SOME NEW CHARACTERIZATION RESULTS ON EXPONENTIAL AND RELATED DISTRIBUTIONS. Communicated by Ahmad Reza Soltani. 1. Introduction

Transcription

1 Bulletn of the Iranan Mathematcal Socety Vol. 36 No. 1 (21), pp SOME NEW CHARACTERIZATION RESULTS ON EXPONENTIAL AND RELATED DISTRIBUTIONS M. TAVANGAR AND M. ASADI Communcated by Ahmad Reza Soltan Abstract. It s well-known that most of the characterzaton results on exponental dstrbuton are based on the soluton of Cauchy functonal equaton and ntegrated Cauchy functonal equaton. Here, we consder the functonal equaton F (x) F (xy) + F (xq(y)), x, xq(y) [, θ), y [, 1], where F and Q satsfy certan condtons, to gve some new characterzaton results on exponental, power and Pareto dstrbutons usng the concepts of condtonal random varables and order statstcs. 1. Introducton Because of mportance of the exponental and geometrc dstrbutons, n many branches of statstcs and appled probablty, a large number of research artcles appear n the lterature characterzng these dstrbutons based on dfferent propertes. The monographs of Galambos and Kotz (1978), Azarlov and Volodn (1986), and Rao and Shanbhag (1994) are devoted to characterzatons of probablty dstrbutons, manly on MSC(2): Prmary: 65F5, 11Dxx; Secondary: 11Y5. Keywords: Condtonal random varable, order statstcs, power dstrbuton, Pareto dstrbuton, functonal equaton. Receved: 9 January 29, Accepted: 28 May 29. Correspondng author c 21 Iranan Mathematcal Socety. 257

2 258 Tavangar and Asad exponental and geometrc dstrbutons. Usually, the problem of characterzng a probablty dstrbuton functon leads to solve a functonal equaton. It s mentoned n Rao and Shanbhag (1986) that most of the characterstc propertes of the exponental and geometrc dstrbutons, based on condtonal expectatons and order statstcs, can be obtaned from ntegrated Cauchy functonal equaton under mnmal assumptons. The monograph of Rao and Shanbhag (1994) provdes a comprehensve study of the applcatons of the ntegrated Cauchy functonal equaton on characterzng exponental and geometrc dstrbutons based on dfferent relatons between ordered random varables. Asad et al. (21) appled ntegrated Cauchy functonal equaton to obtan several characterzaton results on exponental, power and Pareto dstrbutons. Let X be a lfetme (non-negatve) random varable wth cumulatve dstrbuton functon (cdf) F, and survval functon S 1 F. The random varable X s sad to have exponental dstrbuton wth mean λ f S(x) e x/λ, x, λ >, power functon dstrbuton wth parameter vector (α, θ) f ( x ) α F (x), x θ, α >, θ >, θ Pareto dstrbuton wth parameter vector (α, β) f ( ) β α S(x), x β, α >. x These dstrbutons are of partcular nterest n statstcal lterature for ther flexblty to model varous data wth dfferent applcatons. Our purpose here s to gve some characterzaton results on the above dstrbutons. The results are applcatons of a functonal equaton whch s recently solved by Aczel et al. (1999). Ther result s stated n the followng theorem. Theorem 1.1. Among the functons F : [, θ) R + ( [, )), θ (, ], and Q : [, 1] R +, the functonal equaton (1.1) F (x) F (xy) + F (xq(y)), x, xq(y) [, θ), y [, 1], has the trval solutons (1.2) F, Q arbtrary ( 1 f θ < ),

