Weighted maximum product of spacings method for extreme value distributions and its applications in air quality analysis

Abstract

Extreme value distribution (EVD) is important for modeling extreme events, such as drought and haze. There are several methods for estimating the parameters of the EVD. In this study, the parameters of the EVD were estimated using three different methods: the method of moments (MOM), the maximum likelihood estimation (MLE) method, and the maximum product of spacings (MPS) method. The performance of these estimation methods are compared by generating random samples from the EVD with known parameters and then estimating the parameters using each method in a simulation study. The MPS method outperformed the other two methods in terms of parameter estimation accuracy. However, the MPS method has shortcomings, especially for extreme events, because it does not take into account the variance of the spacings between the data points. Thus, the weighted maximum product of spacings (WMPS) approach is introduced in this study to address this issue by using a weight function to account for the variance in the spacings between the data points, which can improve the accuracy of the parameter estimates. The WMPS method was able to provide more accurate estimates than the MPS method. The weight function helps to improve the accuracy and stability of the estimation process by modelling the variance of the data and finding the appropriate slope of the cumulative distribution function (CDF) curve. The local regression method was found to be effective in providing accurate estimates of CDF slopes even when the data were highly skewed or had a large number of outliers. This study has shown that the large sample theory for the maximum product of spacing (MPS) can be extended to the weighted MPS (WMPS) estimator. The asymptotic properties of the WMPS hold where the consistency and normality of the WMPS are proven. In this study, the WMPS method are also applied to estimate parameters for the distribution of extreme air pollution events. Air quality is one of the most important issues that is frequently discussed because it continues to pose a significant global threat to health. Therefore, modelling air pollution is essential for planning and managing air quality because it helps local governments come up with plans to reduce the risk of pollution. Using better estimation methods for the EVD model makes it possible to predict and understand the characteristics of these extreme events and how they affect air pollution more accurately. In this study, EVD is used to fit the distribution of daily maximums of 𝑃𝑀10 data. Both the MPS and WMPS are used to estimate parameters of EVD consisting of Gumbel distribution (Type I), Fréchet distribution (Type II), and Generalized extreme value distribution (GEVD) for these extreme air pollution events. WMPS outperforms MPS in fitting the EVD for all air monitoring stations. The generalized extreme value distribution (GEVD) is the best-fit model for describing daily maximum 𝑃𝑀10 data at seven of the twelve most polluted stations. On the other hand, the 2-parameter Fréchet distribution is the best model for the other five stations. Overall, this study introduces an alternative parameter estimation method which is the WMPS and simulation values have shown that WMPS is suitable to estimate parameters of the EVD family.