Alternative confidence interval estimation for the mean and coefficient of variation in a two-parameter exponential distribution

This paper presents interval estimation for the population mean and coefficient of variation in a two-parameter exponential distribution. The new generalized pivot, profile likelihood function and likelihood ratio statistic are derived and used to construct the confidence intervals. A highlight of this paper is that the generalized and likelihood ratio confidence intervals for the mean and coefficient of variation perform well in terms of coverage probability in many cases. Finally, two real-data applications on the air pollution of particulate matter (PM2.5) and the renewable energy through wind power of Thailand are used for illustration purposes.

in years 2018-2021 Thailand was ranked 45th most polluted country from 117 countries.Under US AQI, this was reported unhealthy in 2018 and unhealthy for sensitive groups in 2019-2021.Main sources of this issue were due to rapid urbanization, enormous transportation increases, construction and commerce [1].According to the Bangkok Air Quality and Air4Thai websites which are the main report channels for PM2.5 air-quality index in Thailand, we observed that the average PM2.5 index per time (hour, day, month or year) was only reported by sample mean estimate, but no confidence interval estimate for the mean and variability of the data were given.In case to do so, the distribution of PM2.5 data is needed to know.Here, we used the PM2.5 dataset of Bang Na district in Bangkok originally reported by the Division of Air Quality and Noise Management Bureau, Pollution Control Department of Thailand.The data were collected from 1 January 2019 to 31 May 2022, totally 41 monthly observations (see plots of the data in the application section).
To fit the probability distribution of this dataset, we used the Akaike Information Criterion (AIC) and Bayesian information criterion (BIC) as the criteria for model selection.They are given in the first part of Table 1.Evidently, the PM2.5 dataset applied here had the best fit to a two-parameter exponential distribution.Wind speed.The second motivated data are related to the renewable energy.Wind power is one of clean energy sources and the fastest-growing renewable energy technologies.Many projects on wind power have been studied and operated both in overseas and in Thailand.It can be observed that wind speed is one of the most important climatic factors affecting to electrical generation [2,3].According to the theory, probability models of wind speeds are often applied to right-skewed distributions.Clark and Dhiman et al. [4,5] described that wind speeds followed a Rayleigh distribution.General researches in Thailand reported that wind speeds fitted a Weibull distribution [6][7][8].In this work, we used the monthly wind speed at 90-m height above the ground observed in October 2018 to March 2020 (18 months) from Nakhon Phanom province located in upper north-eastern Thailand.These data were reported in Suksomporm and others [7,8].Recorded data from the previous papers were shown by graph in the application section.Then, we fitted the distribution of this dataset and showed the results in the lower part of Table 1.It can be seen that a two-parameter exponential distribution was fitted for the wind speed data.Now, we focus on the two-parameter exponential distribution, which has been applied in many areas, such as rainfall data and environmental pollution [9][10][11][12][13][14][15].According to the statistical theory, suppose that X is a random variable from a two-parameter exponential distribution, denoted by X ∼ Exp(λ, θ).The probability density function of X is given as where the observed value x > θ, scale parameter λ > 0, and location parameter ∞ < θ < −∞.The mean and variance of X are E(X) = μ = λ + θ and Var(X) = λ 2 , respectively.
The ratio of standard deviation to the mean, which is the CV of X is defined as τ = λ/(λ + θ).For a special case, if θ = 0 then X is the one-parameter exponential distribution with mean λ.In fact, there have been research studies done confidence intervals for functions of parameter in the two-parameter exponential distribution, see [16][17][18][19][20] for example.Those confidence intervals were derived based on the large-sample method, generalized confidence interval, percentile bootstrap, Bayesian interval and method of variance estimates recovery using maximum likelihood (ML) estimators for λ and θ in model (1).Recently, Krisadab [21] proposed confidence intervals for the mean using the large-sample method and method of variance estimates recovery with the unbiased estimators for λ and θ.It was found that their confidence intervals performed well in simulation studies.In this paper, we then look forward to develop interval estimation using an unbiased estimator in the new approach.
The objective of this study is to construct the confidence intervals for the population mean and CV in the two-parameter exponential distribution.The two main methods are considered: generalized confidence interval with the new pivotal quantity and likelihood ratio confidence interval using the new profile likelihood function.Simulation study aims to investigate the performance of the proposed confidence intervals.Finally, we illustrate the two real datasets on PM2.5 air pollution and wind speed in Thailand, as noted in the beginning of this section, to estimate the mean and CV using the proposed methods.
According to Sarhan [23] and Cohen and Helm [24], the desired estimators having unbiasedness for λ and θ are given by λu = n λ/(n − 1) and θu = (n θ − X)/(n − 1), respectively.These estimators are also called the best linear unbiased estimators.As shown in Krisadab [21], λu and θu have the bias values closer to zero than λ and θ , respectively, in all cases in situation study.The unbiased estimators λu and θu are therefore reasonable used to construct the confidence interval in the next section.

