Monotonicity results and new bounds for the Mills ratio

Zhen-Hang Yang () and Jing-Feng Tian ()
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Zhen-Hang Yang: State Grid Zhejiang Electric Power Company Research Institute
Jing-Feng Tian: North China Electric Power University

Statistical Papers, 2025, vol. 66, issue 1, No 25, 27 pages

Abstract: Abstract The Mills ratio of the standard Gaussian law plays an important role in probability and statistics since its reciprocal is the failure rate function. In the paper, we verify the increasing property of the Turán-type ratio of the higher derivative of the Mills ratio. This result yields two new bounds for the Mills ratio (or failure rate function) having several interesting properties, and improves known results.

Keywords: Gaussian law; Mills ratio; Failure rate function; Monotonicity; Sharp bounds; Primary 60E15; 26A48; Secondary 26D15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00362-024-01646-x

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