A decomposition approach to the asymptotic distribution of the sample variance in discrete models

Document Type : Original Scientific Paper

Authors

Department of Statistics‎, ‎Payme Noor University‎, ‎Tehran‎, ‎Iran

Abstract

Understanding the asymptotic behavior of the sample variance is important in statistical theory and inference. While classical results provide chi-square limiting distributions for continuous populations, discrete random variables often exhibit non-classical behavior, complicating both theoretical analysis and practical applications. In this paper, we propose a decomposition of the sample variance for discrete random variables into two theoretically tractable components, enabling a finer characterization of their stochastic structure. We derive the asymptotic distributions of these components under a specific condition, revealing that certain components converge to chi-square laws - a phenomenon previously observed only in Bernoulli and binomial models with success probability one-half. Extensive Monte Carlo simulations confirm the accuracy of the theoretical approximations and demonstrate their relevance even for moderate sample sizes. These results provide new insights into the interplay between discreteness, variance decomposition, and asymptotic behavior, extending classical chi-square asymptotics to a broader class of discrete models.

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References

Abbasi, N., Keshavarz, N., Yarmohammadi, M. and Saadatmand, A. (2024). A simple method for determining the limiting distribution of sample central moments for two-point and Binomial distributions. Journal of Statistical Modelling: Theory and Applications, 4(1):1–10.
Johnson, N.L., Kemp, A.W. and Kotz, S. (2005). Univariate Discrete Distributions. 3rd edition, Hoboken, NJ: Wiley.
Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics, Vol. 1: Distribution Theory. 4th edition, London: Charles Griffin.
Lehmann, E.L. and Casella, G. (1998). Theory of Point Estimation. 2nd edition, Springer, New York.
Lehmann, E.L. and Romano, J.P. (2005). Testing Statistical Hypotheses. 3rd edition, Springer, New York.
Le, V.H. and Vargas, R. (2025). Copula-based cosimulation for simulating temporal or spatial data in biogeosciences. Journal of Geophysical Research: Biogeosciences, 130(10):e2025JG008802.
Ma, Y., Genton, M.G. and Parzen, E. (2011). Asymptotic properties of sample quantiles of discrete distributions. The Annals of Statistics, 39(3):1380–1409.
Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.
Van der Vaart, A.W. (2000). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 3, Cambridge University Press.
Journal of Statistical Modelling: Theory and Applications
Volume 6, Issue 2
July 2025
Pages 117-126

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History

  • Receive Date: 23 December 2025
  • Revise Date: 05 July 2026
  • Accept Date: 16 July 2026

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