Confidence interval for the population mean
The new generalized confidence interval, profile likelihood (PL) and likelihood ratio confidence intervals for the population mean (μ) are introduced.The procedures are given as follows.

Generalized confidence interval
The method used to construct the generalized confidence interval in this section is referred to the concept of Weerahandi [25].It is based on the notion of a pivotal quantity statistic.Let G = G(X, x, ν) be a function of variable X with observed value x and generic parameter ν.G is called the generalized pivotal quantity if it satisfies the two following conditions.
(1) G has a probability distribution free of unknown parameter.
(2) Replacing X by its observed value x, the observed pivot g(x, x, ν) does not depend on the nuisance parameter.
Then, a computer-based approach is required for deriving the solution.The lower and upper limits obtained from the percentiles of pivot G are referred to as the generalized confidence interval for ν.
In our case, the population mean of X ∼ Exp(λ, θ) is given as μ = λ + θ .Now, the generalized pivots for λ and θ will be first derived.Then, we form the two functions to the new generalized pivotal quantity of μ.These are related to the following theorem.Proof: Since X ∼ Exp(λ, θ) with probability model (1), the log-likelihood function of λ and θ is given by ln According to Rohatgi and Saleh [26], we have that S is the unbiased estimator of λ.Therefore, the moment generating function of W = 2n λu /λ = 2nS/(λ(n − 1)) is derived as , where t < (n − 1)/2n.Hence, W ∼ Gamma(a, b), where a = n−1, b = 2n/(n − 1) and the distribution of W does not depend on λ and θ .W is then the pivotal quantity.This ends of proof.
Using the pivot function proposed in Theorem 2.1, we then rewrite the equivalent of λ, which is the generalized pivot for λ, as Similarly, it is easy to derive that T = 2n( θ − θ)/λ has a chi-square distribution with two degrees of freedom, namely χ 2 df =2 , (for more details see Sangnawakij and Niwitpong [18]).The distribution of T does not depend on θ .So that we have the new generalized pivot for θ: Using ( 3) and ( 4), the generalized pivotal quantity for the parameter of interest or μ is given by Note that ( 5) is free from μ.Moreover, if x is given, g(x, x, μ) = μ which does not depend on nuisance parameter.Hence, G(X, x, μ) is satisfied the conditions of [25].A (1 − α)100% generalized confidence interval for μ is therefore defined by where G(α/2) and G(1 − α/2) are the (α/2)100th and (1 − α/2)100th percentiles of G(X, x, μ), respectively, and α is the significance level.To obtain the lower and upper limits of ( 6), the computational approach is applied.We provide pseudocode for the generalized confidence interval of μ in Algorithm 1.Note that B denotes number of iterations, say 1000 or more.

