640 zyxwvutsrqponmlkj IEEE TRANSACTIONS ON RELIABILITY, VOL. 43, zyxwv NO. 4, 1994 DECEMBER Characterization of Distributions By Relationships Between Failure Rate and Mean Residual Life JosC M. Ruiz Jorge Navarro Murcia University, Murcia Murcia University, Murcia zyxwvutsrq Key Words zyxwvutsrqpo - Characterization,Failure rate, Mean residual life, Renewal process. Reader Aids - General purpose: Widen state of the art Special math needed for explanations: Probability theory Special math needed to use results: Same Results useful to: Statisticians, reliability theoreticians Abstract - The characterizationsuse relations between the failure rate function and conditional expectation. The zyxwvuts theorems pro- ved here extend the results of some authors and can be used in the context of the renewal process. The utility of our results is demonstrated by using them to characterize some common distributions. In the renewal process [ 13, 20 - 221 the Y, associated with X, has a special importance [4], since, for large values of zy x, it represents the life of the process when an operating compo- nent is replaced, upon failure, by another, possessing the same life distribution. A problem posed recently, included in the theory of renewal process, is the characterization of distributions by rela- tionship between various representative functions (failure rate or mean residual life) of X & Y [4, 8, 14, 151. Another prob- lem of characterizationby the relationship between r(x) & m (x) has been discussed in [ l , 16, 17, 191. Our aim in this paper is to characterize X by obtaining its Cdf by relations between r(x) and e(x) or m(x). Section 2 obtains F(x), assuming that v ( t) = e ( t) r (t) is known; this allows us to extend the results of [4, 8, 14, 151. Section 3 characterizes F(x) by the relation: m(x) = k + zy q(x) -r(x), where q(x) is a real function satisfying certain con- ditions. This result extends the results in [l, 16, 17, 191 which are special cases. The theorems of section 3 can be interpreted in the renewal process context. Section 4 gives example applica- tions of these results. All proofs are in the appendix. 1. INTRODUCTION 2. USING RELATIONSHIPS BETWEEN r(x) & e(x) Characterizations of distributions in statistics are impor- tant [2,3,7]. In particular, in reliability theory, given a ran- dom variable X (often representing the life a unit in a certain process), some functions are assigned to it; those functions represent aging and characterize this variable. Among the most used are: failure rate, mean residual life, and mean function truncated on the left (vitality function [lo]). Several characterizations of distributions using these functions have been extensively discussed [5, 6, 9 11 - 13, 18, 231. Notation X a r.v. Y a r.v. withfy(t) = Rx(t)/p Z implies X or Y zyxwvutsrqp f( - 13 P ( .) pdf{X} 9 pmf{X) fu(*), PdfiY), P m f w F( - ), R( *> Cdf{X), Sf{X} Fy(*). Ry(*) Cdf{Y), Sf{U r( e), ry( e) failure rate of [X, yl p, py e( e), ey( e) mean residual life of [X, yl: E{Z-zlZiz} m( e), my( .) mean function of [X, yl truncated on the left: c,k constants. Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue. mean life of [X, yl E(ZIZ2z) = ez(z) + z Lemma 1 [lo]. Let X be a continuous r.v.. Then: “(x) - e’(x) + 1 r(x) = ~ - -. 4x1 m(x) -x 4(2-1) Zkeorem 1. Let X be a continuous r.v. with finite mean p. F(x) is uniquely determined by v(x) = r (x) .e (x) , xE D= {t€ I F(t) < 1): F(x) = 1 - exp [ - sx ’(‘) dt], -Q, Z(t)+p-t Z(t) = v(u) du. si.. 4 Remark 2.1. In the context of the renewal process, if we use [4]: then theorem 1 allows us to characterize distributions by rela- tionshipsoftype: ry(t) = g(t).r(t) orey(t) = h(t).e(t).r Remark 2.2. v(x) =c ing (2-3) gives ry(t) =r(t) which is the result in [4]. a. c = 1. Using (2-2) gives the exponential distribution. Us- r 00 18-9529/94/$4.00 O 1994 IEEE ~~