Hacettepe Jornal of Mathematics and Statistics Volme 44 3 25, 75 733 Eact moments of generalized order statistics from type II eponentiated log-logistic distribtion Deendra Kmar Abstract In this paper some new simple epressions for single and prodct moments of generalized order statistics from type II eponentiated loglogistic distribtion hae been obtained. The reslts for order statistics and record ales are dedced from the relations deried and some ratio and inerse moments of generalized order statistics are also carried ot. Frther, a characterization reslt of this distribtion by sing the conditional epectation of generalized order statistics is discssed. 2 AMS Classification: 62G3, 62E. Keywords: Eact moments, ratio and inerse moments, generalized order statistics, order statistics, pper record ales, type II eponentiated log-logistic distribtion and characterization. Receied 7//23 : Accepted 29/6/24 Doi :.5672/HJMS.2427472. Introdction A random ariable X is said to hae type II eponentiated log-logistic distribtion if its probability density fnction pdf is gien by f αβ/σβ,, α, σ >, β >. σ[ + /σ β ] α+ and the corresponding srial fnction is { } β α, F + σ, α, σ >, β >..2 It is easy to see that αβ F σ[ + /σ β ]f..3 Log-logistic distribtion is considered as a special case of type II eponentiated loglogistic distribtion when α. It is sed in srial analysis as a parametric model where in the mortality rate first increases then decreases, for eample in cancer diagnosis Department of Statistics, Amity Institte of Applied Sciences Amity Uniersity, Noida-2 33, India Email: deendrastats@gmail.com
76 or any other type of treatment. It has also been sed in hydrology to model stream flow and precipitation, and in economics to model the distribtion of wealth or income. Kamps [24] introdced the concept of generalized order statistics gos as follows: Let X, X 2... be a seqence of independent and identically distribted iid random ariables r with absoltely continos cmlatie distribtion fnction cdf F and pdf, f, α, β. Let n N, n 2, k >, m R, be the parameters sch that γ r k + n rm + >, for all r {, 2,..., n }, where M r n jr mj. Then X, n, m, k,...,xn, n, m, k, r, 2,... n are called gos if their joint pdf is gien by n n k γ j [ F i] m f i [ F n] k f n.4 j i on the cone F 2... n F. The model of gos contains as special cases, order statistics, record ales, seqential order statistics. Choosing the parameters appropriately Cramer, [8], we get the ariant of the gos gien in Table. Table : Variants of the generalized order statistics γ n k γ r m r i Seqential order statistics α n n r + α r γ r γ r+ ii Ordinary order statistics n r + ii Record ales i Progressiely type II censored order statistics R n + n r + + n jr Rj Rr Pfeifer s record ales β n β r β r β r+ For simplicity we shall assme m m 2... m n m. The pdf of the r th gos, Xr, n, m, k, r n, is f Xr,n,m,k Cr r! [ F ] γr fg r m F.5 and the joint pdf of Xr, n, m, k and Xs, n, m, k, r < s n, is C s f Xr,n,m,k,Xs,n,m,k, y r!s r! [ F ] m fgm r F where F F, C r h m [h mf y h mf ] s r [ F y] γs fy, < y,.6 i { m+ m+, m ln, m r γ i, γ i k+n im+,
77 and g m h m h m, [,. Theory of record ales and its distribtional properties hae been etensiely stdied in the literatre, Ahsanllah [4], Balakrishnan et al. [2], Nezoro [33], Glick [2] and Arnold et al. [8, 9]. Resnick [35] discssed the asymptotic theory of records. Seqential order statistics hae been stdies by Arnold and Balakrishnan [7], Kamps [24], Cramer and Kamps [9] and Schenk [37], among others. Aggarawala and Balakrishnan [] established recrrence relations for single and prodct moments of progressie type II right censored order statistics from eponential and trncated eponential distribtions. Balasooriya and saw [4] deelop reliability sampling plans for the two parameter eponential distribtion nder progressie censoring. Balakrishnan et al. [3] obtained bonds for the mean and ariance of progressie type II censored order statistics. Ordinary ia trncated distribtions and censoring schemes and particlarly progressie type II censored order statistics hae been discss by Kamps [24] and Balakrishnan and Aggarwala [], among others. Kamps [24] inestigated recrrence relations for moments of gos based on non-identically distribted random ariables, which contains order statistics and record ales as special cases. Cramer and Kamps [2] deried relations for epectations of fnctions of gos within a class of distribtions inclding a ariety of identities for single and prodct moments of ordinary order statistics and record ales as particlar cases. Varios deelopments on gos and related topics hae been stdied by Kamps and Gather [23], Ahsanllah [5], Pawlas and Szynal [34], Kamps and Cramer [22], Ahmad and Fawzy [2], Ahmad [3], Kmar [27, 28, 29] among others. Characterizations based on gos hae been stdied by some athors, Keseling [25] characterized some continos distribtions based on conditional distribtions of gos. Bieniek and Szynal [5] characterized some distribtions ia linearity of regression of gos. Cramer et al. [7] gae a nifying approach on characterization ia linear regression of ordered random ariables. Khan et al. [26] characterized some continos distribtions throgh conditional epectation of fnctions of gos. The aim of the present stdy is to gie some eplicit epressions and recrrence relations for single and prodct moments of gos from type II eponentiated log-logistic distribtion. In Section 2, we gie the eplicit epressions and recrrence relations for single moments of type II eponentiated log-logistic distribtion and some inerse moments of gos are also worked ot. Then we show that reslts for order statistics and record ales are dedced as special cases. In Section 3, we present the eplicit epressions and recrrence relations for prodct moments of type II eponentiated log-logistic distribtion and we show that reslts for order statistics and record ales are dedced as special cases and ratio moments of gos are also established. Section 4, proides a characterization reslt on type II eponentiated log-logistic distribtion based on conditional moment of gos. Two applications are performed in Section 5. Some conclding remarks are gien in Section 6. 2. Relations for single Moments In this Section, the eplicit epressions, recrrence relations for single moments of gos and inerse moments of gos are considered. First we need the basic reslt to proe the main Theorem. 2.. Lemma. For type II eponentiated log-logistic distribtion as gien in.2 and any non-negatie and finite integers a and b with m
78 where and J ja, ασ j p Proof From 2.2, we hae p j/β p, β > j and j,, 2,..., 2. [αa + + p j/β] α i J ja, b J ja, { αα+...α+i, i>, i. j [ F ] a fg b mf d. 2.2 By making the sbstittion z [ F ] /α in 2.3, we get j [ F ] a fd. 2.3 J ja, ασ j z j/β z αa+ j/β dz. ασ j p j/β p z αa+ j/β+p dz p and hence the reslt gien in 2.. 2.2. Lemma. For type II eponentiated log-logistic distribtion as gien in.2 and any non-negatie and finite integers a and b ασj m + b J ja, b p m + b b b b p+ b α b+ σ j b! p J ja + m +, 2.4 j/β p [α{a + m + + } + p j/β], m 2.5 j/β p, m, 2.6 [αa + + p j/β] b+ where J ja, b is as gien in 2.2. Proof: On epanding gmf b [ F m+ m+ ]b binomially in 2.2, we get when m b J ja, b b j [F ] a+m+ fd m + b m + b b b J ja + m +,. Making se of Lemma 2., we establish the reslt gien in 2.5 and when m that J ja, b as b b. Since 2.5 is of the form at m, therefore, we hae
79 where J ja, b A b b A ασ j [α{a + m + + } + p j/β] m + b, 2.7 p p j/β p. Differentiating nmerator and denominator of 2.7 b times with respect to m, we get b J ja, b Aα b +b b b [α{a + m + + } + p j/β]. b+ On applying the L Hospital rle, we hae b lim m J ja, b Aα b +b b b. 2.8 [αa + + p j/β] b+ Bt for all integers n and for all real nmbers, we hae Riz [36] n i n i n n!. 2.9 i Therefore, i b +b b b b!. 2. Now on sbstitting 2. in 2.8, we hae the reslt gien in 2.6. 2.3. Theorem. For type II eponentiated log-logistic distribtion as gien in.2 and r n, k, 2,... and m E[X j r, n, m, k] ασ j C r r!m + r Cr Jjγr, r 2. r! r p+ r p j/β p, β > j and j,, 2,... 2.2 [αγ r + p j/β] where J jγ r, r is as defined in 2.2. Proof. From.5 and 2.2, we hae E[X j r, n, m, k] Cr Jjγr, r r! Making se of Lemma 2.2, we establish the reslt gien in 2.2. Identity 2.. For γ r, k, r n and m r r r!m + r γ r. 2.3 r t γt Proof. 2.3 can be proed by setting j in 2.2. Special Cases
72 i Ptting m, k in 2.2, the eplicit formla for the single moments of order statistics of the type II eponentiated log-logistic distribtion can be obtained as r E[Xr:n] j C r:n +p r ασ j j/β p [αn r + + + p j/β], p where n! C r:n r!n r!, ii Setting m in 2.2, we dedce the eplicit epression for the single moments of pper k record ales for type II eponentiated log-logistic distribtion in iew of 2. and 2.6 in the form E[X j r, n,, k] E[Z k r and hence for pper records E[Z r j ] αk r σ j j ] E[X j Ur ] αr σ j p p p j/β p [αk + p j/β] r p j/β p [α + p j/β] r. Recrrence relations for single moments of gos from.5 can be obtained in the following theorem. 2.4. Theorem. For the distribtion gien in.2 and 2 r n, n 2 and k, 2,..., Proof. σj αβγ r E[X j r, n, m, k] E[X j r, n, m, k] From.5, we hae E[X j r, n, m, k] + jσβ+ αβγ r E[X j β r, n, m, k]. 2.4 Cr r! j [ F ] γr fg r m F d. 2.5 Integrating by parts treating [ F ] γr f for integration and rest of the integrand for differentiation, we get E[X j r, n, m, k] E[X j r, n, m, k] + jcr γ rr! j [ F ] γr gm r F d the constant of integration anishes since the integral considered in 2.5 is a definite integral. On sing.3, we obtain E[X j r, n, m, k] E[X j r, n, m, k] σjcr αβγ rr! j [ F ] γr fgm r F d + σβ+ jc r j β [ αβγ rr! F ] γr fgm r F d and hence the reslt gien in 2.4. Remark 2.: Setting m, k, in 2.4, we obtain a recrrence relation for single moments of order statistics for type II eponentiated log-logistic distribtion in the form E[Xr:n] j E[Xr :n] j + σj αβn r + jσ β+ αβn r + E[Xj β r :n].