Profile likelihood confidence interval
The profile likelihood (PL) method is an approach used to deal with a nuisance parameter [27].This parameter will be eliminated and then the PL function can be used in more elaborate developments in statistical inference later on.In this study, the PL confidence interval for μ is introduced.We prefer to give the PL function directly for μ with the elimination of nuisance parameter θ .Theorem 2.2: Let X ∼ Exp(λ, θ) with model (1).The profile log-likelihood for μ with λ = μ − θ and estimated θ is where θ is the ML estimator for θ.The profile maximum likelihood estimator for μ under ln L(μ) is μpl = X, and its variance is Proof: Since X ∼ Exp(λ, θ) and μ = λ + θ, the log-likelihood function for λ and θ is shown in (2).If we rewrite λ as μ − θ, the log-likelihood for μ and θ is of the form In fact, we are interested in statistical inference for μ, but ln L(μ, θ) still involves θ .Using the PL method, the nuisance parameter is then needed to eliminate.Here we fix θ by its ML estimator, or θ .The profile log-likelihood for μ is established as To find the profile ML estimator for μ, the first partial derivative for ln L(μ) with respect to μ, which maximizes (7), is given by This can be solved for μ as μpl = X, which is the unbiased estimator for μ.The variance of μpl is derived from the inverse of negative of Fisher information I(μ).It is given For the large sample size, μpl has an approximately normal distribution with mean μ and variance Var( μpl ).Substituting μpl into μ, the estimated variance of μpl is obtained as The proof is ended here.
The large-sample interval approximation with the estimated variance introduced in Theorem 2.2 is applied.Based on the central limit theorem, the test statistic for the null hypothesis, H 0 : μ = μ 0 , is given by .
T μ has an approximately standard normal distribution under H 0 .It can be seen that the distribution of T μ does not depend on μ.T μ is then the pivotal quantity.Therefore, a new (1 − α)100% PL confidence interval for μ is given by where z α/2 is the upper (α/2)100th percentile of a standard normal distribution.We note that if the sample size is small (typically n < 30), t α/2,df =n−1 which is the upper (α/2)100th percentile of a t-distribution with n−1 degrees of freedom is used, instead of z α/2 .

Likelihood ratio confidence interval
Another technique for constructing the confidence interval for μ used in this paper is based on a quantity known as the likelihood ratio statistic.Generally, this method provides an approximate confidence interval using the large sample chi-square distribution of the likelihood ratio test.The conventional method used to obtain the likelihood ratio test statistic for a generic parameter ν is defined by where ln L(ν) and ln L(ν) is the (profile) log-likelihood function for ν and maximized value of the log-likelihood function, respectively.The statistic given in ( 9) is an approximately χ 2 df =1 distribution [28].
In our case, we assume that X ∼ Exp(λ, θ) and the population mean is μ = λ + θ .The profile log-likelihood function for μ is presented in (7).Hence, we have the likelihood ratio statistic for the null hypothesis H 0 : μ = μ 0 , as where and are the profile log-likelihoods for H 0 and overall space, respectively.LR μ has an approximately χ 2 df =1 distribution.In interval estimation using (10), a (1 − α)100% two-sided confidence limits are the two values of μ that satisfy the following solution: where The confidence limits obtained from the above solution is guaranteed to cover μ with a given probability and the estimated mean μpl is within the interval.We denote this confidence interval as CI m3 .In computation, the PL confidence interval based on (11) can be obtained from the gosolnp function of the Rsolnp package in the R language [29].

Existing confidence interval
An exact confidence interval for μ derived from the percentiles of a pivotal quantity is considered in this section.We refer this to the confidence interval proposed by Krishnamoorthy and Xia [13].They presented the pivot for a general case cλ + θ, which is given as where c is a known positive constant.The cumulative distribution function of f n,c is related to a chi-square distribution.Here, we note that f n,c is different from our pivotal quantity given in (5).For a special case if c = 1, an exact (1 − α)100% confidence interval for estimating μ = λ + θ is obtained, and given by where f n,1;α/2 and f n,1;1−α/2 are the (α/2)100th and (1 − α/2)100th percentiles of f n,1 .The table of exact percentiles of f n,1 to calculate 90%, 95% and 99% confidence intervals for μ can be seen in [13].In the simulation section of this paper, CI KX will be investigated the performance and used to compare to our proposed confidence intervals.

Confidence interval for the population coefficient of variation
The new confidence intervals for coefficient of variation (τ ) in the two-parameter exponential distribution are proposed.These are constructed based on the four methods: large-sample approximation, generalized pivotal, profile likelihood and likelihood ratio methods.The assumptions of these approaches are described as follows.

Large-sample confidence interval
We first consider the point estimator for τ = λ/(λ + θ).Using the ML estimators for θ and λ, the estimated coefficient of variation is given as τ = 1 − X (1) / X.However, its bias is f (n)λ/(λ + θ), where f (n) = (n − 1)/n.This follows that τ is the biased estimator for τ .Another estimator which is the unbiased estimator for τ is then applied, it is given as τu = τ /f (n).Taking variance to τu , it is accomplished as A simple technique often used to construct the confidence interval is considered here using the large-sample method.Based on the normal approximation, the pivot statistic is given by where Var( τu ) is the estimated variance of τu obtained by substituting λu and θu into λ and θ , respectively, of Var( τu ).Under H 0 : τ = τ 0 , the distribution of T τ is an approximately standard normal distributed.Thus a (1 − α)100% confidence interval for τ is given as Again, we suggest using the percentile value of a t-distribution with n−1 degrees of freedom for the data with small sample sizes.