3 Some new characterzaton results on exponental 259 and (1.3) F (x) c >, x (, θ), F (), Q(y), y (, 1], Q() > arbtrary ( 1 f θ < ). For all other solutons, there exst constants k >, α > such that (1.4) F (x) kx α, Q(y) (1 y α ) 1/α. If θ > 1 and F (1) 1, then k 1. Conversely, all pars of functons of the form (1.2), (1.3) and (1.4) satsfy (1.1). It s mentoned n Aczel et al. (1999) that the functons F and/or Q may map nto the gven ranges; and onto s not assumed. Nether s any regularty (monotoncty, contnuty) supposed. Also, the same result follows when one assumes that Eq. (1.1) holds only for almost all pars (x, y) [, θ) [, 1] (wth respect to planar Lebesgue measure). The remander of the paper s organzed as follows: In Secton 2, we obtan some characterzaton results on exponental, power and Pareto dstrbutons based on functonal equaton n (1.1). Secton 3 deals wth some characterzaton results based on equalty n dstrbuton of some condtonal random varables such as resdual lfe random varable. In ths secton, characterzatons based upon equalty of expectaton of some condtonal random varables are also gven. Secton 4 s devoted to characterzaton results based on dentty of dstrbutons and equalty of expectaton of some functons of order statstcs. The results of ths secton are extensons of the results obtaned recently by Asad (26), and Tavangar and Asad (27). 2. Characterzatons based on relatons satsfed by cumulatve dstrbuton functon Assume that F s a cdf. In what follows, we wll defne the support of F (or a random varable havng cdf F ) as (α(f ), ω(f )), where α(f ) nf{x : F (x) > }, and ω(f ) sup{x : F (x) < 1}. Note that, n general, the former may be, and the latter may be +. If F satsfes (1.1), then, as a consequence of propertes of a dstrbuton functon (such as the rght contnuty), two trval solutons of equaton (1.1) are excluded and we obtan the followng result.

4 26 Tavangar and Asad Theorem 2.1. Let F be any cdf wth support [, θ), θ >. Suppose that Q : [, θ) R +. The functonal equaton (2.1) F (x) F (xy) + F (xq(y)), x, xq(y) [, θ), y [, 1], holds f and only f F s a (rescaled) power functon dstrbuton wth parameter vector (α, θ), for some constant α >, and Q(y) (1 y α ) 1/α, y 1. Remark 2.2. Equaton (2.1) has a probablstc nterpretaton as follows. Let X be a contnuous non-negatve random varable on [, θ) wth cdf F. If there s a functon Q, whch satsfes n the condtons of Theorem 2.1, such that P [X > xy X x] P [X xq(y) X x], x, [, θ), y [, 1], then Q(y) (1 y α ) 1/α, and X has the power functon dstrbuton. Most of the characterzaton results on exponental dstrbuton are based on the Cauchy functonal equaton whch s known n the statstcal lterature as the lack of memory property. A non-negatve random varable X s sad to have the lack of memory property f, for all x, y >, ts survval functon S satsfes S(x + y) S(x)S(y). It s well-known that the only contnuous dstrbuton wth ths property s the exponental survval functon. In the followng theorem we gve a characterzaton result on exponental dstrbuton whch s based upon the functonal equaton n Theorem 2.1. Theorem 2.3. Let F be any cdf wth support R +, and S 1 F. Assume that Q : R + R +. The functonal equaton (2.2) S(x) S(x + y) + S(x + Q(y)), x, y [, ), holds f and only f F s an exponental dstrbuton wth mean λ, for some λ >, and Q(y) λ log(1 e y/λ ), y >. Proof. The f part of the theorem s straghtforward and hence we prove the only f part. To ths end, we defne the cdf G as G(z) S( log z), z [, 1), where S s the survval functon defned n the statement of the theorem. Let u e x, v e y, and Q (v) exp{ Q( log v)}. It s obvous that Q : [, 1] R +. Now, Eq. (2.2) mples that G(u) G(uv) + G(uQ (v)), u, uq (v) [, 1), v [, 1]. That s, the par of functons (G, Q ) satsfes Eq. (2.1) wth θ 1.

