High levels of air pollution are a recurrent problem in many cities/regions around the world (see, for instance, [1] [2] ). This may produce harmful effects on the population’s health, as well as the environment. Among the many pollutants measured in Mexico City’s monitoring network are ozone (O₃), sulphur dioxide (SO₂), carbon monoxide (CO), nitrogen dioxide (NO₂), particulate matter with a diameter smaller than 10 microns (PM₁₀) and those with a diameter smaller than 2.5 (PM_2.5). Pollutant concentrations are measured in parts per million (ppm) in the cases of O₃, SO₂, NO₂, and CO, and in micrograms per cubic meter (μg/m³) when we consider PM₁₀ and PM_2.5. These pollutants are also known as the criterion pollutants which are used to establish the air quality indices in the city. They are chosen because of their possible harmful impact on human health, as well as the environment ( [3] [4] ). For instance, if we have ozone levels above 0.11 ppm, the ill, newborn, and elderly may experience serious health deterioration (see, for example, [5] - [8] , among others). We also know that sulphur and nitrogen dioxides, when in contact with the right level of humidity in the atmosphere, may produce acid rain ( [9] [10] ). Additionally, exposure of pregnant women to CO, PM₁₀, and PM_2.5 may produce adverse effects on the newborn ( [11] - [13] ), and exposure to PM₁₀ and PM_2.5 may cause cardiovascular problems in the population in general and an increase in mortality in at-risk groups ( [14] - [17] ). Therefore, it is very important to study the behavior of these and other pollutants. It is possible to find in the literature several works studying pollution behavior using data from several regions of the world. Among them are [18] - [20] , in particular, it is of interest to study the behavior of concentrations that present extreme values when compared to the remaining measurements. One way of studying this is to use extreme value models.

Extreme value theory ( [21] - [23] ) has been applied in many areas ( [24] ) including the study of several environmental problems. For instance, we have [25] where these models are used to study air pollution with application to the Istanbul data [26] , in which they are used to study the ozone extreme trends using data from one monitoring station in Mexico City [27] , where they are used to predict air pollution threshold exceedances with an application to NO₂, O₃, and CO data obtained from monitoring sites located in Queensland, Australia [28] , which studies the impact of climate change on global flood and precipitation [29] , where trends in temperature extremes in some cities in Mexico are analyzed, and [30] where extreme sea levels in coastal China are studied. In the present study, we are interested in analyzing the joint behavior of pairs of pollutants extreme measurements.

When it comes to studying the distribution of multivariate data we may consider, for instance, the direct use of the multivariate distributions when they are available ( [31] ) or to use of copulas ( [32] ) to obtain the expressions for the corresponding distributions. Copulas have also been used in works where multivariate extreme value models are considered to study environmental problems, for instance, in [33] a copula approach was considered to study multivariate multi-parameter extreme value models with an application to annual flooding data from the Spey basin in the north of Scotland [34] , where a copula model is used to analyze the ranking of the paired data (PM₁₀, CO), (PM₁₀, NO₂), (PM₁₀, O₃), and (PM₁₀, SO₂) with a generalized Pareto distribution and data from the Klang City, Malaysia.

In the present study we consider an extreme value model and a copula to study the association and correlation between pairs of pollutants obtained from the Mexico City monitoring network. We consider the bivariate analysis of the pairs (NO₂, O₃), (SO₂, O₃), (CO, O₃), (PM₁₀, O₃), and (PM_2.5, O₃). The copula used is the Gumbel-Hougaard ( [35] [36] ) considering two extreme value distributions, namely, the Weibull and Fréchet as possible marginal distributions. The novelties are the copula used, as well as the data sets. The differences from previous works also using data from Mexico City are the observational period, the copula used, the data taken into account, and the regional analysis performed.

This work is organized as follows. In Section 2, we present the mathematical and the Bayesian formulations of the model. Section 3 gives an application to the case of Mexico City’s monthly maximum measurements of the six criterion pollutants. In Section 4, a discussion of the results, as well as some additional comments are given. Finally, in Section 5 we conclude. An Appendix placed after the list of references, presents the one-dimensional analysis of the monthly maxima for all pollutants and regions considered in this work, some tables and figures mentioned in the main text, the explicit form of the likelihood functions, and some of the computational details.