Generalized confidence interval
The generalized confidence interval for τ proposed in this section is based on the assumption of Weerahandi [25], as well as described in Section 2.1.1.We then refer the information given in ( 3) and ( 4) to construct the pivotal quantity for τ .So, it is easy to see that the generalized pivot function for τ is given as We note that ( 14) is exactly different from the one suggested in Sangnawakij and Niwitpong [18].Therefore, the new lower and upper percentile limits of G(X, x, τ ) for the generalized confidence interval for τ are obtained from where G τ (α/2) and G τ (1 − α/2) are the (α/2)100th and (1 − α/2)100th percentiles of G(X, x, τ ), respectively.In computation, we use a procedure for finding this confidence interval presented in Algorithm 2.

Profile likelihood confidence interval
The confidence interval for τ is introduced in this section.We construct it using the profile likelihood (PL) function which nuisance parameter θ is eliminated.This function is related to the following theorem.
Proof: From X ∼ Exp(λ, θ), we know that τ = λ/(λ + θ).According to (2), if λ is replaced by τ θ/(1 − τ ), the profile log-likelihood for τ and θ becomes ln However, τ is the parameter of interest and θ is the nuisance parameter.The profile loglikelihood for τ with eliminating θ , denoted as ln L(τ ), is considered.Substituting θ by θ , we have From ( 16), the first partial derivative with respect to τ is given by Maximizing the above equation by equating it to zero, we have the PL estimator τpl = 1 − θ/ X = τ .To find the variance of τpl based on (16), the inverse of Fisher information is applied.It follows that Finally, its estimated variance is obtained by substituting τ in the above equation by τpl .Furthermore, τpl has a normal distribution with mean τ and variance Var( τpl ), as n → ∞.This is ends of proof.
Applying the large-sample approximation, a (1 − α)100% confidence interval for τ is then given by where Var( τpl ) is the estimated variance of τpl .

Likelihood ratio confidence interval
As detailed in Section 2.1.3,the likelihood ratio method is based on the two likelihoods under the null and maximized value.Now, it is used to construct the confidence interval for τ .Using the profile log-likelihood given in ( 16), the likelihood ratio statistic for H 0 : τ = τ 0 is given by is the profile log-likelihood under H 0 and is the maximized value of profile log-likelihood function.Under the null, LR τ has an approximately χ 2 df =1 distribution.To compute a two-sided (1 − α)100% confidence limits for τ , we solve the following equation: where χ 2 1−α,df =1 is the (1 − α)100th percentile of a χ 2 df =1 distribution.The confidence interval obtained from ( 18) is denoted as CI t4 .In simulation study and application, the numerical optimization within the R language can be used.Again, we apply the gosolnp function in computation.

Simulation study
The performance of the three proposed confidence intervals for μ (CI m1 − CI m3 ) and four confidence intervals for τ (CI t1 − CI t4 ) presented in Section 2 are evaluated in terms of coverage probability and expected length using simulations in several scenarios.The settings used in this study are set to cover possible situations.The data in a two-parameter exponential distribution are generated based on the inverse transform method using the R statistical package [30].In this step, we sample the data X from X = θ − λ log(1 − u), where u is generated from a continuous uniform distribution, U(0, 1).The related parameters of the distribution are given in the following.
The sample size (n) considered in simulations is given as 15, 20, 30, 50 and 100.Each scenario is repeated 5000 times.For the generalized pivot method, number of iterations is B = 3000.The criteria for performance comparison is based on the coverage probability (CP) and expected length (EL).They are estimated by 5, 000 and EL = 5,000 respectively, where ν is a generic parameter.In fact, ν is corresponding to the parameter of interest, μ or τ in our study.c(L(X) ≤ ν ≤ U(X)) is the number that ν lies within the lower limit L(X) and upper limit U(X) of a confidence interval.We prefer the confidence interval that has the coverage probability greater than or close to the nominal level of 0.95 with a short expected length.

