On the asymptotic distribution of a general measure of monotone dependence. (English) Zbl 0862.62014

Summary: The asymptotic normality of a class of statistics, including Gini’s index of cograduation and Spearman’s rank correlation coefficient, is proved. The asymptotic normality is stated under a large class of alternatives including the bivariate distributions corresponding to a condition of lack of association introduced in this paper. The problem of testing the hypothesis of lack of association and of constructing confidence intervals for the population index of cograduation are also considered.


62E20 Asymptotic distribution theory in statistics
62H20 Measures of association (correlation, canonical correlation, etc.)
62G10 Nonparametric hypothesis testing
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[1] CAMBANIS, S., SIMONS, G. and STOUT, W. 1976. Inequalities for EEk X, Y when the marginals are fixed. Z. Wahrsch. Verw. Gebiete 36 285 294. Z. · Zbl 0325.60002 · doi:10.1007/BF00532695
[2] CIFARELLI, D. M. and REGAZZINI, E. 1974. Ancora sull’indice di cograduazione del Gini. Technical Report Serie III, No. 5, Istituto di Matematica Finanziaria dell’Universita, Torino. Z.
[3] CIFARELLI, D. M. and REGAZZINI, E. 1977. On a distribution-free test of independence based on Z Gini’s rank association coefficient. In Recent Developments in Statistics J. R. Barra,. F. Brodeau, G. Romier, B. Van Cutsem, eds. 375 385. North-Holland, Amsterdam. Z. · Zbl 0369.62043
[4] CIFARELLI, D. M. and REGAZZINI, E. 1990. Some contributions to the theory of monotone dependence. Technical Report 90.17, CNR-IAMI, Milano. Z.
[5] CONTI, P. L. 1993. Una classe di misure di dipendenza monotona tra due variabili continue: teoria descrittiva e problemi inferenziali non parametrici. Ph.D. dissertation, Dipartimento di Statistica, Probabilita e Statistiche Applicate, Universita La Sapienza, Roma. Z.
[6] GAENSSLER, P. and STUTE, W. 1979. Empirical processes: a survey of results for independent and identically distributed random variables. Ann. Probab. 7 193 243. Z. · Zbl 0402.60031 · doi:10.1214/aop/1176995085
[7] GINI, C. 1914. Di una misura delle relazioni tra le graduatorie di due caratteri. Tipografia, Cecchini, Roma. Z.
[8] HARDY, G. H., LITTLEWOOD, J. E. and POLy A, G. 1929. Some simple inequalities satisfied by ćonvex functions. Messenger of Math. 58 145 152. Z.
[9] HILDEBRANDT, T. H. 1963. Introduction to the Theory of Integration. Academic Press, New York. Z. · Zbl 0112.28302
[10] HOEFFDING, W. 1948. A class of statistics with asy mptotically normal distribution. Ann. Math. Statist. 19 293 325. Z. · Zbl 0032.04101 · doi:10.1214/aoms/1177730196
[11] KIMELDORF, G. and SAMPSON, A. R. 1978. Monotone dependence. Ann. Statist. 6 895 903. Z. · Zbl 0378.62059 · doi:10.1214/aos/1176344262
[12] KIMELDORF, G. and SAMPSON, A. R. 1987. Positive dependence orderings. Ann. Inst. Statist. Math. 39 113 128. Z. · Zbl 0617.62006 · doi:10.1007/BF02491453
[13] NEUHAUS, G. 1971. On weak convergence of stochastic processes with multidimensional parameter. Ann. Math. Statist. 42 1285 1295. Z. · Zbl 0222.60013 · doi:10.1214/aoms/1177693241
[14] SALVEMINI, T. 1951. Sui vari indici di cograduazione. Statistica 11 133 154. Z.
[15] SEN, P. K. 1960. On some convergence properties of U-statistics. Calcutta Statist. Assoc. Bull. 10 1 18. Z. · Zbl 0109.12504
[16] TCHEN, A. H. 1980. Inequalities for distributions with given marginals. Ann. Probab. 8 814 827. Z. · Zbl 0459.62010 · doi:10.1214/aop/1176994668
[17] YANAGIMOTO, T. and OKAMOTO, M. 1969. Partial orderings of permutations and monotonicity of a rank correlation statistic. Ann. Inst. Statist. Math. 21 489 506. · Zbl 0208.44704 · doi:10.1007/BF02532273
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