The Annals of Statistics

On the asymptotic distribution of a general measure of monotone dependence

Donato Michele Cifarelli, Pier Luigi Conti, and Eugenio Regazzini

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Abstract

In this paper 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 Section 3. The problems of testing the hypothesis of lack of association and of constructing confidence intervals for the population index of cograduation are also considered.

Article information

Source
Ann. Statist., Volume 24, Number 3 (1996), 1386-1399.

Dates
First available in Project Euclid: 20 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032526975

Digital Object Identifier
doi:10.1214/aos/1032526975

Mathematical Reviews number (MathSciNet)
MR1401856

Zentralblatt MATH identifier
0862.62014

Subjects
Primary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62G10: Hypothesis testing 62E20: Asymptotic distribution theory

Keywords
Empirical processes functional limit theorem index of cograduation lack of association (= indifference) monotone dependence $U$-statistics

Citation

Cifarelli, Donato Michele; Conti, Pier Luigi; Regazzini, Eugenio. On the asymptotic distribution of a general measure of monotone dependence. Ann. Statist. 24 (1996), no. 3, 1386--1399. doi:10.1214/aos/1032526975. https://projecteuclid.org/euclid.aos/1032526975


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References

  • 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.
  • 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.
  • 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.
  • CIFARELLI, D. M. and REGAZZINI, E. 1990. Some contributions to the theory of monotone dependence. Technical Report 90.17, CNR-IAMI, Milano. Z.
  • 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.
  • 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.
  • GINI, C. 1914. Di una misura delle relazioni tra le graduatorie di due caratteri. Tipografia, Cecchini, Roma. Z.
  • HARDY, G. H., LITTLEWOOD, J. E. and POLy A, G. 1929. Some simple inequalities satisfied by ´ convex functions. Messenger of Math. 58 145 152. Z.
  • HILDEBRANDT, T. H. 1963. Introduction to the Theory of Integration. Academic Press, New York. Z.
  • HOEFFDING, W. 1948. A class of statistics with asy mptotically normal distribution. Ann. Math. Statist. 19 293 325. Z.
  • KIMELDORF, G. and SAMPSON, A. R. 1978. Monotone dependence. Ann. Statist. 6 895 903. Z.
  • KIMELDORF, G. and SAMPSON, A. R. 1987. Positive dependence orderings. Ann. Inst. Statist. Math. 39 113 128. Z.
  • NEUHAUS, G. 1971. On weak convergence of stochastic processes with multidimensional parameter. Ann. Math. Statist. 42 1285 1295. Z.
  • SALVEMINI, T. 1951. Sui vari indici di cograduazione. Statistica 11 133 154. Z.
  • SEN, P. K. 1960. On some convergence properties of U-statistics. Calcutta Statist. Assoc. Bull. 10 1 18. Z.
  • TCHEN, A. H. 1980. Inequalities for distributions with given marginals. Ann. Probab. 8 814 827. Z.
  • YANAGIMOTO, T. and OKAMOTO, M. 1969. Partial orderings of permutations and monotonicity of a rank correlation statistic. Ann. Inst. Statist. Math. 21 489 506.
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