Let $d\ge 1$ and $N\ge 1$ be natural numbers representing, respectively, the number of pollutants considered in the analysis and the number of observations of each one of them during the observational period $\left[\mathrm{0,}T\right]$, $T>0$. Denote by $Z_t^{\left(i\right)}$ the measurement of the ith pollutant at time t; $i=\mathrm{1,2,}\cdots \mathrm{,}d$; $t=\mathrm{1,2,}\cdots \mathrm{,}T$. Let $Z^{\left(i\right)}=\left\{Z_t^{\left(i\right)}:t=1,2,\cdots ,T\right\}$ be the process recording the ith pollutant concentration values throughout the observational period, $i=\mathrm{1,2,}\cdots \mathrm{,}d$. We assume that the ith pollutant has, respectively, distribution and density functions $F_i(\cdot )$ and $f_i(\cdot )$; $i=\mathrm{1,2,}\cdots \mathrm{,}d$. Denote by $F_{ij}\left(\cdot \mathrm{,}\cdot \right)$ and $f_{ij}\left(\cdot \mathrm{,}\cdot \right)$ the joint distribution and density functions of the ith and jth pollutants, respectively, $i\mathrm{,}j=\mathrm{1,2,}\cdots \mathrm{,}d$; $i\ne j$. We assume that $F_{ij}\left(\cdot \mathrm{,}\cdot \right)$ is given by a copula function denoted by $C_{\theta }\left(\cdot \mathrm{,}\cdot \right)$ with $F_i(\cdot )$ and $F_j(\cdot )$ the marginal distributions, i.e., $F_{ij}\left(\cdot \mathrm{,}\cdot \right)=C_{\theta }\left(F_i(\cdot )\mathrm{,}{F}_j(\cdot )\right)$; $i\ne j$; $i\mathrm{,}j=\mathrm{1,2,}\cdots \mathrm{,}d$. Due to the nature of the correlation between pairs of measurements, in the present case we assume a Gumbel-Hougaard ( [35] - [38] ) copula for the pairs of distribution functions. Hence, we take

$C_{\theta }\left(u\mathrm{,}v\right)=\exp \left(-{\left[{\left(-\log u\right)}^{\theta }+{\left(-\log v\right)}^{\theta }\right]}^{1/\theta }\right)\mathrm{,}$(1)

where $\theta \in \left[\mathrm{1,}\infty \right)$ is the association parameter. (Note that when $\theta =1$ we have independence of the two random variables whose joint distribution we are trying to model).

Remark. Even though there are many different types of copula, we have decided to use the Gumbel-Hougaard because this copula is suitable when we have a positive sample correlation between the two sets of measurements we are studying. This is the case here (see Figure A1 in Appendix A).

When $F_{ij}\left(\cdot \mathrm{,}\cdot \right)$ is given by (1), the corresponding joint density function $f_{ij}\left(\cdot \mathrm{,}\cdot \right)$ is given by

$f_{ij}\left(x\mathrm{,}y\right)=f_i\left(x\right)f_j\left(y\right)c_{\theta }\left(F_i\left(x\right)\mathrm{,}{F}_j\left(y\right)\right)\mathrm{,}$(2)

where

$\begin{array}{l}{c}_{\theta }\left(u\mathrm{,}v\right)=\frac{{\left(-\log u\right)}^{\theta -1}{\left(-\log v\right)}^{\theta -1}}{uv}\exp \left(-{\left[{\left(-\log u\right)}^{\theta }+{\left(-\log v\right)}^{\theta }\right]}^{\frac{1}{\theta }}\right)\\ \times \left({\left[{\left(-\log u\right)}^{\theta }+{\left(-\log v\right)}^{\theta }\right]}^{\frac{2}{\theta }-2}+\left(\theta -1\right){\left[{\left(-\log u\right)}^{\theta }+{\left(-\log v\right)}^{\theta }\right]}^{\frac{1}{\theta }-2}\right)\mathrm{.}\end{array}$

In the present study we are using the monthly maximum measurements, hence two well known extreme values distributions will be considered when using a copula in order to obtain the joint distributions of pairs of maxima. These distributions are the Fréchet (α, σ) (F) and the Weibull (α, σ) (W) which are given as follows ( [39] [40] ). Random variables X and Y are said to have Fréchet (α, σ) and Weibull (α, σ) distributions, respectively, if their distribution and density functions are of the following forms,

