Best unbiased graphical estimators of location-scale distribution parameters: application to the Pozzuoli’s bradyseism earthquake data

Pasquale Erto,Antonio Lepore

doi:10.1007/s10651-016-0356-9

Abstract

Graphical techniques are recommended for critical applications because they allow for visual analysis and helpful understanding of the results, also for non-statisticians. However, “graphical” estimation methods are often criticized because they are less efficient with respect to “analytical” methods. This paper proposes a new general graphical method that leads to the best linear unbiased estimators of location-scale distribution parameters. Therefore, the reputation of graphical methods is raised and their strategic use is encouraged. An applicative example analyzes the earthquake magnitudes registered during the serious 1983–1984 bradyseismic crisis in Campi Flegrei, Italy. This graphical analysis will certainly work as a strategic reference picture to which the data, arising from the further bradyseismic crises expected in the next future, can be compared.

Highlights

As confirmed by the renewed interest appeared in the recent literature (Rigdon and Basu 1989; Makkonen 2006, 2008a; de Haan 2007; Cook 2011, 2012; Kim et al 2012; Erto and Lepore 2013; Fuglem et al 2013; Makkonen and Pajari 2013; LozanoAguilera et al 2014) practitioners are used to exploiting modern software that adopts graphical estimation methods via probability papers, even if there is a variety of effective analytical methods available, such as Maximum Likelihood and Bayesian techniques
The Root Mean Square Deviation (RMSD) and the BIAS modulus of the proposed estimators (12) of location and scale parameters are compared (Tables 2, 3) with the usual estimators obtained through the ordinary leastsquare method (i.e., σ(i, j) = σ if i = j and zero otherwise) and the classical plotting positions (Table 1) as well as with the Maximum Likelihood Estimators (MLEs)
On the basis of theoretical considerations, a new probability paper based on the generalized least-squares method is proposed

Summary

Introduction

As confirmed by the renewed interest appeared in the recent literature (Rigdon and Basu 1989; Makkonen 2006, 2008a; de Haan 2007; Cook 2011, 2012; Kim et al 2012; Erto and Lepore 2013; Fuglem et al 2013; Makkonen and Pajari 2013; LozanoAguilera et al 2014) practitioners are used to exploiting modern software that adopts graphical estimation methods via probability papers, even if there is a variety of effective analytical methods available, such as Maximum Likelihood and Bayesian techniques. Especially in critical applications, the graphical estimation gives the unique opportunity to share statistical information with non-statisticians (e.g., by allowing a visual check of the fit of the chosen model and by giving helpful understanding of the consequent conclusions). X N , the basic problem of graphical methods is how to establish the estimate Fi of the cumulative distribution function (cdf) FX x(i) (i.e., the plotting position) that can ensure a required property (e.g., unbiasedness) for the resulting estimators of the distribution parameters. 2, a new graphical method is proposed that allows best linear unbiased estimation of location-scale distribution parameters. 3 exploits Monte Carlo simulation in the case of Gumbel parent distribution in order to confirm the unbiasedness of the resulting estimators of the distribution parameters as well as to compare the proposed solution to classical methods.

A new Gumbel probability paper

A critical application: the Pozzuoli’s bradyseism

December

VIII IX X XI XII XIII

Conclusions