5 Some new characterzaton results on exponental 261 Therefore, usng Theorem 2.1, we have G(x) x α, x [, 1), and Q (y) (1 y α ) 1/α, y [, 1], for some constant α >. Ths means that F s an exponental cdf wth mean λ 1/α, and Q s as stated n the theorem. Hence, the proof s complete. Remark 2.4. The probablstc nterpretaton of Eq. (2.2) s as follows. Let X be contnuous non-negatve random varable on (, ) wth survval functon S. If there exsts a functon Q whch satsfes n condtons of Theorem 2.3, and P [X x + y X x] P [X < x + Q(y) X x], x, y >, then Q(y) λ log(1 e y/λ ), for some λ >, and X has an exponental dstrbuton wth mean 1/λ. It s also worth notng that Q(y) here s equal to Q(y) λ y r(x)dx, where r(x) f(x)/f (x) s the reversed hazard rate, and f s the densty of F. The next theorem gves a characterzaton of the Pareto dstrbuton whch follows from Theorem 2.1. The proof, beng the same as the proof of Theorem 2.3, s omtted. Theorem 2.5. Let F be any cdf wth support [β, ), and S 1 F. Suppose that Q : [1, ) R +. Then, the functonal equaton S(x) S(xy) + S(xQ(y)), x, xq(y) [β, ), y [1, ), holds f and only f F s a Pareto dstrbuton wth parameter vector (α, β), for some constant α >, and Q(y) (1 y α ) 1/α, y Characterzatons based on condtonal random varables Gven any cdf F, the quantle (or generalzed nverse) functon F s defned by F (u) nf{x : F (x) u}, u (, 1). It s known that F, n general, does not preserve the relaton < ;.e., t s not true that x < y F (x) < F (y). For any cdf F, the followng lemma, whch can be proved easly, provdes some results on F, whch we wll use n the sequel.

6 262 Tavangar and Asad Lemma 3.1. For any cdf F, () t < F (u) F (t) < u, () u F (t) F (u) t. Let X be an arbtrary random varable wth cdf F. Then, F (X) as well as S(X) 1 F (X) have unform U(, 1) dstrbutons. Hence, usng the probablty ntegral transform, we conclude that F [S(X)] s dstrbuted as F, where F s the quantle functon. That s X d F [S(X)], where d stands for equalty n dstrbuton. A natural queston that arses s whether there exsts a strctly decreasng functon Q, for whch the relaton X d Q(X) holds. In ths secton, we obtan some solutons to ths queston for some specal condtonal random varables (such as the resdual lfe random varable). Frst, we show that when the random varable of nterest s the resdual lfe random varable, the functon Q s unque and the underlyng dstrbuton s exponental. Ths s gven by the followng theorem. Theorem 3.2. Let X be a non-negatve random varable wth the survval functon S. Suppose that Q : R + R + s a strctly decreasng functon. Let also X t [X t X > t] be the resdual lfe random varable. Then X t d Q(Xt ), for almost all t R + (wth respect to Lebesgue measure) wth S(t) >, f and only f S s the survval functon of an exponental random varable wth mean λ, for some constant λ >, and Q(y) λ log(1 e y/λ ), y >. Proof. Frst, note that under the assumpton S(t) >, the condtonal random varable X t s well-defned. Also, note that snce every monotone functon s measurable, Q(X t ) s a random varable. We have, P [Q(X t ) x] 1 P [X t > Q 1 (x)] 1 P [X > t + Q 1 (x)]. P [X > t] Let U be a random varable wth unform U(, 1) dstrbuton. From the probablty ntegral transform, we have X d F (U). Now, t follows

7 Some new characterzaton results on exponental 263 from Lemma 3.1 that and P [Q(X t ) x] 1 P [U > F (t + Q 1 (x))] P [U > F (t)] P [X t x] We need to prove that (3.1) 1 S(t + Q 1 (x)), S(t) P [X t + x] P [X > t] 1 {S(t + x) + F (t + x) F ((t + x) )}. S(t) P [X t x] S(t + x). S(t) Let D {x R + F has jump at x} denote the set of dscontnuty ponts of F whch s known to be countable. If D s an empty set, then the result s trval. Hence, let D {d 1, d 2,...}, and defne the set E s, 1, 2,..., as E {(t, x) R + R + t+x d } {(d x, x) x [, d ]}. It s easy to observe that the E are measurable sets of planar Lebesgue measure zero whch, n turn, mples that D s a set of planar Lebesgue measure zero. Therefore, Eq. (3.1), and consequently the followng equaton hold for almost all pars (t, x) R + R + wth respect to planar Lebesgue measure: S(t) S(t + x) + S(t + Q 1 (x)). Now the result follows from Theorem 2.3. Remark 3.3. It can be easly shown that Theorem 3.2 holds f we replace X t d Q(Xt ) wth X t d [Q(X t) X > t], where Q meets the requrements of the theorem. A smlar result characterzng a (rescaled) power functon dstrbuton s as follows. Theorem 3.4. Let X be a non-negatve random varable wth support [, θ) havng the cdf F. Assume that Q : [, 1] R + s a strctly decreasng functon. Let also X (t) [t 1 X X t]. Then, X (t) d Q(X(t) ), for almost all t [, θ] (wth respect to Lebesgue measure) wth F (t) >, f