$F_X^{\left(F\right)}\left(x\right)=\exp \left[-{\left(\frac{x}{\sigma }\right)}^{-\alpha }\right]\text{and}{f}_X^{\left(F\right)}\left(x\right)=\frac{\alpha }{\sigma }{\left(\frac{x}{\sigma }\right)}^{-\alpha -1}\exp \left[-{\left(\frac{x}{\sigma }\right)}^{-\alpha }\right],$(3)

and

$F_Y^{\left(W\right)}\left(y\right)=1-\exp \left[-{\left(\frac{y}{\sigma }\right)}^{\alpha }\right]\text{and}{f}_Y^{\left(W\right)}\left(y\right)=\frac{\alpha }{\sigma }{\left(\frac{y}{\sigma }\right)}^{\alpha -1}\exp \left[-{\left(\frac{y}{\sigma }\right)}^{\alpha }\right],$(4)

with $x,y,\alpha ,\sigma >0$. Taking $b={\left(1/\sigma \right)}^{\alpha }$ we may rewrite (4) in the equivalent form,

$F_Y^{\left(W\right)}\left(y\right)=1-\exp \left[-b{y}^{\alpha }\right]\text{and}{f}_Y^{\left(W\right)}\left(y\right)=b\alpha y^{\alpha -1}\exp \left[-b{y}^{\alpha }\right].$(5)

The latter will be considered in the computer codes used to estimate the parameters of the models. In both distributions, the vector of parameters to be estimated is $\phi =\left(\alpha \mathrm{,}\sigma \right)$.

Denote by $\left(Z_t^{\left(i\right)}\mathrm{,}{Z}_t^{\left(j\right)}\right)$ the random vector recording the joint pollutants concentrations, $i\mathrm{,}j=\mathrm{1,2,}\cdots \mathrm{,}d$; $i\ne j$; $t=\mathrm{1,2,}\cdots \mathrm{,}T$. In the present work, $\left(Z_t^{\left(i\right)}\mathrm{,}{Z}_t^{\left(j\right)}\right)$ will record the monthly maximum concentrations of pollutants i and j. Let $\left({\alpha }_i\mathrm{,}{\sigma }_i\right)$ and $\left({\alpha }_j\mathrm{,}{\sigma }_j\right)$ denote the vector of parameters of the distribution functions $F_i(\cdot )$ and $F_j(\cdot )$, respectively. We also assume that $F_{ij}\left(\cdot \mathrm{,}\cdot \right)$ is given by the copula (1) with $f_{ij}\left(\cdot \mathrm{,}\cdot \right)$ the associated density function. Hence, the vector of parameters to be estimated is $\phi =\left({\alpha }_i\mathrm{,}{\sigma }_i\mathrm{,}{\alpha }_j\mathrm{,}{\sigma }_j\mathrm{,}\theta \right)$.

Estimation of the parameters will be performed under the Bayesian point of view ( [41] ) using information provided by the so-called posterior distribution of the vector of parameters. If $\varphi$ is the vector of parameters of a model describing a data set $D$, then the posterior distribution of $\varphi$, denoted by $P\left(\varphi \mathrm{<mo>\; |\; </mo>}D\right)$ is such that $P\left(\varphi \mathrm{<mo>\; |\; </mo>}D\right)\propto L\left(D\mathrm{<mo>\; |\; </mo>}\varphi \right)P\left(\varphi \right)$, where $L\left(D\mathrm{<mo>\; |\; </mo>}\varphi \right)$ and $P\left(\varphi \right)$ are, respectively, the likelihood function of the model and the so-called prior distribution of $\varphi$. Estimation of the parameters is made easier by using the software OpenBugs ( [42] [43] ) since we only need to specify the likelihood function of the model and the prior distributions of the parameters.

In the present case the observed data are given by $D^{\left(i\mathrm{,}j\right)}=\left\{\left(z_t^{\left(i\right)}\mathrm{,}{z}_t^{\left(j\right)}\right)\mathrm{,}t=\mathrm{1,2,}\cdots \mathrm{,}T\right\}$, $i\mathrm{,}j=\mathrm{1,2,}\cdots \mathrm{,}d$; $i\ne j$. The prior distributions of the parameters will be specified when the model is applied to the Mexico City data. In all cases we will assume prior independence of the parameters. The likelihood function when the ith and jth pollutants are taken into account is $L\left(D^{\left(i\mathrm{,}j\right)}\mathrm{<mo>\; |\; </mo>}\phi \right)={\displaystyle \underset{t=1}{\overset{T}{\prod }}}{f}_{ij}\left(z_t^{\left(i\right)}\mathrm{,}{z}_t^{\left(j\right)}\right)\mathrm{,}$ with $f_{ij}\left(\cdot \mathrm{,}\cdot \right)$ the appropriate joint density function given by the copula. The particular forms of the bivariate density functions obtained using the copula are given in Appendix B.

