Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 1 (23) 135 142 Characterizations on Heavy-tailed Distributions by Means of Hazard Rate Chun Su 1, Qi-he Tang 2 1 Department of Statistics and Finance, University of Science and Technology of China, Hefei 2326, China (E-mail: suchun@ustc.edu.cn 2 Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 118 WB Amsterdam, Netherland (E-mail: q.tang@uva.nl) Abstract Let F () be a distribution function supported on [, ), with an equilibrium distribution function F e(). In this paper we shall study the function r e() =( ln F e()) = F ()/ F (u) du, which is called the equilibrium hazard rate of F. By the limiting behavior of r e() we give a criterion to identify F to be heavy-tailed or light-tailed. Two broad classes of heavy-tailed distributions are also introduced and studied. Keywords Equilibrium distribution, Hazard rate, heavy-tailed distribution. 2 MR Subject Classification 6E5, 62E99 1 Introduction Let F be a distribution function (d.f.) with support on [, ). According to [3], we say F,orits corresponding non-negative random variable (r.v.) X, is heavy-tailed if it has no eponential moments, i.e., e t df () =, for any t>. (1.1) Write K as the class of such heavy-tailed distributions (see [2]). If there eists some t > such that g(t) := e t df () <, for <t<t, (1.2) then F, or its corresponding non-negative r.v. X, is called light-tailed, where g(t) is called its moment generating function. Give a d.f., in order to determine that it is heavy-tailed or light-tailed, the criteria may be various. We introduce in Section 2 a function r e () (see (2.3) below), so called the equilibrium hazard rate function of F. In this paper the function r e () will act as one of the most important tools to characterize the heavy-tailed d.f.s. In terms of the limiting properties of r e (), we give a criterion by which we can decide heavy-tailedness or light-tailedness of a d.f. F.Thenwefirst introduce two classes of heavy-tailed distributions, say M and M. Listedattheendofthis paper are two eamples, which show that M and M are really broad classes of heavy-tailed distributions. This is just the partial motivation of the present research. Manuscript received February 12, 21. Revised January 29, 22. Supported by the National Natural Science Foundation of China (No.17181) & Special Foundation of USTC.
136 C. Su, Q.H. Tang 2 Notation of Function r e () Throughout, for a r.v. X, it s d.f. is denoted by F () =P (X ) anditstailbyf () = 1 F () =P (X >). Suppose that { P (X ) = 1, F () >, ; <EX= df() = F () d = µ<. (2.1) In view that, if X denotes the r.v. of a risk variable, it is usually non-negative and has a finite epectation EX <, we discuss our problems under condition (2.1) above. Write V () = F (u) du, V () = F (u) du and F e () = 1 µ V (). One easily sees that F e () is also a distribution on [, ), which is called the equilibrium distribution of F, or the integrated tail distribution of F. In the contet of applied probability, it can be used to describe ruin probabilities of renewal risk model, tail distributions of ladder heights of random walk, and limiting distributions of waiting time and busy period of queueing model, and so on. See [1,2,7] for details. As Pitman [6] suggested, in studying heavy-tail properties of F, one uses R() = ln F () rather than F itself. Clearly, R() is non-decreasing on [, ), and R() =, R( ) =. Further, for F with a density function f, R() is absolutely continuous and its first derivative is given by r() =R () = f() F (). (2.2) In contet of reliability theory R() defined above is termed the hazard function, whereas r() is its hazard rate. For the equilibrium distribution F e of d.f. F, its hazard function is R e () = ln F e (). Since F e() =F ()/µ at each continuous point of F,wehave,almost everywhere with respect to the Lebesgue measure, the hazard rate function of F e () equals r e () = F () F () =,, (2.3) F (u) du V () which is also called the equilibrium hazard rate function of F. Generally speaking, r e () possesses some nice properties. In reliability theory, one needs to consider the mean ecess function of a life-span variable X, i.e., m() =E(X X > ), ; see, for eample, Definition 6.2.3 in [2]. Clearly, m() = udf(u) F () F () = F (u) du F () = V () F () = 1 r e (). In the realm of reliability theory or medicine, m() is referred as the mean residual life function; in insurance business, m() can be interpreted as the epected claim size in the unlimited layer over priority. Herewecallm() the mean ecess loss function. When X is heavy-tailed, one may reasonably assume lim m() =, i.e. lim r e() =.
