THE EXISTENCE OF NEGATIVE MCMENTS OF CONTINUOUS DISTRIBUTIONS WALTER W. PIEGORSCH AND GEORGE CASELLA. Biometrics Unit, Cornell University, Ithc, NY 14853 BU-771-M * December 1982 Abstrct The question of existence of negtive moments of continuous probbility density function is explored. Sufficient conditions for both existence nd nonexistence of negtive moments re given. The conditions re esy to check nd, lthough they do not constitute necessry nd sufficient condition for existence, they will usully decide the cse for most commonly encountered density functions. For positive densities, necessry nd sufficient condition for existence (which is much hrder to verify) is lso presented. *rn the Biometrics Unit Series, Cornell University, Ithc, NY 14853.
THE EXISTENCE OF NEGATIVE MCMENTS OF CONTINUOUS DISTRIBUTIONS WALTER W. PIEGORSCH AND GEORGE CASELlA Biometrics Unit, Cornell University, Ithc, NY 14853 1. Introduction. In n introductory (post-clculus) probbility nd sttistics course, students spend much time in evlution of expecttions of rndom vribles. For the most prt, such evlutions re confined to positive moments of well-behved distributions nd, hence, the question of existence of these moments is rrely n issue. With the possible exception of the Cuchy distribution, the existence of t lest two positive moments is usully foregone conclusion. This is not the cse, however, if one is ttempting the evlution of negtive moments, for here nonexistence is much more common occurrence. In mny prcticl pplictions one is led quite nturlly to the evlution of negtive moments of rndom vrible. We cite three simple exmples. EXAMPLE 1.1. The exponentil distribution with prmeter A, given by ( ) -AX f x = Ae, (1.1) is often used s model for life-testing situtions. If one observtion is tken from this distribution, the stndrd method of mximum likelihood estimtion would led to using 1/x s n estimte of A Does the expected vlue, E(l/x), exist? EXAMPLE 1.2. In simple liner regression, one ssumes the model y =CX + sx. Usully, interest is in predicting y from x using fitted eqution y =+ " " sx A A where, under stndrd ssumptions, ex nd S re normlly distributed. Sometimes, however, the interest is in inverse regression; predicting x from y. The AMS 1980 Subject Clssifictions: Primry 60E99; Secondry 26Dl0.
-2- nturl estimte of x, for given y, is then (y-)/. Does E[(y-)/s] exist? EXAMPLE l.3. If smple is being drwn from popultion with men 1...1. nd stndrd devition cr, mesure of the mount of vrition in the popultion (which is prticulrly populr in the engineering sciences) is the coefficient of vrition cr/ii.j.j A nturl estimte of this quntity is s/lxl, where :X nd s re the smple men nd smple stndrd devition, respectively. Does E(s/lxl) exist? In generl, the theory behind the existence of negtive moments is quite difficult, nd not nerly s complete s tht involving positive moments. There, one is led into the (quite elegnt) theory of chrcteristic functions, the Fourier trnsforms of the density function. However, by restricting ttention to distributions with continuous density functions, the tools vilble to the student in n introductory course re sufficient to estblish some firly strong results; certinly strong enough to del with most continuous distributions tht re likely to be encountered. In Section 2 we present two theorems, one giving sufficient condition for existence of the first negtive moment, nd the other condition giving sufficient condition for nonexistence. Both conditions re esy to check; they do not involve evluting n integrl. Although these theorems re strong enough to hndle most common distributions, they do not provide necessry nd sufficient condition for the existence of the first negtive moment. In Section 3 we present distribution which is not covered by these theorems, nd prove theorem which gives necessry nd sufficient condition for the existence of the first negtive moment of positive rndom vrible. Section 4 discusses some immedite generliztions to higher order negtive moments.