Several versions of the model will be considered. Hence, several criteria will be used to aid in the selection of the best model to describe the data. One of them is the deviance information criterion (DIC) ( [44] ) and another is the Bayes factor ( [45] ) through the marginal likelihood function (MLF). They are described as follows. The deviance information criterion is given by $\text{DIC}=D\left(\bar{\varphi }\right)+2p_D$, with $D\left(\bar{\varphi }\right)=-2\log \left[L\left(D\mathrm{<mo>\; |\; </mo>}\bar{\varphi }\right)\right]+C$ the deviance evaluated at the posterior mean $\bar{\varphi }$ of the parameter $\varphi$, C a constant, and $p_D$ the effective number of parameters in the model which is given by $p_D=\bar{D}\left(\varphi \right)-D\left(\bar{\varphi }\right)$, with $\bar{D}\left(\varphi \right)=\text{E}\left[D\left(\varphi \right)\right]$ the posterior mean deviance. The smaller the value of the DIC the better suited is the model to describe a data set $D$. The Bayes discrimination method may be described as follows. From [45] the marginal likelihood function of a data set $D$ for Model $l$, $l=\mathrm{1,2,}\cdots \mathrm{,}J$, is given by $V_l={\displaystyle \int L\left(D\mathrm{<mo>\; |\; </mo>}{\varphi }^{\left(l\right)}\right)P\left({\varphi }^{\left(l\right)}\right)\text{d}{\varphi }^{\left(l\right)}}$, where ${\varphi }^{\left[l\right]}$ is the vector of parameters for Model $l$ and $P\left({\theta }^{\left[l\right]}\right)$ is the joint prior distribution of ${\varphi }^{\left(l\right)}$. The Bayes discrimination method prefers model i to model j if $V_j/V_i<1$.

Remark. For completeness, in Appendix we also present an individual analysis of each criterion pollutant considered here. In this case only the univariate distributions are taken into account. Hence, in the univariate analysis we will be modeling one set of data at time. Hence, in this case the process is $Z^{\left(i\right)}$, $i=\mathrm{1,2,}\cdots \mathrm{,}d$, and the vector of parameters to be estimated is just $\phi =\left(\alpha \mathrm{,}\sigma \right)$, and the observed data are denoted by the set $D^{\left(i\right)}=\left\{z_t^{\left(i\right)}t=\mathrm{1,2,}\cdots \mathrm{,}T\right\}$. In addition to the selection criteria described above, in the one-dimensional case we will also use the graphical fit between estimated and empirical density functions associated with a data set, as well as the p-value of the Kolmogorov-Smirnov test. The likelihood function of the model when we are using the ith pollutant data is given by $L\left(D^{\left(i\right)}\mathrm{<mo>\; |\; </mo>}\phi \right)={\displaystyle {\prod }_{t=1}^T}{f}_i\left(z_t^{\left(i\right)}\right)$, where $\phi$ is the vector of parameters of either the Fréchet or the Weibull distribution with $f_i(\cdot )$ the corresponding density function and $D^{\left(i\right)}$ is the set of the monthly maxima of the pollutant taken into account.

Application will be made to measurements of so-called criterion pollutants collected at Mexico City monitoring network ( http://www.aire.cdmx.gob.mx/default.php?opc=%27aKBh%27 ). This network comprises several monitoring stations placed throughout the metropolitan area. Measurements in each monitoring station are obtained minute by minute and the averaged hourly result is reported at each station. The data actually used in the application are the monthly maximum measurements. Depending on the pollutant we have an observational period. Hence, in the cases of ozone, NO₂, SO₂, and CO the data used were collected from January 1990 to December 2021 giving a total of $T=384$ monthly maxima; from January 1995 to December 2021 in the case of PM₁₀, giving a total of $T=324$ measurements; and from August 2003 to December 2021 for the PM_2.5 data with a total of $T=221$ values. When pollutants with different observational periods are used in the bivariate study, the observational period taken into account is the intersection of the observational periods of the pollutants involved. Whenever necessary we will associate the pollutants O₃, NO₂, SO₂, CO, PM₁₀, and PM_2.5 with the integer numbers 1, 2, 3, 4, 5, and 6, respectively.

