*4.2. Air Pollution*

The concentration of average air pollutants has been used in epidemiological surveillance as an indicator of the level of atmospheric contamination. Among its associated adverse effects in humans, diseases such as bronchitis are found. The distribution of this concentration has a bias to the right, and is always positive. It is typically assumed that the data on air pollutant concentrations are uncorrelated and independent and thus they do not require the diurnal or cyclic trend analysis, see [32]. The data set studied in this section corresponds to daily measures of ozone levels (in *ppb* = *ppm* × 1000) in the city of New York, USA, from May to September, 1973, collected by the New York State Conservation Department. The sample mean, standard deviation, asymmetry and kurtosis coefficients are given, respectively, by 42.129, 32.987, 1.209 and 1.112.

### 4.2.1. FBS versus the BS and SBS Distributions

Maximum likelihood estimators, their estimated standard errors (in parenthesis), for the BS, SBS and FBS models, were computed, the results are: BS model: *α*ˆ = 0.982 (0.064) and *β*ˆ = 28.031 (2.265). SBS model: *α*ˆ = 1.270 (0.235), *β*ˆ = 14.831 (4.019) and *λ*ˆ = 1.066 (0.533). FBS: *α*ˆ = 5.160 (0.481), *β*ˆ = 78.000 (0.008), ˆ*δ* = 3.991 (0.050) and *λ*ˆ = −9.135 (2.417). Forthisdataset,thelog-likelihoodratiostatisticstotestBSvsFBSandSBSvsFBSaregivenby

 −2 log(<sup>Λ</sup>1) = 12.812 and − 2 log(<sup>Λ</sup>2) = 5.828,

which are greater than the corresponding 5% critical values from the chisquare distribution, which are 5.99 (with 2 df) and 3.84 (with one df), respectively. So, we can conclude that the unimodal FBS model provides a better fit to this dataset than BS and FBS models.

### 4.2.2. FBS versus the Extended BS (EBS) Model

Fitting the five-parameter EBS model, *EBS*(*<sup>α</sup>*, *β*, *σ*, *ν*, *<sup>λ</sup>*), proposed in [16] as the best for this dataset, and whose point estimates for the parameters and summaries for comparison are equal to

$$EBS(0.780, 0.596, 3.618, -3.539, -0.096),$$

*CAIC* = 1111.154 and *BIC* = 1106.154. On the other hand, for the FBS model, it was obtained *CAIC* = 1108.396 and *BIC* = 1104.396. Then, according to the CAIC and BIC criteria, FBS model presents the best fit to this data set dealing with the daily ozone level concentration in the atmosphere.

Figure 5 depicts the histograms and the fitted density curves for the data set studied and empirical cdf for variable daily ozone level concentration in the atmosphere, while the dashed line corresponds to the cfd for FBS model.

**Figure 5.** (**a**) Plots for FBS, (solid line), BS (dotted line), SBS (dashed line) and EBS (dotted and dashed line) models. (**b**) Empirical cdf with estimated FBS cdf (dashed line).
