Optimality in weighted L₂-Wasserstein goodness-of-fit statistics

In Del Barrio, Cuesta-Albertos, Matran and Rodriguez-Rodriguez (1999) and Del Barrio, Cuesta-Albertos and Matran (2000), the authors introduced a new class of goodness-of-fit statistics based on the L₂-Wasserstein distance. It was shown that the desirable property of loss of degrees-of-freedom holds only under normality. Furthermore, these statistics have some limitations in their applicability to heavier-tailed distributions. To overcome these problems, the use of weight functions in the statistics was proposed and investigated by De Wet (2000), De Wet (2002) and Csörgő (2002). In the former the issue of loss of degrees-of-freedom was considered and in the latter the application to heavier-tailed distributions. In De Wet (2000) and De Wet (2002) it was shown how the weight functions could be chosen in order to retain the loss of degrees-of-freedom property separately for location and scale families. The weight functions that give this property, are the ones that give asymptotically optimal estimators for respectively the location and scale parameters – thus estimation optimality. In this paper we show that in the location case, this choice of “estimation optimal” weight function also gives “testing optimality”, where the latter is measured in terms of approximate Bahadur efficiencies

Fisher's information and the class of f-divergences based on extended Arimoto's entropies

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Tsallis’ entropies — Axiomatics, associated f-divergences and Fisher’s information

In a previous paper, de Wet and Österreicher (2016) showed how Arimoto’s extended class of entropies generates a family of f-divergences leading to approximation results and finally to Fisher’s information in a limiting way. In the current paper, the so-called Tsallis class of entropies is used in a similar fashion to generate a new family of f-divergences with analogous properties. The approximation properties are proved in a form which significantly generalizes the corresponding results in the above mentioned paper

Prediction error estimation of the survey-weighted least squares model under complex sampling

Linear modelling with the objective to predict a future response is ubiquitous in statistical analysis. Methods such as cross-validation and the bootstrap are well known for estimating the predictive performance of a model fitted to i.i.d. data. However, many large-scale surveys make use of a complex sampling design where the data are no longer i.i.d. and sampling weights are assigned to each observation to account for this. This paper shows how the cross-validation and bootstrap methods need to be adapted to evaluate the predictive performance of the survey-weighted least squares model. The investigation of the performance of the different prediction error estimation methods is evaluated through a simulation study. The Income and Expenditure Survey 2005/2006 of Statistics South Africa will form the basis of the analysis. The simulation study will also investigate whether the model’s predictive performance is improved through the truncation of outlier sampling weights. For this purpose, two new thresholds, viz. the 1.5IQR and Hill, are introduced. It was found that the bootstrap estimator of prediction error achieved lower mean squared error while the K-fold cross-validation estimator achieved lower bias. Further improvement was observed using the 1.5IQR and Hill truncated sampling weights

Bells as memorials in South Africa to the Great (1914-18) War

The contribution of South Africa to the allied cause in the Great War, and the sacrifice of so many lives of the White and Coloured populations, is memorialised on bells of the Cape Town carillon, on ringing, and on clock and other bells. The contribution of the Black population awaits recognition. Restoration of the Cape Town carillon so that it can again be played effectively, would be a fitting memorial to those who lost their lives in the non-combative roles that were open to the majority population of South Africa. Completion of the ring at what is now Queenstown cathedral would also be a fitting tribute to the bravery and unstinting service of so many South Africans during the Great War.Colin Lewis was Professor of Geography at Rhodes University, Grahamstown, South Africa from 1989 until his retirement at the end of 2007. In 1990, with the strong support of the incumbent Vice-Chancellor, Dr Derek Henderson, he instigated the Certificate in Change Ringing (Church Bell Ringing) in the Rhodes University Department of Music and Musicology - the first such course to be offered in Africa. Since that date he has lectured in the basic theory, and taught the practice of change ringing. He is the Ringing Master of the Cathedral of St Michael and St George, Grahamstown, South Africa

Robust Estimation of Pareto-Type Tail Index through an Exponential Regression Model

In this paper, we introduce a robust estimator of the tail index of a Pareto-type distribution. The estimator is obtained through the use of the minimum density power divergence with an exponential regression model for log-spacings of top order statistics. The proposed estimator is compared to an existing estimator for Pareto-type tail index based on fitting an extended Pareto distribution with the minimum density power divergence. A simulation study is conducted to assess the performance of the estimators under different contaminated samples from different distributions. The results show that the proposed estimator has better mean square errors and less sensitivity to an increase in the number of top order statistics. In addition, the estimation of the exponential regression model yields estimates of second-order parameters that can be used for estimation of extreme events such as quantiles and exceedance probabilities. The estimators are illustrated with a practical dataset on insurance claims