8 264 Tavangar and Asad and only f F s a (rescaled) power functon dstrbuton wth parameter vector (α, θ), for some α >, and Q(y) (1 y α ) 1/α, y 1. Proof. Along the lnes of the proof of Theorem 3.2, we get P [X (t) x] 1 F (tx) F (t), for almost all pars (t, x) [, θ] [, 1] wth respect to planar Lebesgue measure, and P [Q(X (t) ) x] F (tq 1 (x)), F (t) for all (t, x) [, θ] [, 1]. Now, the result follows from Theorem 2.1. We can now state the next result characterzng the Pareto dstrbuton. The proof s smlar to the ones gven for the above theorems and hence s omtted. Theorem 3.5. Let X be a non-negatve random varable wth support [β, ), and denote by S ts survval functon. Assume that Q : [, 1] R + s a strctly decreasng functon. Let also X [t] [tx 1 X > t]. Then, X [t] d Q(X[t] ), for almost all t [β, ) (wth respect to Lebesgue measure) wth S(t) >, f and only f S s the survval functon of a Pareto dstrbuton wth parameter vector (α, β), for some α >, and Q(y) (1 y α ) 1/α, y 1. In the followng theorem, we prove some results characterzng exponental, power and Pareto dstrbutons based on some condtonal expectatons. Theorem 3.6. Let X be a non-negatve random varable havng a contnuous cdf F, and survval functon S. () Assume that the support of F s (, θ), and lm t F (t)/t α exsts for constant α >. Then, (3.2) E{X X t} E{(t α X α ) 1/α X t}, t θ, f and only f F s a (rescaled) power functon dstrbuton wth parameter vector (α, θ).

9 Some new characterzaton results on exponental 265 () Assume that the support of F s (, ), and lm t e t/λ S(t) exsts for constant λ >. Then, (3.3) E{X t X > t} E{ λ log(1 e (X t)/λ ) X > t}, t, f and only f F s an exponental dstrbuton wth mean λ. () Assume that the support of F s (β, ), and lm t t α S(t) exsts for constant α >. Then, (3.4) E{X 1 X > t} E{(t α X α ) 1/α X > t}, t β, f and only f F s a Pareto dstrbuton wth parameter vector (α, β). Proof. () The f part s easy to verfy. To prove the only f part, note that Eq. (3.2) s equvalent to: (3.5) t {F (t) F (x) F ((t α x α ) 1/α )}dx, t θ. Snce F s contnuous, gven any t [, θ], there exsts a pont u t (, t) such that (3.6) F (t) t α uα t F (u t ) t α u α + tα u α t F ((t α u α t ) 1/α ) t t α t α u α t Let µ t be a probablty measure whch s concentrated on two ponts only such that t puts mass u α t /t α at pont t u t, and mass 1 u α t /t α at pont t (t α u α t ) 1/α. Let H(t) F (t)/t α, t [, θ]. Then, Eq. (3.6) can be wrtten as: H(t) t H(t u)µ t (du). It follows from Theorem 1 of Fosam and Shanbhag (1997) that H(t) s a postve constant ndependent of t. Hence, the proof s complete. () The proof of the f part s straghtforward and hence we prove the only f part. To ths end, note that Eq. (3.3) s equvalent to {S(t) S(t + x) S(t λ log(1 e x/λ ))}dx, t, whch, after makng approprate transformatons, can be wrtten as: s {S( log s) S( log u) S( log(s α u α ) 1/α } 1 du, s 1, u.

