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On approximation and estimation of distribution function of sum of independent random variables

Author

Listed:
  • N. N. Midhu

    (IQVIA)

  • Isha Dewan

    (Indian Statistical Institute)

  • K. K. Sudheesh

    (Indian Statistical Institute)

  • E. P. Sreedevi

    (Maharaja’s College)

Abstract

In this paper, we obtain an approximation for the distribution function of sum of two independent random variables using quantile based representation. The error of approximation is shown to be negligible under some mild conditions. We then use the approximation to obtain a non-parametric estimator for the distribution function of sum of two independent random variables. The exact distribution of the proposed estimator is derived. The estimator is shown to be strongly consistent and asymptotically normally distributed. Extensive Monte Carlo simulation studies are carried out to evaluate the bias and mean squared error of the estimator and also to assess the approximation error. We also compare the performance of the proposed estimator with other estimators available in the literature. Finally, we illustrate the use of the proposed estimator for estimating the reliability function of a standby redundant system.

Suggested Citation

  • N. N. Midhu & Isha Dewan & K. K. Sudheesh & E. P. Sreedevi, 2024. "On approximation and estimation of distribution function of sum of independent random variables," Statistical Papers, Springer, vol. 65(3), pages 1437-1467, May.
    • Handle: RePEc:spr:stpapr:v:65:y:2024:i:3:d:10.1007_s00362-023-01413-4
    DOI: 10.1007/s00362-023-01413-4
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    References listed on IDEAS

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    1. Gordon Willmot & Jae-Kyung Woo, 2007. "On the Class of Erlang Mixtures with Risk Theoretic Applications," North American Actuarial Journal, Taylor & Francis Journals, vol. 11(2), pages 99-115.
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    3. R. Arabi Belaghi & M. Noori Asl & H. Bevrani & William Volterman & N. Balakrishnan, 2018. "On the distribution-free confidence intervals and universal bounds for quantiles based on joint records," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(12), pages 2969-2978, June.
    4. Buonocore, Aniello & Pirozzi, Enrica & Caputo, Luigia, 2009. "A note on the sum of uniform random variables," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 2092-2097, October.
    5. Kordecki, Wojciech, 1997. "Reliability bounds for multistage structures with independent components," Statistics & Probability Letters, Elsevier, vol. 34(1), pages 43-51, May.
    6. Saralees Nadarajah & Xiao Jiang & Jeffrey Chu, 2015. "A saddlepoint approximation to the distribution of the sum of independent non-identically beta random variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 69(2), pages 102-114, May.
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