72 Remark 2.2: Ptting m, in Theorem 2.4, we get a recrrence relation for single moments of pper k record ales from type II eponentiated log-logistic distribtion in the form σj E[X k Ur αβk j ] E[X k Ur j ] + jσβ+ αβk E[Xk Ur j β ]. Inerse moments of gos from type II eponentiated log-logistic distribtion can be obtain by the following Theorem. 2.5. Theorem. For type II eponentiated log-logistic distribtion as gien in.2 and r n, k, 2,..., Proof. E[X j β r, n, m, k] From.5, we hae p E[X j β r, n, m, k] σ j β p Γ p!γ j p r β i Now letting t [ F ] /α in 2.7, we get Since E[X j β r, n, m, k] k B j β, β > j. 2.6 + p+ j/β αγ i C r r r r!m + r j β [ F ] γ r fd. 2.7 σ j β C r r r!m + r +p r Γ j β p!γ j p β p m + + n r + + p + j/β αm + b a b a a where Ba, b is the complete beta fnction. Therefore, E[X j β r, n, m, k] σj β C r m + r Γ Γ,. Ba + k, c Bk, c + b 2.8 α{k+n rm+}+p+ j/β αm+ α{k+nm+}+p+ j/β αm+ Γ j p β p p!γ j p β 2.9 and hence the reslt gien in 2.6. Special Cases iii Ptting m, k in 2.9, we get inerse moments of order statistics from type II eponentiated log-logistic distribtion as; E[X j β r:n ] σj β n! n r! p Γ j Γ[αn r + + p + j/β] β. p p!γ j Γ[αn p + + p + j/β] β
722 i Ptting m in 2.6, to get inerse moments of k record ales from type II eponentiated log-logistic distribtion as; p Γ j E[X j β Ur ] β σj β r. p p!γ j p + p+ j/β β αk Recrrence relations for inerse moments of gos from.2 can be obtained in the following theorem. 2.6. Theorem. For type II eponentiated log-logistic distribtion and for 2 r n, n 2 k, 2,..., σj β E[X j β r, n, m, k] E[X j β r, n, m, k] αβγ r j βσβ+ + E[X j 2β r, n, m, k], β > j. 2.2 αβγ r Proof. The proof is easy. Remark 2.3: Setting m, k in 2.2, we obtain a recrrence relation for inerse moments of order statistics for type II eponentiated log-logistic distribtion in the form σj β E[Xr:n j β ] E[X j β j βσβ+ αβn r + r :n] + αβn r + E[Xj 2β r:n ]. Remark 2.4: Ptting m, in Theorem 2.6, we get a recrrence relation for inerse moments of pper k record ales from type II eponentiated log-logistic distribtion in the form σj β E[X k Ur αβk j β ] E[X k Ur j β ] + 3. Relations for prodct moments j βσβ+ E[X k Ur αβk j 2β ]. In this Section, the eplicit epressions and recrrence relations for single moments of gos and ratio moments of gos are considered. First we need the following Lemmas to proe the main reslt. 3.. Lemma. For type II eponentiated log-logistic distribtion as gien in.2 and any non-negatie integers a, b, c with m where J i,ja, b, c J i,ja,, c α 2 σ i+j Proof: From 3.2, we hae where p q p+q j/β p j/β q [αc + + p j/β] [αa + c + 2 + p + q {i + j/β}], 3. i y j [ F ] a f[h mf y h mf ] b [ F y] c fydyd. 3.2 J i,ja,, c G i [ F ] a fgd, 3.3 y j [ F y] c fydy. 3.4