-3-2. Conditions for Existence nd Nonexistence. The difficulties experi -1 enced in clculting expressions such s E(X ) will occur ner X= 0 For exmple, if X hs discrete probbility density function (pdf) with positive mss t X= 0, E(x-1 ) will be infinite. Cho nd Strwdermn [1] sidestep this problem by dding vlue to the rndom vrible which mkes it positive with probbility one. We will only be concerned here with the continuous cse, whic~ in some respects, is more complex. Let X be rndom vrible with continuous pdf, f(x). For ny Y, 5 > 0 we cn write CD -y 0 5 CD E(x- 1 ) = J x1 -oo -oo -y 0 5 f(x)dx = [J + J + J + JJ[x1 f(x)dx] (2.1) -Y oo Clerly, both I x 1 f(x)dx nd I x 1 f(x)dx re finite (since x 1 -oo 5-1. " these regions), hence E(X ) <"" if nd only if is bounded over nd 5 J x" 1 f(x)dx<"" 0 (2.2) for some y, 5>0. Note thte(x- 1 )<oo if nd only if E(lxl" 1 )<oo. We hve the following simple condition for nonexistence of E( lxl" 1 ) THEOREM 2.1. Let f(x) be~ continuous pdf on (, b), -oos:<b:5:c:o (f(x) =0 on (-co, ] U [b,=)). If ~ of the following conditions is true: then E(lxl- 1 ) =oo. (i) <O<b nd f(o)>o (ii) =O nd li~ f(x) > 0 x->0 (iii) b=o nd lim - f(x) > 0 x->0
-4- Proof. If (i) is true, from the continuity of f(x) there exists E, 5>0 such tht f(x) > E when XE ( -5,5), nd 5 cn be chosen such tht ( -5, 5) c (, b) Then, using (2.1), we hve 0 5 5 E(lxl- 1 ):::: I lxj- 1 f(x)dx+ I lxl- 1 f(x)dx::::2ej x 1 dx=o:> (2.3) -5 0 0 If (ii) is true, there exists E, 5> 0 such tht f(x) > E when xe(o, 5), nd 5 cn be chosen to stisfy 0< 5< b. Then 5 5 E ( I X 1 1 ) :::: I x 1 1 f ( x) dx :::: E J x 1 dx = = (2.4) 0 0 The proof is similr for conditien (iii). This theorem shows, in prticulr, tht continuous pdf on (-ex>,=) will not, in generl, hve first negtive moment. (One could, of course, construct distribution continuous on (-o:>,o:>), with f(o)=o, such tht the first negtive moment does exist, but this would be rther uncommon distribution.) The fr more interesting cse, to which we will restrict ttention, is tht of positive rndom vribles. If f(x) is continuous pdf on (O,oo) it is nturl to inquire, in light of Theorem 2.1, if lim+ f(x) = 0 is sufficient condition for E(x- 1 ) < CX) This is not the x... o cse, for it is the rte t which f(x) pproches zero tht is importnt. THEOREM 2.2. :::: 0 such tht -1 then E(X )<<». If f(x) is ~ continuous pdf on (O,oo), nd there exists lim[f(x) /x] < oo X-+0 (2.5) Proof. If ;:::O stisfies (2.5), then there exists finite constnt M. l nd 5>0 such tht lf(x)/x. S:M when o:::;;;xs: 5. Hence,
-5-0 0 0 J X 1 f(x)dx = J x-l[f(x)/x]dx~mj x-ldx 0 0 0 M =-o <eo, (2.6) so, using (2.2), E(x-1 ) exists. Theorem 2.2 hs n immedite corollry for differentible density functions. COROLLARY 2.l. Let f(x) be~ continuous pdf on (O,c:o), with f(o) =0. If f(o) exists nd is finite, then E(X-l)<c:o. Proof. From the definition of derivtive, f(o) = lim f(x)-f(o) X l. fix\ = J.m ~ x... O X (2.7) since f(o) = 0. The existence of E(X-l) now follows from Theorem 2.2 with = 1. As mentioned before, the results of this section re strong enough to determine the existence or nonexistence of first negtive moments for mny common distributions. We illustrte this with some exmples. EXAMPLE 2.1. Let f(x) be the uniform density on (,b), i.e., 1 f(x) = b- 0 if otherwise < x<b (2.8) If < 0< b, then f( 0) = (b- rl > 0 nd hence, from Theorem 2.1, E( I xl l) =(X) If either =O or b=o, it gin follows from Theorem 2.1 tht E(jxl 1 ):c:o. If [, b] does not contin zero, then E( I xl- 1 ) <eo. EXAMPLE 2.2. If f(x) is the norml density with men ~ nd stndrd devition cr, f(x) = 1 e-i(x-~)2jcr2 (2rrcr)i -c:~< x< co, (2.9)
