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When a distributional model is chosen, the analytic relation between its shape parameters and the values taken by some kurtosis indexes, especially if they are unconventional, is rarely known. In addition, different indexes may provide contrasting evidence about the level of global kurtosis, when the parameters of the model are varied. That happens because just few parameters act “plainly” on kurtosis, namely so as to produce consistent modifications of the shape of the graph on both its sides. Many parameters, instead, affect kurtosis along with a change of the skewness of the distribution, that is by “inflating” a single side of the graph (usually a tail) at the expense of the other. Thanks to some relevant examples, this paper tries to provide general indications to recognize the two kinds of parameters above and to interpret their effect on the classical Pearson’s standardized fourth moment and on some lesser known kurtosis indexes. Specifically, it is shown that only a decomposed analysis of indexes can help to understand their apparent contradictions, especially when some of them are too sensitive to changes in the tails. Finally, some applications are provided.

The influence function of Gini's gamma (also known as Gini's cograduation index) is derived. This function is shown to be bounded so as to claim that the considered measure is B-robust. After defining a Fisher consistent version of Gini's gamma for the bivariate Normal model, a closer examination of the related influence function is given, by reporting its plot and by computing its gross error sensitivity. Finally, some results of a simulation study are reported to evaluate how robustness can be combined with efficiency for the considered measure.

As known, the least-squares estimator of the slope of a univariate linear model sets to zero the covariance between the regression residuals and the values of the explanatory variable. To prevent the estimation process from being influenced by outliers, which can be theoretically modelled by a heavy-tailed distribution for the error term, one can substitute covariance with some robust measures of association, for example Kendall's tau in the popular Theil–Sen estimator. In a scarcely known Italian paper, Cifarelli [(1978), ‘La Stima del Coefficiente di Regressione Mediante l'Indice di Cograduazione di Gini’, Rivista di matematica per le scienze economiche e sociali, 1, 7–38. A translation into English is available at http://arxiv.org/abs/1411.4809 and will appear in Decisions in Economics and Finance] shows that a gain of efficiency can be obtained by using Gini's cograduation index instead of Kendall's tau. This paper introduces a new estimator, derived from another association measure recently proposed. Such a measure is strongly related to Gini's cograduation index, as they are both built to vanish in the general framework of indifference. The newly proposed estimator is shown to be unbiased and asymptotically normally distributed. Moreover, all considered estimators are compared via their asymptotic relative efficiency and a small simulation study. Finally, some indications about the performance of the considered estimators in the presence of contaminated normal data are provided.

A path analysis method for causal systems based on generalized linear models is proposed by using entropy. A practical example is introduced, and a brief explanation of the entropy coefficient of determination is given. Direct and indirect effects of explanatory variables are discussed as log odds ratios, i.e., relative information, and a method for summarizing the effects is proposed. The example dataset is re-analyzed by using the method.

The classical goodness-of-fit problem, in the case of a null continuous and completely specified distribution, is faced by a new version of the Girone–Cifarelli test (see Girone, 1964; Cifarelli, 1974 & 1975). This latter test was introduced for the two-sample problem and showed a substantial gain of power over other common tests based on the empirical distribution function, notably over the Kolmogorov–Smirnov test. First, the problem of the re-definition of the Girone–Cifarelli test-statistic is considered, by reviewing the literature on the subject. A classical remark by Anderson (1962) is shown to be useful to choose the integrating function in the newly defined test-statistic. The sample properties of such a test-statistic are then studied. A table of critical values is obtained by simulation; moreover, the asymptotic null distribution is considered and its accuracy as an approximation of the finite distribution is discussed. Finally, a simulation study, considering a wide set of distributions mostly used in applications, is conducted to compare the proposed test with its classical competitors. The study gives some indications to locate such situations where the Girone-Cifarelli test performs at its best, notably over the Kolmogorov–Smirnov test.