Performance of confidence interval for the mean
The simulated coverage probabilities and expected lengths of the 95% confidence intervals for μ are shown in Figure 1 (see the corresponding values in Table 1 of the Supplemental Material).The results show that coverage probabilities of CI m2 are much lower than 0.95 in many situations, especially when n < 100.This confidence interval has the limitation in estimating μ if n < 100.The coverage probabilities of CI m1 are greater than those of CI m2 in general cases.However, it still has coverage probability lower than 0.95 in many cases as well.This shows that CI m1 is not consistent.In contrast to the likelihood ratio method, CI m3 provides the coverage probabilities greater than the target probability level at 0.95 in all situations in the study.This means that CI m3 can cover or estimate the true mean very well.CI KX developed by Krishnamoorthy and Xia [13] also has a good performance in terms of coverage probability, as their coverages are close to 0.95 in many cases.However, the coverage probabilities of CI KX are lower than those of CI m3 for all cases in the study.Next, we consider the expected length of the confidence interval for μ.As shown in Figure 1 (lower part), when sample size is increased, the expected length of each confidence interval decreases.CI m2 has the shortest expected length compared to the others.CI m3 provides the length slightly greater than CI m1 and CI KX , respectively.Although the expected length of CI m3 seems to be greater than that of the compared methods, it is important to note that CI m3 is satisfied the target probability level in general cases, reflecting that CI m3 can precisely estimate the true value with the acceptable length of interval.In conclusion, CI m3 performs precisely as well as CI KX .We therefore recommend CI m3 as an alternative confidence interval to estimate the population mean in the two-parameter exponential distribution.

Performance of confidence interval for the coefficient of variation
Figures 2 and 3 show the simulation results of the estimators for τ .The corresponding values are given in Table 2 of the Supplemental Material.Since there has not been research investigated the performance of point estimators τ and τu = τpl , it is addressed in this section.We can see from Figure 2 that τu provides the bias values very close to zero than τ in all cases in the study.Furthermore, it decreases to zero for large sample size.In other word, τu is an asymptotically unbiased estimator for τ .τu is then reasonable to use for constructing the confidence interval for τ .Focusing on interval estimation, the results given in Figure 3 show that coverage probabilities of CI t1 are satisfied the nominal probability level at 0.95 if τ > 0.3.Those of CI t2 are greater than and close to 0.95 in general cases.However, CI t3 cannot work well, as its coverage probability is smaller than the other estimators and lower than 0.95, especially when τ ≥ 0.7.The coverage probabilities of CI t4  are greater than 0.95 and the compared confidence intervals when τ < 0.7.From Figure 3 (lower part), the expected lengths of all interval estimators decrease if the sample size is increased.When τ < 0.7, the expected lengths of these approaches do not much differ.
For τ ≥ 0.7, CI t3 and CI t4 have shorter expected length than CI t1 and CI t2 , but the former two confidence intervals are low coverage in such a case.In summary, CI t4 is superior to estimate the true CV in the two-parameter exponential distribution when the dispersion of data is small to moderate level (τ < 0.7).If the large spread of data is observed (τ ≥ 0.7), CI t1 and CI t2 are the suitable methods, where the former estimator has the explicit formula in use, while the latter confidence interval needs computer programming in computation.

Data application
The two real-data examples mentioned in the introduction section are detailed again in this section.They are used to illustrate and compare the analytical methods proposed in this paper.
Example 4.1 (PM2.5 in Thailand): PM2.5 is the particulate matter in the air with diameter of less than 2.5 microns (μm 3 ).Earlier in 2019, Thailand had hardly faced a critical haze situation and air quality issue.The problem is found in many areas of Thailand, both city and suburb, especially in the winter (November to March) [31].PM2.5 may not cause immediate harm to health; however, cumulative exposure over time can lead to various health disorders [32].From the statement and problem, it is important to use a suitable statistical tool to measure the impact of PM2.5 air pollution.In this work, the PM2.5 (μg/m 3 ) per month in Bang Na district of Bangkok observed from January 2019 to May 2022 are applied.The data with 41 observations are shown by graphs and given in Figure 4(a,b).It is clear that this dataset has the right-skewed distribution.Fitting the best distribution using AIC and BIC as noted in Table 1 are explored.The two-parameter exponential distribution fits for this dataset.We conclude that inferential statistic based on the two-parameter exponential distribution should be used in the analysis.