10 266 Tavangar and Asad wth α 1/λ. Defnng G(s) S( log s), s 1, we get an ntegral equaton smlar to Equaton (3.5) wth F replaced by G. The result then follows usng the same arguments as the ones used to prove part () of the theorem. () The f part s easy to verfy and hence we prove the only f part. One can show that Eq. (3.4) s equvalent to 1 {S(t) S(t/x) S([t(1 x α )] 1/α )}dx, t β. Let G(z) S(1/z), < z < 1/β. Upon makng approprate transformatons, t s easly seen that Eq. (3.5) holds wth F, and θ replaced by G, and 1/β, respectvely. Ths mples that G s the cdf of a (rescaled) power functon dstrbuton wth parameter vector (α, 1/β). In vew of the relaton between S and G, the proof s then complete. 4. Characterzatons based on order statstcs Let X 1, X 2,..., X n be ndependent random varables wth a common cdf F. The order statstcs relatve to X are denoted by X 1:n, X 2:n,..., X n:n. Order statstcs have many applcatons n dfferent branches of appled probablty and statstcs such as relablty, lfe-testng, goodness of ft tests, etc. (see, for example, Davd and Nagaraja, 23). In the statstcs lterature, a large number of research work s devoted to characterzatons of probablty dstrbutons based on order statstcs. (Among others, we refer the reader to Rao and Shanbhag, 1994, Azlarov and Volodn, 1986, and Asad, et al., 21). In ths secton, we gve some characterzatons of the exponental, power, and Pareto dstrbutons based on order statstcs. The specalzed versons of the results of ths secton have already appeared n Asad (26), and Tavangar and Asad (27). Before gvng the man results of ths secton, we frst prove the followng lemma. Lemma 4.1. Let X 1:n, X 2:n,..., X n:n be the order statstcs from any cdf F. Then, () The survval functon of [X r:n X 1:n > t], 1 r n, s gven by r 1 Ḡ r,n (x t) ( n ) {1 θ t (x)} {θ t (x)} n, x > t, where θ t (x) S(x)/S(t), and S(x) 1 F (x), and

11 Some new characterzaton results on exponental 267 () The cdf of [X r:n X n:n t] s gven by n ( ) n H r,n (x t) {ϕ t (x)} {1 ϕ t (x)} n, x t, r where ϕ t (x) F (x)/f (t). Proof. () The proof follows from the fact that [X r:n X 1:n > t], r 1, 2,..., n, can be consdered as the order statstcs from condtonal random varable [X X > t] wth survval functon S(x)/S(t), x > t. () The proof follows by notng that [X r:n X n:n t], r 1, 2,..., n, are the order statstcs from condtonal random varable [X X t] wth cdf F (x)/f (t), x t. Now, we can prove the followng theorem. Theorem 4.2. Let X 1:n, X 2:n,..., X n:n denote the order statstcs from any cdf F wth support R +. Let S 1 F and assume that Q : R + R + s a strctly decreasng functon. Then, (4.1) [X r:n t X 1:n > t] d [Q(X n r+1:n t) X 1:n > t], for some 1 r n, and for almost all t R + (wth respect to Lebesgue measure) wth S(t) >, f and only f Q(y) λ log(1 e y/λ ), y >, and F s an exponental dstrbuton wth mean λ, for some λ >. Proof. Frst, we prove the only f part of the theorem. Usng Lemma 4.1, we get r 1 ( ) n P [X r:n t x X 1:n > t] {1 θ t (t + x)} {θ t (t + x)} n +{G r,n (t + x t) G r,n ((t + x) t)}, where G r,n (x t) s the cdf of the condtonal random varable [X r:n X 1:n > t], and θ t (t + x) S(t + x)/s(t). In vew of what we have observed n the proof of Theorem 3.2, we have, n ( ) n P [X r:n t x X 1:n > t] {θ t (t + x)} {1 θ t (t + x)} n n r+1 θt(t+x) 1 B(r, n r + 1) zn r (1 z) r 1 dz,