-6- -1 then f( 0) > 0 nd E(X ) does not exist. This exmple covers the sitution outlined in Exmples 1.2 nd 1.3 (if the popultion is ssumed to be norml), nd shows tht neither of the nturl estimtes presented there hve finite expecttion. EXAMPLE 2.3. The gmm distribution with prmeters r>o nd A.>O hs pdf f(x) r A. r-1 -t..x = ---- x e, 0 :s; X< CXl r(r) (2.10) where r( ) denotes the gmm function. (The exponentil distribution of Exmple l.l is specil cse, hving r = l.) It is esy to check tht f(o) > 0 if r :s; 1, showing the nonexistence of E(x-1) if r :s; l If r> 1, then lim f(x) = 0 0+ X X-> (2.11) showing tht E(X- 1 ) < =if nd only if r> 1. Although Theorems 2.1 nd 2.2 do not provide one necessry nd sufficient condition for the existence of the first negtive moment, the bove exmples show tht they re strong enough to decide the cse for mny common distributions. Perhps their min virtue lies in the simplicity of their conditions; one need only evlute the density function to see if the theorem pplies. In the next section, we shll investigte density for which these theorems fil to give n nswer, nd prove necessry nd sufficient condition for existence of the first negtive moment. Unfortuntely, the simplicity in checking the condition is forfeit, for we no longer cn stte it in terms of the density function.
-7-3. A Necessry nd Sufficient Condition. The estblishment of necessry nd sufficient conditions for the existence of moments is, in generl, difficult endevor. Indeed, when deling with positive moments, one is Quickly led into the theory of chrcteristic functions (see, e.g., [3] or [4]). The theory of the existence of negtive moments cnnot be tied in with tht of chrcteristic functions, so, in sense, is less elegnt. However, for the cse of continuous, positive density functions, we cn present reltively simple necessry nd sufficient condition. This condition, unfortuntely, is not nerly s esy to verify s those in Section 2, for it is in terms of n integrl rther thn limit. As fr s we know, one cnnot express necessry nd sufficient condition in terms of limit. We begin with n exmple of density which is not covered by the theorems of Section 2. EXAMPLE 3.1. For ny constnt, 0< < 1, define the density f (x) by f ~x) = [log(l/x) T 1 / J [log(l/t) Jl dt 0< x< 0 = 0 otherwise (3.1) It is esy to check tht, for every, 0< < 1, lim f (x) = 0 x-+0 f(x) lim = co for every > 0 x-+0+ X (3.2) hence, the theorems of Section 2 do not pply to f (x) We cn, however, estblish the nonexistence of the first negtive moment by noting
