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A class of models for uncorrelated random variables

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  • Ebrahimi, Nader
  • Hamedani, G.G.
  • Soofi, Ehsan S.
  • Volkmer, Hans

Abstract

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.

Suggested Citation

  • Ebrahimi, Nader & Hamedani, G.G. & Soofi, Ehsan S. & Volkmer, Hans, 2010. "A class of models for uncorrelated random variables," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1859-1871, September.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:8:p:1859-1871
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    References listed on IDEAS

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    1. de Paula, Aureo, 2008. "Conditional Moments and Independence," The American Statistician, American Statistical Association, vol. 62, pages 219-221, August.
    2. Hamedani, G. G. & S. Key, Eric & Volkmer, Hans, 2004. "Solution to a functional equation and its application to stable and stable-type distributions," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 1-9, August.
    3. Hamedani, G. G. & Volkmer, H. W., 2009. "Letter to the Editor," The American Statistician, American Statistical Association, vol. 63(3), pages 295-295.
    4. Shaw, W.T. & Lee, K.T.A., 2008. "Bivariate Student t distributions with variable marginal degrees of freedom and independence," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1276-1287, July.
    5. Rodríguez-Lallena, José Antonio & Úbeda-Flores, Manuel, 2004. "A new class of bivariate copulas," Statistics & Probability Letters, Elsevier, vol. 66(3), pages 315-325, February.
    6. Jones, M. C., 2002. "A dependent bivariate t distribution with marginals on different degrees of freedom," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 163-170, January.
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    Cited by:

    1. Schennach, Susanne M., 2019. "Convolution without independence," Journal of Econometrics, Elsevier, vol. 211(1), pages 308-318.
    2. Ebrahimi, Nader & Jalali, Nima Y. & Soofi, Ehsan S., 2014. "Comparison, utility, and partition of dependence under absolutely continuous and singular distributions," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 32-50.
    3. Vexler, Albert & Zou, Li, 2022. "Linear projections of joint symmetry and independence applied to exact testing treatment effects based on multidimensional outcomes," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    4. Chen, Xiaohong & Linton, Oliver & Yi, Yanping, 2017. "Semiparametric identification of the bid–ask spread in extended Roll models," Journal of Econometrics, Elsevier, vol. 200(2), pages 312-325.
    5. Alessandra Carleo & Carlo Domenico Mottura & Luca Passalacqua, 2011. "The mathematical framework underlying the "scenarios" approach for derivate transactions by italian local authorities," Departmental Working Papers of Economics - University 'Roma Tre' 0127, Department of Economics - University Roma Tre.

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