Abstract
Under some restriction, we establish the minimum volume confidence region for parameters of Pareto distribution, which can be applied to complete samples and, as well as left, right or doubly censored samples. It is not only computationally convenient, but also almost as accurate as the best confidence region in the literature, the computation of which is difficult in the double or left censoring case.
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Acknowledgements
This research is partially supported by the National Natural Science Foundation of China (NSFC, Grant No. 11561073). The author would like to thank Editors and anonymous referees for valuable comments, corrections and suggestions.
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Appendices
Appendix A: Proof of Theorem 1
Proof
Let \(C^T\) be any level \(1-\alpha \) confidence set of \(\theta \), satisfying \({\int }_{\theta \in C^t}~dt\le r_k(\theta )|C^\theta |\). Then for any \(\theta \in \Theta \),
It follows from \(P[\theta \in C^T_k]=1-\alpha \) that
where
by definition of \(C^T_k\), and \(\overline{C}\) denotes the complementary set of C. Thus, \(|C_k^\theta |\le |C^\theta |\) for \(\theta \in \Theta \), which implies that \(|C^T_k|\le |C^T|\). The proof is complete. \(\square \)
Appendix B: R code for computing \(k_{\alpha },\ k_1,\ k_2,\ k_1(x),\ k_2(x) \) in (4) and (5)
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Jiang, F., Zhou, J. & Zhang, J. Restricted minimum volume confidence region for Pareto distribution. Stat Papers 61, 2015–2029 (2020). https://doi.org/10.1007/s00362-018-1018-9
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DOI: https://doi.org/10.1007/s00362-018-1018-9