Abstract
In this paper we study convolution residuals, that is, if \(X_1,X_2,\ldots ,X_n\) are independent random variables, we study the distributions, and the properties, of the sums \(\sum _{i=1}^lX_i-t\) given that \(\sum _{i=1}^kX_i>t\), where \(t\in \mathbb R \), and \(1\le k\le l\le n\). Various stochastic orders, among convolution residuals based on observations from either one or two samples, are derived. As a consequence computable bounds on the survival functions and on the expected values of convolution residuals are obtained. Some applications in reliability theory and queueing theory are described.
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Acknowledgments
We thank two reviewers whose comments on previous versions of this paper led to a significant improvement of the results and the presentation. The research of Moshe Shaked is supported by NSA grant H98230-12-1-0222. The research of Baha-Eldin Khaledi is financially supported by Research Department of Islamic Azad University, Kermanshah Branch, Kermanshah, Iran.
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Amiripour, F., Khaledi, BE. & Shaked, M. Stochastic orderings of convolution residuals. Metrika 76, 559–576 (2013). https://doi.org/10.1007/s00184-012-0404-x
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DOI: https://doi.org/10.1007/s00184-012-0404-x