A new rank correlation measure $\beta_n$ is proposed, so as to develop a nonparametric test of independence for two variables. $\beta_n$ is shown to be the symmetrized version of a measure earlier proposed by Borroni and Zenga (2007). More specifically, $\beta_n$ is built so that it can take the opposite sign, without changing its absolute value, when the ranking of one variable is reversed. Further, the meaning of the population equivalent of $\beta_n$ is discussed. It is pointed out that this latter association measure vanishes not only at independence but, more generally, at {\em indifference}, that is when the two variables do not show any ``tendency" to positive or negative dependence. The null distribution of $\beta_n$ needs an independent study: hence, the finite null variance and a table of critical values are determined. Moreover, the asymptotic null distribution of $\beta_n$ is derived. Finally, the performance of the test based on $\beta_n$ is evaluated by simulation. $\beta_n$ is shown to be a good competitor of some classical tests for the same problem.

CART (Classification and Regression Trees) is a non-parametric tree-structured recursive partitioning method, introduced by Breiman et al. (1984), to predict a response variable $Y$ on the basis of $p$ predictors: $X_1,\ldots,X_p$ observed on a learning sample of $N$ units. In this paper the case where the response $Y$ is an ordered categorical variable with $k$ levels $y_1\prec \ldots\prec y_k$ is considered. The aim of the classification tree is thus to predict the level of $Y$ on the basis of the vector $\mathbf{X}$ of the $p$ explanatory variables. Instead of accommodating existing algorithm to the ordinal classification task, some papers have faced directly the problem of identifying a suitable rule to grow ordinal classification trees. We will consider these papers into details and propose some improvements based on a known decomposition result regarding Gini's mean difference.

A common problem in control chart analyses is dealing with autocorrelated data. This problem is very often faced by fitting a suitable time-dependence model to data and by building the chosen control chart on its residuals based on the assumption that the stochastic process, of which the observed dataset is considered as a finite realization, is gaussian. In this paper the chance of dealing with autocorrelated non- gaussian data in control charts is analyzed. In particular, two cases will be considered: the case of uncorrectly modeled autocorrelated data and the case where time- dependence in data is disregarded.

The finite null expectation and variance of a rank correlation measure proposed by Borroni and Zenga (2007) are considered. A formal proof of their expressions was never provided, even if some partial results were published in a technical report (Borroni, 2006), which had a limited circulation. This paper aims at filling this gap by reporting a detailed proof.

The classical goodness-of-fit problem, in the case of a null continuous and completely specified distribution, is faced by a new version of the Girone-Cifarelli test. This latter test was introduced for the two-sample problem and showed a substantial gain of power over other common tests based on the empirical distribution function, like the Kolmogorov-Smirnov test. After considering the problem of the definition of the test-statistic in the goodness-of-fit framework, this paper deals with the sample properties of the test and compares it with its classical competitors. The superiority of the Girone-Cifarelli test is shown via a simulation study considering symmetric and skewed distributions.

Recently Borroni and Zenga (2007) proposed a new nonparametric test for independence; the corresponding test-statistic Dn is based on Gini’s mean difference computed on the total ranks obtained on the basis of two rankings of a sample of n units. The considered test is a competitor of classical tools such as Sperman’s rho, Kendall’s tau and Gini’s cograduation index. While some research work has been done to evaluate the performance of Dn in small samples, this paper reports some further asymptotical results. The Pitman asymptotic relative efficiency (ARE) of Dn with respect to its competitors is computed for the Fairlie-Gumbel-Morgenstern distribution. In addition, some simulation results are reported for other families of distribution where the computation of the ARE poses some technical difficulties.

Some results to evaluate the performance of a nonparametric test of independence introduced by Borroni and Zenga (2007) are considered. Monte Carlo simulations are reported and the Pitman asymptotic relative efficiency of the test, compared to other classical competitors, is computed in the case of the Fairlie-Gumbel-Morgenstern distribution.

A new rank correlation index, which can be used to measure the extent of concordance or discordance between two rankings, is proposed. This index is based on Gini's mean difference computed on the totals ranks corresponding to each unit and it turns out to be a special case of a more general measure of the agreement of $m$ rankings. The proposed index can be used in a test for the independence of two criteria used to rank the units of a sample, against their concordance/discordance. It can then be regarded as a competitor of other classical methods, such as Kendall's tau. The exact distribution of the proposed test-statistic under the null hypothesis of independence is studied and its expectation and variance are determined; moreover, the asymptotic distribution of the test-statistic is derived. Finally, the implementation of the proposed test and its performance are discussed.