Characterizations on Heavy-tailed Distributions by Means of Hazard Rate 137 In what follows, let t + F =sup {t : } e t df () <, a + R() = lim inf, α R() F = lim sup. It is well known (see, for eample, Theorem 2.3.1 and Theorem 2.5.1 in [7]) that Lemma 2.1. If a + >, thenf is light-tailed and t + F = a+ ;ifα F =,thenf is heavy-tailed. In this paper we obtain: Theorem 2.1. For distribution F satisfying (2.1), we have (1) lim r e() =implies that F is heavy-tailed, i.e. (1.1) holds; (2) lim inf r e() > implies that F is light-tailed, i.e. (1.2) holds for some t >. Proof. (1) For arbitrary ε>, there eists > such that r e () = F () F (u) du <ε, for, and R e () R e ( )=logf e ( ) log F e () = r e (u) du ε( ), which means that and therefore that R e () R e ( ) ε,, R e () lim =. From this fact and Lemma 2.1, we know that F e is heavy-tailed. Since F is heavy-tailed if and only if F e is heavy-tailed, we conclude the proof of (1). (2) In fact, in this case, we have t = t + F = a+ = lim inf R() = lim inf r e(u) du >, and our conclusion follows immediately from Lemma 2.1. Now we introduce two classes of heavy-tailed distributions. Definition 2.1. F M, if the distribution F has the property that lim r e() =. Definition 2.2. F M, if the distribution F has the property that lim sup r e () <. Theorem 2.1 (1) implies M M K.
138 C. Su, Q.H. Tang 3 On Classes M and M For the convenience of further discussions, some important subclasses of heavy-tailed d.f.s are given here; for more details the readers are referred to [2]. 1. L (Long-tailed): F belongs to L if and only if F ( y) lim =1 F () for any y> (or equivalently for y =1); 2. D (Dominant-tailed): F belongs to D if and only if lim sup F (y) F () < for any <y<1 (or equivalently for y =1/2); 3. S (Subeponential): F belongs to S if and only if F lim n () = n F () for any n 2 (or equivalently for n =2),whereF n denotes the n-fold convolution of F with the corresponding tail F n =1 F n. 4. C (Consistently-varying): F belongs to C if and only if F (y) lim lim sup y 1 F () It is well known that (see [2]), F (y) =1, or equivalently, lim lim inf y 1 F () =1. L D S L K, D K, D S, S D. (3.1) The regular property of tail distribution in the class C was first introduced and dubbed a intermediate regular varying property by Jelenković &Lazar [4] when they considered some problems in queueing system and their applications. Schlegel [8] also applied the tail property in the class C to derive an asymptotics for ruin probabilities in perturbed risk models. It is easy to prove that C L D. Now we discuss the relationships among the eisting heavy-tailed classes S, L and D and the two heavy-tailed classes M and M defined in Definitions 2.1 and 2.2. Theorem 3.1. For the distributions satisfying condition (2.1), we have (i) S L M; (ii) D M. Note that (i) and (ii) together show that L D M, which indicates that M includes all heavy-tailed distributions used in practice (see (3.1)). For the distributions satisfying condition (2.1), we can rewrite (3.1) as follows: L D S L M K, D M M K, D S, S D.
