Statistics and Probability Letters 85 (2014) 114–121 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Rényi’s residual entropy: A quantile approach Asok K. Nanda a , P.G. Sankaran b , S.M. Sunoj b,∗ a Department of Mathematics and Statistics, Indian Institute of Science Education and Research, Kolkata, India b Department of Statistics, Cochin University of Science and Technology, Cochin, India article info Article history: Received 14 March 2013 Received in revised form 20 November 2013 Accepted 22 November 2013 Available online 4 December 2013 MSC: 62N05 90B25 Keywords: Rényi’s entropy function Reliability measures Residual lifetime Quantile function Stochastic ordering abstract In the present paper, we introduce a quantile based Rényi’s entropy function and its resid- ual version. We study certain properties and applications of the measure. Unlike the resid- ual Rényi’s entropy function, the quantile version uniquely determines the distribution. © 2013 Elsevier B.V. All rights reserved. 1. Introduction The notion of entropy, later extended to information theory and statistical mechanics, was originally developed by physicists in the context of equilibrium thermodynamics. In 1865, Rudolf Julius Emanuel Clausius, one of the founders of thermodynamics coined the term entropy derived from the Greek word en-trepein which means energy turned to waste, although the concept was introduced by him in the year 1850 in the context of classical thermodynamics. Later, a statistical basis to entropy was given by Ludwig Boltzmann, Willard Gibbs and James Clerk Maxwell (see, Nanda and Das (2006)). A general concept of entropy to quantify the statistical nature of lost information in phone-line signals mathematically was developed by Shannon (1948), an electrical engineer from Bell Telephone Laboratory. In the last few years, the literature on information theory has grown quite voluminous. Apart from communication theory, information theory has found lot of applications in many social, physical and biological sciences, viz ., economics, statistics, accounting, language, psychology, ecology, pattern recognition, computer sciences, fuzzy sets etc. (see, Taneja (2001)). Let X be an absolutely continuous nonnegative random variable (rv) representing the lifetime of a component with cumulative distribution function (CDF) F (t ) = P (X ≤ t ) and survival function (SF) ¯ F (t ) = P (X > t ) = 1 − F (t ). The measure of uncertainty (Shannon, 1948) is defined by H(X ) = H(f ) =−  ∞ 0 (ln f (x)) f (x)dx =−E (ln f (X )), (1.1) ∗ Corresponding author. E-mail address: smsunoj@gmail.com (S.M. Sunoj). 0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.11.016