Abstract
In the present paper, we study the limit distributions for the central and intermediate order statistics under a nonlinear normalization of the form \(\exp \{u_{n}(|\log |x||)^{v_{n}}\) \(\mathcal {S}(\log |x|)\}\) \(\mathcal {S}(x),\) \(~u_{n},v_{n}>0,\mathcal {S}(x)=\) sign (x), which is called exponential norming. We derive all limit types for the central (the quantiles) and intermediate order statistics. It is revealed that under this transformation the log-normal and negative log-normal distributions are possible limits of the central order statistics, while the normal distribution is no longer a possible limit of the quantiles. Some illustrative examples are given.
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Acknowledgements
The authors are grateful to the Editor-in-Chief, Professor Dipak K. Dey, and the referees for their useful suggestions that have substantially improved the presentation.
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Barakat, H.M., Omar, A.R. Limit Theorems for Order Statistics with Variable Rank Under Exponential Normalization. Sankhya A 85, 771–783 (2023). https://doi.org/10.1007/s13171-021-00275-y
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DOI: https://doi.org/10.1007/s13171-021-00275-y
Keywords and phrases
- Nonlinear normalization
- Exponential norming
- Central order statistic
- Quantiles
- Intermediate order statistics.