Example 4.2 (Wind speed in Thailand):
Wind power is one of the important alternative energy sources in many countries.In Thailand, it was amounted to an installed production capacity as of the end of 2014.In more details, wind energy investment in Thailand played a major role of alternative energy investment, shared 30.4%, followed by biomass, biofuels, municipal solid waste, biogas solar and small hydro power, shared 22.5%, 18.0%, 10.5%, 9.6%, 8.7% and 0.3%, respectively [33].The survey found that potential wind power sources located in Thailand are in the East, South and Central.Wind speed is a main factor affecting wind power generation and largely determines the amount of electricity generated by a turbine.On the other hand, it can be said that higher wind speeds generate more power.In this paper, wind speeds (m/s) per month (n = 18) collected in Nakhon Phanom province of Thailand are illustrated.The data are shown in Figure 4(c-d).Fitting of four distributions related to the principle of wind speed [5] are then considered, as given in Table 1.Both and BIC values from the two-parameter exponential distribution are the smallest values compared to the other probability models.Therefore, it fits for the wind speed dataset used in this example.The methods proposed in this paper can be used in computation.
Data analysis.Using the two datasets given above, the estimated parameters for μ and τ based on the proposed methods are presented in Table 2. On average, PM2.5 in the period of study is 20.657 μg/m 3 .It has the CV of 60.6%.The mean of wind speed is 4.26 m/s with the CV of 19.2%.Next, interval estimation is considered.We can see that our data are matched simulation setting given in simulation study.According to simulation results, CI m3 performs well to estimate μ and CI t4 is useful when τ < 70% for any sample size.We conclude that the 95% confidence interval for mean of PM2.5 is CI m3 = (17.801,25.725) μg/m 3 and that for CV of PM2.5 is CI t4 = (51.9%,66.6%).This reports that PM2.5 air quality index is good with moderate deviation.
Finally, the analysis is on the wind speed.The monthly average wind speed is given as CI m3 = (4.021,4.836) m/s.The 95% confidence interval for CV is CI t4 = (12.6%,26.7%), reflexing small deviation.To confirm that the likelihood ratio confidence intervals, CI m3 and CI t4 , converge and can be solved for the lower and upper limits for μ and τ .We illustrate using the graph displayed in Figure 5.The red-dash lines are lower and upper limits, while the black-dash line shows the point estimate.Our methods succeed.

Conclusion
This paper proposes alternative confidence intervals based on the two main approaches: generalized pivot and likelihood ratio methods.These can be alternative and useful tools to estimate the mean (μ) and coefficient of variation (τ ) in the two-parameter exponential distribution.For mean estimation, the likelihood ratio confidence interval using the new profile likelihood function (CI m3 ) provides the coverage probabilities greater than and close to the target probability level with the suitable expected length in all cases of the study.The results from CI m3 are similar to those of CI KX , introduced by Krishnamoorthy and Xia [13].These two confidence intervals are suggested for estimating μ.The generalized confidence interval using the new pivot function (CI m1 ) can be used when sample size is larger than 20.For estimating τ , the profile-likelihood ratio confidence interval (CI t4 ) is satisfactory in terms of coverage probability when levels of variation in data are small to moderate.For extremely large deviation case, the data probably contain extreme values or outliers.The approach based on a simple likelihood would be sensitive to this data type.Reasonably, the two proposed methods based on likelihoods (CI t3 and CI t4 ) have a limitation to estimate the coefficient of variation in this case.For the robust method to reduce the influence of outliers, see [34].As the research has demonstrated, the proposed generalized confidence interval (CI t2 ) outperforms the other proposed confidence intervals.It can be used to estimate the coefficient of variation when the spread of values in the dataset is large.However, the confidence interval from generalized pivot and profile likelihood ratio methods has no closed-form solution.We then provides a guidance of computer code in the R programming language along with the example data.It is available in the Supplemental Material.

Figure 2 .
Figure 2. Bias of the two estimated coefficients of variation for τ ( τ and τu ) using simulations.

Figure
FigureCoverage probability and expected length of the 95% confidence intervals for τ (CI t1 , CI t2 , CI t3 and CI t4 ) using simulations.

Figure 4 .
Figure 4. Plot (left) and histogram (right) of the two data on PM2.5 per month and wind speed per month in Thailand.

Figure 5 .
Figure 5.The 95% profile likelihood confidence limits for mean (left) and coefficient of variation (right) using the two data examples.

Table 1 .
Fitting the four distributions for PM2.5 pollution and wind speed data in Thailand.
*Note that * denotes the smallest AIC or BIC value.

Table 2 .
Estimated mean and coefficient of variation, and 95% confidence intervals using two data examples.