-8-5 dx = li~ J xlogcl/x) dx = Eli~{ -log[log(l/x)]} ~... 0 -tv = lim log log(l/e) -log[log(l/5)] =eo -to+ (3.3) In generl, condition involving only limits will not yield necessry nd sufficient condition for existence of moments; the condition must itself involve n integrl. Feller [2, Sec. v.6] presents theorem which gives necessry nd sufficient condition for the existence of the first positive moment. We present similr theorem, which gives necessry nd sufficient condition for existence of the first negtive moment. IHEOREM 3.1. Let f(x) be ~ continuous density ~ ( O,(X)) E(X-l) < oo if (X) J 1 - F(x)dx< (X) 0 x2 (3.4) X where F(x) =I f(t)dt is the distribution function ssocited with the density function f. 0 Proof. The proof follows from estblishing the identity J ~ f(x)dx = f..!. F(x)dx, 0 0 ~ (3. 5) in the sense tht one side is finite if nd only if the other side is finite. For ny 5>0, ech integrnd in (3.5) hs finite integrl over [5,(X)), so we cn use integrtion by prts to estblish C10 CO I I 1 -F(5) 1 X 5 _5> - f(x)dx = + - F(x)dx 5 5 x- (3.6) Now suppose the left-hnd side of (3. 5) is finite. This implies tht f(o) = 0,
-9- nd from LHospitls rule, we hve lim F(5) = lim f(5) = 0 0--+0 5 0--+0 (3.7) Thus, letting 5-.0 in (3.6) estblishes (3.5) nd the only if prt of the theorem. Now, suppose tht the right-hnd side of (3.5) is finite. From (3.6), we hve for every 5> 0, = = J ~ f(x)dxs; J _!_ F(x)dx 5 5 x?- (3.8) Letting 5... 0 in (3.8) estblishes E(X- 1 )<=, nd the proof is complete. As mentioned before, the coildi tion given in Theorem 3.1 is not nerly s esy to verify s the conditions in Section 2. In fct, in mny cses, it is just s esy to check directly whether E(X- 1 ) exists, rther thn to check the condition of Theorem 3.1. Fortuntely, most common density functions re well enough behved to be covered by the theorems in Section 2; only the more pthologicl ones, such s (3.1), will not be covered. But the density in (3.1) does serve purpose: it illustrtes tht the ~uestion of existence of negtive moments is delicte one. 4. Generliztions to Higher Order Moments. The theorems presented in Sections 2 nd 3 ~uickly generlize to cover higher order negtive moments, i.e., E(jxj-t3), t3>1. The proofs re similr to those lredy presented, nd will be omitted. Anlogous to the theorems of Section 2, we hve the following two theorems: THEOREM 4.1. Let f(x) be ~ continuous pdf on (, b), -= s; < b s; =. If ~ of the following conditions is true:
. -10- (i) 0 (, b) nd f(o) > 0 (ii) =O nd li~ f(x) > 0 x...o (iii) b=o nd lim x-->0 - f(x) > 0 THEOREM 4.2. Let f(x) be ~ continuous pdf on (O,co) If there exists CX> 0 such tht 1. f(x) ~ lo~ cx+f3-l c:o X-+ X (4.1) The integrtion-by-prts rgument used in Theorem 3.1 remins vlid for higher order moments. Thus, proof similr to tht of Theorem 3.1 cn be used to estblish the following theorem. THEOREM 4.3. Let f(x) be continuous density function on (O,oo). E(X-13) < ~:~:~ if nd only if where F(x) = J f(t)dt X 0 Acknowledgment. c:o J (F(x)/x13 + 1 )dx< c:o 0 Science Foundtion Grnt No. MCSBl-02541. The second uthors reserch is supported by Ntionl (4.2)
,. I. f -ll- References ~ l. Cho, M. T. nd w. E. Strwdermn, Negtive moments of positive rndom vribles, J, Amer. Sttist. Assoc., 67 (l972) 429-431. 2. Feller, w., An Introduction to Probbility Theory nd Its Applictions, Vol. II, 2nd ed., John Wiley nd Sons, New York, 1971. 3. Pitmn, E. J. G., On the derivtives of chrcteristic function t the origin, Ann. Mth. Sttist., 27 (1956) 1156-ll60. 4. Zygmund, A., A remrk n chrcteristic functions, Ann. Mth. Sttist., l8 (l947) 272-276.