12 268 Tavangar and Asad for all (t, x) R + R +, except on a set of planar Labesgue measure, where the last equalty s from the relaton between bnomal sums and the ncomplete beta functon n whch B(.,.) denotes the complete beta functon (see, for example, Davd and Nagaraja, 23). On the other hand, we can use Lemma 4.1 agan to obtan: P [Q(X n r+1:n t) x X 1:n > t] 1 P [X n r+1:n > t + Q 1 (x) X 1:n > t] n r ( ) n {1 1 θt (t + Q 1 (x)) } { θt (t + Q 1 (x)) } n n ( ) n {1 θt (t + Q 1 (x)) } { θt (t + Q 1 (x)) } n n r+1 1 θt(t+q 1 (x)) Now, from these results and Eq. (4.1), we obtan: 1 B(r, n r + 1) zn r (1 z) r 1 dz. θ t (t + x) 1 θ t (t + Q 1 (x)), for almost all (t, x) R + R +, whch leads to Eq. (2.2). Now, the result follows from Theorem 2.3. The f part of the theorem s easy to verfy and hence s omtted. Hence, the proof s complete. The followng theorem proves a characterzaton of the power functon dstrbuton. Theorem 4.3. Let X 1:n, X 2:n,..., X n:n be the order statstcs from any cdf F wth support [, θ]. Assume that Q : [, 1] R + s a strctly decreasng functon. Then, [X r:n X n:n t] d [t Q(t 1 X n r+1:n ) X n:n t], for some 1 r n, and for almost all t [, θ] (wth respect to Lebesgue measure) wth F (t) >, f and only f Q(y) (1 y α ) 1/α, y 1, and F s a (rescaled) power functon dstrbuton wth parameter vector (α, θ), for some constant α >.

13 Some new characterzaton results on exponental 269 Proof. To prove the only f part of the theorem, note that one can apply Lemma 4.1 to obtan n ( ) n P [X r:n x X n:n t] 1 {ϕ t (x)} {1 ϕ t (x)} n r +{H r,n (x t) H r,n (x t)}, where ϕ t (x) F (x)/f (t), and H r,n (x t) denotes the cdf of [X r:n X n:n t]. In vew of the arguments already made, one can conclude that for almost all (t, x) [, θ] [, θ] (wth respect to planar Lebesgue measure), n ( ) n P [X r:n x X n:n t]1 {ϕ t (x)} {1 ϕ t (x)} n r r 1 ( n n n r+1 1 ϕt(x) Also, we have, P [ t Q(t 1 X n r+1:n ) x X n:n t ] n n r+1 ϕt(tq 1 (x/t)) ) {ϕ t (x)} {1 ϕ t (x)} n, ( ) n {1 ϕ t (x)} {ϕ t (x)} n, 1 B(r, n r + 1) zn r (1 z) r 1 dz. ( ) n {ϕ t (tq 1 (x/t))} {1 ϕ t (tq 1 (x/t))} n 1 B(r, n r + 1) zn r (1 z) r 1 dz. Now, the result follows from Theorem 2.1. The f part of the theorem s trval and hence ts proof s omtted. The proof s now complete. Remark 4.4. The specal case of Theorem 4.3, when Q(y) 1 y and the underlyng dstrbuton s contnuous, s nvestgated by Asad (26), and Tavangar and Asad (27). Theorem 4.5. Let X 1:n, X 2:n,..., X n:n denote the order statstcs based on any cdf F wth support [β, ). Let also S 1 F. Assume that

14 27 Tavangar and Asad Q : [, 1] R + s a strctly decreasng functon. Then, [ ] [ ( ) ] t d t X 1:n > t Q X 1:n > t, X r:n X n r+1:n for almost all t [β, ), (wth respect to Lebesgue measure) wth S(t) >, f and only f Q(y) (1 y α ) 1/α, y 1, and F s a Pareto dstrbuton wth parameter vector (α, β), for some constant α >. Proof. Usng Lemma 4.1, we can wrte [ ] t r 1 ( ) n P > x X 1:n > t 1 {θ t (t/x)} n {1 θ t (t/x)} X r:n and P [ ( Q t X n r+1:n n r ) ( n {G r,n (t/x t) G r,n ((t/x) t)}, n ( n 1 1 n r+1 θt(t/x) ] > x X 1:n > t 1 θt(t/q 1 (x)) 1 ) {θ t (t/x)} {1 θ t (t/x)} n 1 B(r, n r + 1) zn r (1 z) r 1 dz. ) {θ t (t/q 1 (x))} n {1 θ t (t/q 1 (x))} 1 B(r, n r + 1) zn r (1 z) r 1 dz. The result then follows from Theorem 2.5. (we omt the detals). The followng theorem gves a characterzaton of the rescaled beta dstrbuton. The proof s smlar to the proof of Theorem 4.3 and hence s omtted. Theorem 4.6. Let X 1:n, X 2:n,..., X n:n denote the order statstcs from any cdf F wth support [, θ). Let also S 1 F. Assume that Q : [, 1] R + s a strctly decreasng functon. Then, [X r:n X 1:n > t] d [1 (1 t) Q((1 t) 1 (1 X n r+1:n )) X 1:n > t],