Characterizations on Heavy-tailed Distributions by Means of Hazard Rate 139 We ll give two eamples at the end of this section to illustrate that M is really larger than L D and that M is really larger than D. Proof. (i) The assertion that S L is well known. Here we only give a direct proof for L M.GivenF Lwhich satisfies condition (2.1), we have, for any l>, r e () = F () V () = F () F (u) du F () +l F (u) du F () lf ( + l). First setting and then l, we get lim r e() = and thus F Mis proved. (ii) Let F Dsatisfy condition (2.1). By definition lim sup F (/2) / F () <, orequivalently for some c >, It follows that for any >, Therefore sup > V () = 1 F (/2) F () F (u) du c for all. (3.2) 2 r e () =sup F () c > V () and F M. F (u) du F (2) 1 c F (). Since F e is widely used in many fields of applied probability, net we shall determine to which heavy-tailed class F e belongs if F is in M or M. We first present the following lemma. Lemma 3.1. If r e ( ) δ holds for some > and δ>, then we have inf r e () δ 1/δ 2, (3.3) where, in the trivial case 1/δ <, theinf takes value on. Proof. For t 1/δ, wehave V ( t) = F (u) du + V ( ) t tf ( t)+ 1 δ F ( ) Therefore r e ( t) =F ( t)/v ( t) δ 2. ( t + 1 ) F ( t) 2 δ δ F ( t). Theorem 3.2. For the distributions satisfying condition (2.1), we have (i) F M F e L; (ii) F M F e C F e D; (iii) F M = F e S. Proof. (i) It is easy to see that, for the distribution F satisfying condition (2.1), we have { } F () =F ( )ep r e (u) du,. (3.4)
14 C. Su, Q.H. Tang So, if F M, then it is easy to prove that F e ( y) lim =1 F e () for any y>, hence F e L. Conversely, we suppose by contradiction that F e Lbut F M, i.e., lim sup r e () =2δ, say, for some δ>. Then there eists some sequence {a n,n 1} satisfying <a n as n such that r e (a n ) >δfor any n N. Without loss of generality, we assume a 1 > 1/δ. Thus by Lemma 3.1, an F e (a n 1/δ) F e (a n ) a = n 1/δ F (u) du an F (u) F e (a n ) V (a n ) a n 1/δ V (u) du 1 2, n N. It follows that lim sup F e ( 1/δ) F e () F e (a n 1/δ) lim 3 F e (a n ) 2, which contradicts to F e L. This completes the proof of (i) in Theorem 3.2. (ii) Suppose that F M, then for any <y<1, we have 1 F e(y) F e () =1+ F (t) dt y y F (t) dt 1+ F (t) dt y ( y F (t) dt y 1. 1) 1 y yf (y) (3.5) From the definition of M,weget y F (t) dt y lim lim sup =, y 1 1 y yf (y) which, together with (3.5), implies F e (y) lim lim sup y 1 F e () =1, i.e., F e C. The proof for F e C F e Dis trivial. Let s prove F e D F M. Obviously, for any fied <y<1, we have F () 1 F (t) dt y 1 F (t) dt 1 y = F (t) dt 1 y ( F e (y) F e () 1 ). Hence the conclusion F M follows from the definition of the class D. (iii) Since M Mand L D S (see (3.1)), we may infer from (i) and (ii) that (iii) holds. The following corollary is a direct consequence of Theorem 3.2. Corollary 3.1. For the distribution F satisfying condition (2.1), we have F D= F M = F e L D= F e S.
Characterizations on Heavy-tailed Distributions by Means of Hazard Rate 141 In view of the results obtained so far, the following questions naturally arise: Whether or not there eists some d.f. F,whichbelongstoM or M, but doesn t belong to D L?Inother words, is class M larger than L D and that class M larger than D? Our answers to both questions are affirmative. Eample 3.1. Let τ be a geometric r.v. distributed by P (τ = n) =(1 q)q n,where <q<1andn. Then, the d.f. F of X = τ p, p>1, satisfies F M, F / D L. Proof. It is well known that, see Appendi A3 of [9], if G D, then { } γ G =: sup γ γ dg() < <. Thus, for the F above, we have F / D since γ F = in our case. Secondly, note that (n +1) p n p as n. Thus, we have that, for any fied M>, lim sup F ( M) F () = lim sup n F (n p M) F (n p ) = lim sup n P (τ n) P (τ >n) = 1 > 1, (3.6) q from which the statement F / Lfollows. Finally we aim at the assertion F M. It is easy to see from (3.6) that, for any M>, lim sup F () V () lim sup Letting M in the above yields +M F () F (u) du F () lim V () =, which is the definition of F M. Eample 3.2. Let X be a r.v. distributed by lim sup F () MF ( + M) = 1 Mq. p n = P (X =2 nα )=c n β 2 nα, n, where α>1, β>1andc > is such that p n =1. Then the d.f. F of r.v. X satisfies n=1 F M, F / D L. Proof. Clearly, for any large > and some n() such that 2 (n 1)α < 2 nα, F () =P (X 2 nα ) P (X =2 nα ). The proof of F / Lis similar to that presented in Eample 3.1, whereas the rest of the proof is straightforward. Acknowledgement. would like to thank Professor J.A. Yan for his interest in this work. His valuable suggestions helped us discover Eample 3.1, which is simpler than that presented in the previous version of this paper.
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