In the present work we have studied the behavior of five pairs of pollutants obtained from the Mexico City monitoring network. The data modeled were the paired ozone monthly maximum measurements and the monthly maxima of four other pollutants part of the so-called criterion pollutants which are used to classify the air quality of the Mexico City metropolitan area. In order to study the joint behavior of these pairs of pollutants a two-dimensional distribution was constructed using a copula. Additionally, analysis was performed using the data collected in the different areas of Mexico City. Results show that depending on the region and pair of pollutants, different marginal distributions should be used.

We may see by the results that the estimated distributions, as well as the estimated association parameters are good approximations to the empirical/observed behavior of the data. Given this good approximation, the probabilities of interest may be approximated by a good degree. Hence, obtaining the odds of having exceedances of certain environmental thresholds (such as those used to declare environmental alerts in the city) and therefore, the possibility of an alert may be made with good accuracy.

The model used here could be considered as a first exploratory action in order to evaluate the air quality in future days and hence, evaluate the measures that could be taken in order to possibly avoid environmental emergencies, as well as inform the population of the risks.

The authors thank an anonymous reviewer for the comments that helped to improve the presentation of the results in this work. This work is part of JAVM Ph.D. Thesis developed at the Benemérita Universidad Autónoma de Puebla, Puebla, México. The authors thank CONAHCyT-Mexico.

In this appendix we present the plots of the monthly maxima of five of the criterion pollutants against the ozone monthly maxima, i.e., the CO, NO₂, SO₂, PM₁₀, and PM_2.5 monthly maxima plotted against the ozone’s.

In this appendix we present the expressions for the likelihood functions in the cases where both $F_i(\cdot )$ and $F_j(\cdot )$ are Fréchet (α, σ) distributions; both are Weibull (α, σ) distributions; and the case where $F_i(\cdot )$ is Weibull (α, σ) and $F_j(\cdot )$ is Fréchet (α, σ).

Both $F_i(\cdot )$ and $F_j(\cdot )$ are Fréchet (α, σ) distributions

In the case where $F_i(\cdot )$ and $F_j(\cdot )$ are both Fréchet (α, σ) distributions, let $\left({\alpha }_i\mathrm{,}{\sigma }_i\right)$ and $\left({\alpha }_j\mathrm{,}{\sigma }_j\right)$ be their respective parameters. Their expressions are given by (3). Since the joint distribution function is given by the copula (1), the joint density given by (2) is

$\begin{array}{c}{f}_{ij}\left(x,y\right)=\left(\frac{{\alpha }_i{\alpha }_j}{{\sigma }_i{\sigma }_j}\right){\left(\frac{x}{{\sigma }_i}\right)}^{-{\alpha }_i-1}{\left(\frac{y}{{\sigma }_j}\right)}^{-{\alpha }_j-1}\exp \left(-{\left[{\left(\frac{x}{{\sigma }_i}\right)}^{-{\alpha }_i\theta }+{\left(\frac{y}{{\sigma }_j}\right)}^{-{\alpha }_j\theta }\right]}^{\frac{1}{\theta }}\right)\\ \left\{{\left[{\left(\frac{x}{{\sigma }_i}\right)}^{-{\alpha }_i\theta }+{\left(\frac{y}{{\sigma }_j}\right)}^{-{\alpha }_j\theta }\right]}^{\frac{2}{\theta }-2}+\left(\theta -1\right){\left[{\left(\frac{x}{{\sigma }_i}\right)}^{-{\alpha }_i\theta }+{\left(\frac{y}{{\sigma }_j}\right)}^{-{\alpha }_j\theta }\right]}^{\frac{1}{\theta }-2}\right\}.\end{array}$

Both $F_i(\cdot )$ and $F_j(\cdot )$ are Weibull (α, b) distributions

In the case where $F_i(\cdot )$ and $F_j(\cdot )$ are both Weibull (α, b) distributions, let $\left({\alpha }_i\mathrm{,}{b}_i\right)$ and $\left({\alpha }_j\mathrm{,}{b}_j\right)$ be their respective parameters. Their expressions are given by (5). Since the joint distribution function is given by the copula (1), the joint density given by (2) is

