Statistical Methodology 10 (2013) 93–102 Contents lists available at SciVerse ScienceDirect Statistical Methodology journal homepage: www.elsevier.com/locate/stamet Some new applications of the total time on test transforms N. Unnikrishnan Nair, P.G. Sankaran ∗ Department of Statistics, Cochin University of Science and Technology, Cochin-682 022, India article info Article history: Received 29 September 2011 Received in revised form 13 July 2012 Accepted 18 July 2012 Keywords: Total time on test transform Ageing criteria Bathtub models Quantile functions IFR ordering abstract The concept of total time on test transforms (TTT) is well known for its applications in different fields of scientific study. In this article we present four applications of TTT in reliability theory. First we characterize ageing criteria such as IFRA and NBU in terms of TTT. Then we utilize an iterated version to construct bathtub shaped hazard quantile functions and corresponding lifetime models. Further, an index is developed for numerically measuring the extent of IFR-ness of a life distribution. Finally we demonstrate how the distributional properties such as kurtosis and skewness can be derived from the TTT. © 2012 Elsevier B.V. All rights reserved. 1. Introduction The concept of the total time on test transform (TTT) was introduced and developed in the early seventies; see for example [3,2]. When several units are simultaneously put under test to ascertain their life lengths, some units may fail during the test while others may survive it. The sum of all the completed and incomplete life lengths constitutes the total time on test statistic, and the limit of this statistic as the number of units increases indefinitely is called the TTT. For a non-negative continuous random variable with distribution function F (x), the TTT is defined as T (u) =  Q (u) 0 [1 − F (t )]dt (1.1) where Q (u) = inf[x|F (x) ≥ u], 0 ≤ u ≤ 1 ∗ Corresponding author. E-mail addresses: unnikrishnannair4@gmail.com (N.U. Nair), sankaran.p.g@gmail.com, sankaranpg@yahoo.com (P.G. Sankaran). 1572-3127/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.stamet.2012.07.003