15 Some new characterzaton results on exponental 271 for some 1 r n, and for almost all t [, θ] (wth respect to Lebesgue measure) wth S(t) >, f and only f Q(y) (1 y β ) 1/β, y 1, and F s a rescaled beta dstrbuton of the form F (x) 1 (1 x/θ) β, x < θ, for some constant β >. By mposng some restrctons on the underlyng cdf, one can obtan the followng result whch s stronger than that gven n ths secton. The proof s omtted, snce t follows easly from proofs of Theorems 4.2, 4.3, and 4.5, and the argument made for the proof of Theorem 3.6. Theorem 4.7. Let X 1:n, X 2:n,..., X n:n denote the order statstcs based on a contnuous cdf F and survval functon S. () Assume that the support of F s (, ), and lm e t/λ S(t) exsts as t, for some constant λ >. Then, E{X r:n t X 1:n > t} E{ λ log(1 e (X n r+1:n t) ) X 1:n > t}, t >, for some 1 r n, f and only f F s an exponental dstrbuton wth mean λ. () Assume that the support of F s (, θ), and lm F (t)/t α exsts as t, for some constant α >. Then, E{X r:n X n:n t} E{(t α X α n r+1:n) 1/α X n:n t}, t θ, for some 1 r n, f and only f F s a power functon dstrbuton wth parameter vector (α, θ). () Assume that the support of F s (β, ), and lm t α S(t) exsts as t, for some α >. Then, E{X 1 r:n X 1:n > t} E{(t α X α n r+1:n )1/α X 1:n > t}, t β, for some 1 r n, f and only f F s a Pareto dstrbuton wth parameter vector (α, β). Acknowledgments The authors would lke to thank an anonymous referee and the Chef Edtor, Prof. S. Azam, for ther valuable comments. We are grateful to the offce of graduate studes of the Unversty of Isfahan for ther support. The partal support of the Ordered and Spatal Data Center of Excellence of Ferdows Unversty of Mashhad s also acknowledged.

16 272 Tavangar and Asad References [1] J. Aczel, R. Ger and A. Jara, Soluton of a functonal equaton arsng from utlty that s both separable and addtve, Proceedng of the Amercan Mathematcal Socety 127 (1999) [2] M. Asad, On the mean past lfetme of the components of a parallel system, Journal of Statstcal Plannng and Inference 136 (26) [3] M. Asad, C.R. Rao and D.N. Shanbhag, Some unfed characterzaton results on the generalzed Pareto dstrbuton, Journal of Statstcal Plannng and Inference 93 (21) [4] T.A. Azarlov and N.A. Volodn, Characterzaton Problems Assocated wth the Exponental Dstrbuton, Sprnger-Verlag, [5] H.A. Davd and H.N. Nagaraja, Order Statstcs, John Wley & Sons, New Jersey, 23. [6] E.B. Fosam and D.N. Shanbhag, Varants of the Choquet-Deny theorem wth applcatons, Journal of Appled Probablty 34 (1997) [7] J. Galambos and S. Kotz, Characterzaton of Probablty Dstrbutons, Lecture Notes n Mathematcs, Vol. 675, Sprnger, Berln, [8] C.R. Rao and D.N. Shanbhag, Recent results on characterzaton of probablty dstrbutons: a unfed approach through extensons of Deny s theorem, Advances n Appled Probablty 18 (1986) [9] C.R. Rao and D.N. Shanbhag, Choquet-Deny type functonal equatons wth applcatons to stochastc models, John Wley & Sons, Ltd, Chchester, [1] M. Tavangar and M. Asad, Generalzed Pareto dstrbutons characterzed by generalzed order statstcs, Communcatons n Statstcs: Theory and Methods 36(7) (27) M. Tavangar Department of Statstcs, Unversty of Isfahan, 81744, Isfahan, Iran Emal: mahd.tavangar@gmal.com M. Asad Department of Statstcs, Unversty of Isfahan, 81744, Isfahan, Iran Emal: m.asad@stat.u.ac.r