$\begin{array}{c}{f}_{ij}\left(x,y\right)=\left({\alpha }_i{\alpha }_j\right)\left(b_i{b}_j\right)\left(x^{{\alpha }_i-1}{y}^{{\alpha }_j-1}\right)\exp \left(-b_i{x}^{{\alpha }_i}-b_j{y}^{{\alpha }_j}\right)\\ \frac{{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta -1}{\left(-\log \left[1-{\text{e}}^{-b_j{y}^{{\alpha }_j}}\right]\right)}^{\theta -1}}{\left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\left[1-{\text{e}}^{-b_j{y}^{{\alpha }_j}}\right]}\\ \exp \left(-{\left[{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta }+{\left(-\log \left[1-{\text{e}}^{-b_j{y}^{{\alpha }_j}}\right]\right)}^{\theta }\right]}^{\frac{1}{\theta }}\right)\\ \{{\left[{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta }+{\left(-\log \left[1-{\text{e}}^{-b_j{y}^{{\alpha }_j}}\right]\right)}^{\theta }\right]}^{\frac{2}{\theta }-2}\\ +\left(\theta -1\right){\left[{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta }+{\left(-\log \left[1-{\text{e}}^{-b_j{y}^{{\alpha }_j}}\right]\right)}^{\theta }\right]}^{\frac{1}{\theta }-2}\},\end{array}$

with $b_k={\sigma }_k^{-1/{\alpha }_k}$, $k=i\mathrm{,}j$.

$F_i(\cdot )$ is a Weibull ( ${\alpha }_i$, $b_i$) distribution and $F_j(\cdot )$ is a Fréchet ( ${\alpha }_j$, ${\sigma }_j$)

In the case where $F_i(\cdot )$ is a Weibull (α, b) distribution and $F_j(\cdot )$ is a Fréchet (α, σ), let $\left({\alpha }_i\mathrm{,}{b}_i\right)$ and $\left({\alpha }_j\mathrm{,}{\sigma }_j\right)$ be their respective parameters. Their expressions are given by (5) and (3), respectively. Since the joint distribution function is given by the copula (1), the joint density given by (2) is

$\begin{array}{c}{f}_{ij}\left(x,y\right)=\left(\frac{b_i{\alpha }_i{\alpha }_j}{{\sigma }_j^{-{\alpha }_j\theta }}\right)\left(x^{{\alpha }_i-1}{y}^{-{\alpha }_j\theta -1}\right)\exp \left(-b_i{x}^{{\alpha }_i}\right)\frac{{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta -1}}{1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}}\\ \exp \left(-{\left[{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta }+{\left(\frac{y}{{\sigma }_j}\right)}^{-{\alpha }_j\theta }\right]}^{\frac{1}{\theta }}\right)\end{array}$

$\begin{array}{c}\{{\left[{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta }+{\left(\frac{y}{{\sigma }_j}\right)}^{-{\alpha }_j\theta }\right]}^{\frac{2}{\theta }-2}\\ +\left(\theta -1\right){\left[{\left(-\log \left[1-{\text{e}}^{-b_i{x}^{{\alpha }_i}}\right]\right)}^{\theta }+{\left(\frac{y}{{\sigma }_j}\right)}^{-{\alpha }_j\theta }\right]}^{\frac{1}{\theta }-2}\},\end{array}$

with $b_k={\sigma }_k^{-1/{\alpha }_k}$, $k=i\mathrm{,}j$.

In this appendix we present the models, computational details, results, as well as the observed (empirical) and estimated density functions for all pollutants and regions in the one-dimensional models. In all cases, both Fréchet (α, σ) and Weibull (α, σ) distributions were assumed to describe the behavior of the monthly maximum measurements. Once the parameters have been estimated the graphical representation of the density functions, DIC, MLF, and p-value were used to select the distribution that best fit the observed data.

The prior distributions of the parameters of the models are given as follows. In the cases of both Fréchet and Weibull distributions, the parameter α has as a prior distribution a uniform U(0, 10) in all cases. When we consider the parameter b in the Weibull distribution, it has as a prior distribution a U(0, 10) in all cases. If we take into account the parameter σ of the Fréchet distribution, the values of the hyperparameters of the prior distribution are more heterogeneous. Hence, when we have CO, in all regions, the parameters σ will have as prior distribution a uniform U(0, 10). It has a uniform U(0, 100) in the cases of SO₂ and regions CE, SE, and SW; and of PM_2.5 in all regions with the exception of region NE. The uniform prior distribution U(0, 150) is used in the cases of NO₂ in all regions;