Secondo fascicolo del 2006
Dependence measures based on partial and total orderings
by Francesca Greselin
The aim of this paper is to propose a new operational measure for evaluating the degree of dependence existing between two nominal categorical variables. Given an r×c table T, representing bivariate statistical data, our approach to measure the strength of this relation is based on the consideration of the class ℱ of all contingency tables with the same given margins as T. Once a partial or total ordering of dependence in ℱ (as defined in Greselin and Zenga [2004b]) has been given, the relative position assumed by T in ℱ can be a meaningful measure of dependence. Some desirable properties of these indexes are presented: by construction they are normalized, coherent with each level of ordering and attain extreme values in extreme dependence situations. They are invariant to permutation of rows and columns in the table and to transposition (as qualitative variables classification requires), and, finally they show a sort of stability behaviour with respect to similar populations. Furthermore, their straightforward interpretability is compared with the classical interpretation of some well-known normalized indexes. Interesting remarks arise when the comparison is carried out on the discussion of their values, particularly on the extreme dependence situations.
Keywords: partial ordering of dependence, association measure, dependence measure.
Gini’s cograduation index for the estimation of the coefficients of a quadratic regression model
by Claudio Giovanni Borroni
Least square estimates of regression parameters may become unreliable when some outliers affect the data. This fact forces to search for different methods of estimation, some of which consist of substituting ranks to observations to avoid influences by extremes values. Cifarelli (1978) proposes one of such methods to estimate the slope parameter of a linear regression model, by using Gini’s cograduation index. In this paper Cifarelli’s method is generalized to the case of quadratic regression, that is to the estimation of the parameters β and γ of the model Yi = α + β xi + γ xi2 + εi (i = 1 ,…, n) where the xi’s are supposed to be known constants and the εi’s are iid error terms following an unknown distribution. The parameters β and γ are estimated in a two-step procedure, each time by searching for the value which sets to zero the Gini’s cograduation index between the residuals of the model and the values of the explanatory variable (in analogy with a related property of the least square method involving covariance). A simulation study is provided to test the performance of the proposed method, which proves to be often superior to other known related methodologies.
Keywords: rank methods, quadratic regression, Gini’s cograduation index, Theil-Sen estimator.
Merging Gini’s Indices under Quadratic Loss
by S. E. Ahmed, R. Ghori, M. N Goria, A. Hussein
The Gini index is perhaps one of the most used indicators of economic and social condition. This article develops simultaneous estimation strategies of the Gini indices when samples are taken from several sources. In particular, we propose large sample test statistics for homogeneity of the indices. The null and non-null distributions of the proposed test statistics are derived. Further, a shrinkage estimator is suggested. The asymptotic bias and risk of the proposed estimator is derived and compared with benchmark estimator.
Keywords: Gini index, testing, shrinkage estimation, local alternatives, asymptotic quadratic risk.
Modelling dynamics and uncertainty in assessment of quality standards for fine particulate matters
by Alessandro Fassò, Orietta Nicolis
In order to assess compliance with air quality standards, European regulations prescribe to monitoring the concentration of particulate matters and to control both annual and daily averages. The measurement accuracy varies according to monitor type, temperature and pollution level, often in a complex nonlinear manner. Consequently, comparisons, threshold exceedances interpretation and compliance assessment are often difficult.
In this paper, we consider the displaced dynamical calibration (DDC) model which is able to calibrate biased readings by using displaced data obtained by reference instruments. Moreover, we discuss the uncertainty of annual averages of daily concentrations. An application to the Northern Italy air quality network allows us to draw some empirical conclusions.
Keywords: PM10 , measurement instruments, calibration, state-space modelling, quality standard assessment.
Statistical calibration of psychometric tests
by Francesca De Battisti, Silvia Salini, Alberto Crescentini
A calibration procedure is generally performed in order to correctly translate the personal traits observed through a psychometric test into numerical values. The calibration process ensures the objectivity of the measure instruments. Psychological measures are usually of indirect type, they are obtained as a result of a statistical inference process. Statistical calibration makes use of particular models, based on the inversion of the previous mentioned indirect measures. The Rasch model can be considered one of this model.
Keywords: statistical calibration, intelligence test, Rasch analysis.