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 b and c of the model Yi = a + b xi + c xi^2 + ei (i = 1 ,..., n) where the xi's are supposed to be known constants and the ei's are iid error terms following an unknown distribution. The parameters b and c 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.

In this paper the performance of a new nonparametric test proposed by Borroni and Zenga (2003) for the independence of two criteria is discussed. The test-statistic is based on Gini's mean difference computed on the total ranks assigned to each sampled unit according to the chosen criteria of sorting. The performance of the test is measured by simulating its power function via Monte Carlo methods when it is applied as a one-sided test of independence against concordance. At this aim, after assuming that the two criteria of ranking are based on the values taken by two quantitative variables, a wide range of bivariate models is set for the two populations. The choice of the simulated models reflects the common situation of sampling from non-Normal populations, usually faced in economic applications. The reported results show that the new proposed test has often good performances and that it can be considered as a natural competitor of other common nonparametric tests.

A very important problem in time series analysis is testing for randomness against serial dependence. Classical parametric methods, commonly based on the autocorrelation coefficient, can be misleading when the underlying distributional assumptions are not fulfilled. In this paper the use of a nonparametric measure of serial dependence, based on Gini's cograduation index, is discussed. More specifically, the finite and asymptotic properties of this test-statistic are discussed; the test is then compared with its competitors via asymptotic relative efficiency.

In this paper a new nonparametric test for randomness against serial dependence is developed. This test is based on a serial version of Gini's cograduation index (see Gini, 1954). The expectation and variance of the proposed test-statistic under the null hypothesis of the test are derived. Moreover, some indications about the exact distribution of the test-statistic under randomness are given. Finally, the proposed test is compared, via simulation, with the test based on the Spearman-Wald-Wolfowitz serial rank correlation coefficient (see Wald and Wolfowitz, 1943)

In questo lavoro ci si propone di confrontare le performance di alcuni test non parametrici per l’uguaglianza di due popolazioni. Il confronto viene effettuato simulando le rispettive funzioni di potenza tramite elaboratore. Poiché tale procedura richiede una specificazione dell’ipotesi alternativa del test, si sceglie di considerare il caso particolare di popolazioni distribuite secondo leggi asimmetriche.

A new nonparametric test for the equality of two populations is proposed. The test is a generalization of the Girone-Cifarelli test (see Girone (1964) and Cifarelli (1974, 1975)) which has been shown to be occasionally more powerful than other nonparametric tests, such as the Kolmogorov-Smirnov test. The test is based on a measure of dissimilarity between the sequences of ranks corresponding to the samples drawn from the two populations. The test can be adapted to verify the hypothesis of equality against one-sided and two-sided alternatives. Exact and asymptotic critical values of the test are provided. The asymptotic distribution of the test-statistic under H_0 shows an interesting link with brownian motion in [0,1].

Hierarchical partition models (see Malec and Sedransk, 1992, Consonni and Veronese, 1995) aim at finding an optimal grouping (partition) of a set of experiments regarding a target variable. In this class of models the partition is regarded as an unknown parameter, and one of the main goals is computing the posterior distribution over the class of the possible partitions. This problem has been addressed in Sampietro and Veronese (1998), where a Metropolis-Hastings algorithm is applied. In this paper the performance of an alternative procedure, based on the logic of genetic algorithms, is evaluated. The results of the two approaches are compared, even if a conjoint use of them is to be advised.

Consider a two-way contingency table displaying the joint distribution of two categorical variables X and Y. A frequent need in the analysis of this kind of data is to group and collapse some rows of the table, in such a way that the collapsed table reproduces the association between X and Y in the best way. To solve this problem, it is hence necessary to compute a measure of association (for instance the likelihood-ratio statistic G2) for every possible grouping of the rows of the original table. Unfortunaltely, this can hardly be done if the number of categories to be grouped is large; hence some approximate solutions must be searched (see Siatkowski and Krajewsky, 1989). We now propose a new method, based on the logic of genetic algorithms, to determine such approximations. The proposed algorithm can be applied when the number of group in the collapsed table is fixed or variable. The performance of the algorithm is tested on a dataset regarding rose varieties and reasults are compared with those obtained by Siatkowski and Krajewsky (1989)