Limit Theorems for Order Statistics with Variable Rank Under Exponential Normalization

H. M. Barakat () and A. R. Omar ()
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H. M. Barakat: Zagazig University
A. R. Omar: Al-Azhar University, Girls Branch

Sankhya A: The Indian Journal of Statistics, 2023, vol. 85, issue 1, No 32, 783 pages

Abstract: 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 $\exp \{u_{n}(|\log |x||)^{v_{n}}$ S ( log | x | ) } $\mathcal {S}(\log |x|)\}$ S ( x ) , $\mathcal {S}(x),$ u n , v n > 0 , 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.

Keywords: Nonlinear normalization; Exponential norming; Central order statistic; Quantiles; Intermediate order statistics.; 62G32; 62G30 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13171-021-00275-y

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