(1 − α)% two-sided confidence interval for the CV of an IG distribution based on fiducial inference is given by (16)${\displaystyle C{I}_{FCI}=\left[L_{FCI},U_{FCI}\right]=\left[{\theta }^t\left(\alpha /2\right),{\theta }^t\left(1-\alpha /2\right)\right],}$ where θ^t(α/2) and θ^t(1 − α/2) denotes the 100(α/2) -th and 100(1 − α/2) -th percentiles of θ^t, respectively.

Algorithm 3 The Gibbs sampling

(1) Take the initial values (MLEs) of parameters (μ⁽⁰⁾, λ⁽⁰⁾)

(2) Generate ${\mu }^{\left(t\right)}\sim IG\left({\hat{\mu }}^{\left(t-1\right)},n{\hat{\lambda }}^{\left(t-1\right)}\right)$

(3) Generate ${\lambda }^{\left(t\right)}\sim \left({\hat{\lambda }}^{\left(t-1\right)}/n\right){\chi }_{n-1}^2$

(4) Repeat Step 2–3, T times, where T is the number of MCMC replications.

(5) Burn-in 1,000 samples and compute the parameter of interest.

(6) Compute the 95% confidence interval based on the fiducial inference Eq. (16).

The F-HPDCI method

Hear, the HPD credible interval of the parameters are obtained by using the MCMC method (Chen & Shao, 1999). It is assumed that each value inside the interval has a higher posterior density than any of the values outside of it (Box & Tiao, 2011).

Hence, the 100(1 − α)% two-sided confidence interval for the CV of an IG distribution based on the F-HPDCI is given by (17)${\displaystyle C{I}_{F-HPDCI}=\left[L_{F-HPDCI},U_{F-HPDCI}\right].}$

The following algorithm is used to construct confidence interval using the F-HPDCI method for the CV of an IG distribution:

Algorithm 4

(1) Take the initial values (MLEs) of parameters (μ⁽⁰⁾, λ⁽⁰⁾)

(2) Use algorithm 3 to calculate the parameter of interest.

(3) Compute the 95% confidence interval based on F-HPDCI according to Eq. (17).

Example 1

PM 2.5 from the Din Daeng district of Bangkok were collected by the Pollution Control Department, Thailand (http://air4thai.pcd.go.th/webV2/download.php) due to this area having a high traffic volume (Table 2). Figure 3 exhibits a Q–Q plot of the PM 2.5 data, indicating that an IG distribution is suitable for this dataset. Before computing the confidence intervals, the minimum Akaike information criterion (AIC) and Bayesian information criterion (BIC) were first used to test the best-fitting distribution for these data. These two criteria are respectively defined as (20)${\displaystyle AIC=-2lnL+2k,}$ and (21)${\displaystyle BIC=-2lnL+2kln\left(n\right),}$ where L is a likelihood function, k is the number of parameters, and n is the number of recorded measurements.

It was found that the PM 2.5 data fit an IG distribution because the AIC and BIC values for this distribution were smaller than the other tested distributions (normal, lognormal, Cauchy, exponential, and Weibull) (Table 3). The summary statistics were computed: $n=31,\hat{\mu }=53.1229,\hat{\lambda }=337.9519,$ and CV = 0.3965. The 95% confidence intervals for the CV of the PM 2.5 dataset by using the five methods are reported in Table 4.These results agree with the simulation results for n = 30 in that the F-HPDCI method performed the best in terms of coverage probability and average length, which supports the simulation results.

Example 2

PM 2.5 data from the Bang Khun Thian district, Bangkok, were collected by the pollution Control Department, Thailand (http://air4thai.pcd.go.th/webV2/download.php) due to many factors contributing to PM 2.5 pollution in this area (e.g., road construction, car repair shops, and small and large factories) (Table 5). The summary for statistics were n = 31, $\hat{\mu }=56.6161,\hat{\lambda }=225.1443$, and CV = 0.5015. A Q–Q plot of these data determining the appropriateness of using an IG distribution is shown in Fig. 4. Furthermore, it was found that the AIC and BIC values for the IG distributions were lower than other tested distributions (Table 6), and thus it provided the best fit for the data. The 95% confidence intervals for the CV of this dataset by using the five methods are reported in Table 7. In agreement with the simulation results for n = 30 the F-HPDCI method provided the best confidence interval performance in terms of coverage probability and average length.