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The Gini methodology in statistical inference and related area has got much attention in recent years. In this paper, using a characterisation based on the Gini index we develop a nonparametric test for testing exponentiality against new better than used in expectation class. We derive the exact null distribution of the proposed test statistic and then calculate the critical values for different sample sizes. Asymptotic properties of the test statistic are discussed. The test is compared with some other test by evaluating Pitman's asymptotic efficacy. We also discuss how to incorporate right-censored observations in our study. A simulation study is presented to demonstrate the performance of the testing method. Finally, we illustrated our test procedure using two real data sets. [ABSTRACT FROM AUTHOR]

Statistics based on spacings, or the gaps between points, have been widely used in many contexts, primarily in testing goodness of fit. This paper derives Edgeworth-type asymptotic expansions for the sum of functions ofs-step spacings wheres, the order of spacings, may increase together with the sample sizen. Whensis fixed, it is known that only the Greenwood test, based on the sum of squares of these spacings, is first-order asymptotically efficient. In contrast, it is shown here that ifsgoes to infinity, there exist many other tests which are first-order efficient. We introduce and study the second-order efficiency of such tests and show that ifsis sufficiently large relative ton, the Greenwood test is no longer second-order efficient. Interestingly, we see that the common phenomenon of first-order efficiency implying second-order efficiency does not hold true in this situation. [ABSTRACT FROM AUTHOR]

Nonparametric procedures for testing and estimation of the shape matrix in the case of multivariate elliptic distribution are considered. Testing for sphericity is an important special case. The tests and estimates are based on the spatial sign and rank covariance matrices. The estimates based on the spatial sign covariance matrix and symmetrized spatial sign covariance matrix are Tyler's [A distribution-free M-estimator of multivariate scatter, Ann. Statist. 15 (1987), pp. 234-251] shape matrix and and Dumbgen's [On Tyler's M-functional of scatter in high dimension, Ann. Inst. Statist. Math. 50 (1998), pp. 471-491] shape matrix, respectively. The test based on the spatial sign covariance matrix is the sign test statistic in the class of nonparametric tests proposed by Hallin and Paindaveine [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707-2756]. New tests and estimates based on the spatial rank covariance matrix are proposed. The shape estimates introduced in the paper play an important role in the inner standardisation of the spatial sign and rank tests for multivariate location. Limiting distributions of the tests and estimates are reviewed and derived, and asymptotic efficiencies as well as finite-sample efficiencies of the proposed tests are compared with those of the classical modified John's [Some optimal multivariate tests, Biometrika 58 (1971), pp. 123-127; The distribution of a statistic used for testing sphericity of normal distributions, Biometrika 59 (1972), pp. 169-173] test and the van der Waerden test (Hallin and Paindaveine, [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707-2756]). The symmetrised spatial sign- and rank-based estimates and tests seem to have a very high efficiency in the multivariate normal case, and they are much better than the classical estimate (shape matrix based on the regular covariance matrix) and test (John's test) for distributions with heavy tails. [ABSTRACT FROM AUTHOR]

Kendall's τ for univariate samples is extended to the multivariate case using unit direction vectors in place of signs. Depending on whether the design matrix is fixed, this statistic yields a test for the multivariate analysis of variance or multivariate regression. Test of independence based on this statistic are easy, efficient, and robust. Its asymptotic properties and its Pitman efficiency are studied. In addition, some useful algebra in multiple dimensions is presented. [ABSTRACT FROM AUTHOR]