723 By setting z [ F y] /α in 3.4, we find that G ασ j p p j/β p[ F ] c++{p j/β}/α. [αc + + p j/β] On sbstitting the aboe epression of G in 3.3, we get J i,ja,, c ασ j p p j/β p [αc + + p j/β] i [ F ] a+c++{p j/β}/α fd. 3.5 Again by setting t [ F ] /α in 3.5 and simplifying the reslting epression, we derie the relation gien in 3.. 3.2. Lemma. For the distribtion as gien in.2 and any non-negatie integers a, b, c J i,ja, b, c α2 σ i+j m + b m + b b b p q b p+q+ b J i,ja + b m +,, c + m + 3.6 j/β p [α{c + m + + } + p j/β] i/β q, m 3.7 [α{a + c + m + b + 2} + p + q {i + j/β}] p+q α b+2 σ i+j b! j/β p i/β q [αc + + p j/β] b+ [αa + c + 2 + p + q {i + j/β}], m p q where J i,ja, b, c is as gien in 3.2. Proof: When m, we hae [h mf y h mf ] b m + [ F m+ F y m+ ] b b m + b b b [ F y] m+ [ F ] b m+. Now sbstitting for [h mf y h mf ] b in eqation 3.2, we get J i,ja, b, c m + b b b J i,ja + b m +,, c + m +. Making se of the Lemma 3., we derie the relation gien in 3.7. When m, we hae J i,ja, b, c as b b. On applying L Hospital rle, 3.8 can be proed on the lines of 2.6. 3.8 3.3. Theorem. For type II eponentiated log-logistic distribtion as gien in.2 and r < s n, k, 2,... and m
724 E[X i r, n, m, kx j s, n, m, k] r C s r r!s r!m + r J i,jm + m +, s r, γ s 3.9 α 2 σ i+j C s r r!s r!m + s 2 p q s r Proof: From.6, we hae E[X i r, n, m, kx j s, n, m, k] s r p+q++ r j/β p i/β q [αγ s + p j/β][αγ r + p + q {i + j/β}], β > mai, j and i, j,, 2,.... 3. C s r!s r! i y j [ F ] m f g r m F [h mf y h mf ] s r [ F y] γs fydyd. 3. On epanding gm r F binomially in 3., we get E[X i r, n, m, kx j s, n, m, k] r r C s r!s r!m + r J i,jm + m +, s r, γ s. Making se of the Lemma 3.2, we derie the relation in 3.. Identity 3.: For γ r, γ s, k, r < s n and m Proof. s r s r At i j in 3., we hae C s r!s r!m + s 2 r s r!m + s r γ s. 3.2 s tr+ γt s r r s r + γ s γ r. Now on sing 2.3, we get the reslt gien in 3.2. At r, 3.2 redces to 2.3. Special cases: i Ptting m, k in 3., the eplicit formla for the prodct moments of order statistics of the type II eponentiated log-logistic distribtion can be obtained as EX i r:nx j s:n α 2 σ i+j C r,s:n s r r s r p q p+q++ n s j/β p [αn s + + + p j/β] i/β q [αn r + + + p + q {i + j/β}].
725 where, C r,s:n n! r!s r!n s!. iiptting m in 3., we dedce the eplicit epression for the prodct moments of pper k record ales for the type II eponentiated log-logistic distribtion in iew of 3.9 and 3.8 in the form E[X k Ur i X k Us j ] αk s σ i+j p q p+q j/β p [αk + p j/β] s r i/β q [αk + p + q {i + j/β}] r and hence for pper records EXUrX i j Us αs σ i+j p+q j/β p i/β q [α + p j/β] s r [α + p + q {i + j/β}]. r Remark 3. p q At j in 3., we hae E[X i r, n, m, k r ασ i C s r!s r!m + s 2 s r r q s r q++ i/β q γ s [αγ r + q i/β]. 3.2 Making se of 3.2 in 3.3 and simplifying the reslting epression, we get E[X i ασ i C r r r, n, m, k q+ r!m + s r q i/β q [αγ r + q i/β], as obtained in 2.2. Making se of.6, we can derie recrrence relations for prodct moments of gos from.2. 3.4. Theorem. For the gien type II eponentiated log-logistic distribtion and n N, m R, r < s n σj αβγ s E[X i r, n, m, kx j s, n, m, k] E[X i r, n, m, kx j s, n, m, k] + jσβ+ αβγ s E[X i r, n, m, kx j β s, n, m, k]. 3.4 Proof: From.6, we hae where I E[X i r, n, m, kx j s, n, m, k] C s r!s r! i [ F ] m fg r m F Id, 3.5 y j [ F y] γs [h mf y h mf ] s r fydy.
726 Soling the integral in I by parts and sbstitting the reslting epression in 3.5, we get E[X i r, n, m, kx j s, n, m, k] E[X i r, n, m, kx j s, n, m, k] jc s γ sr!s r! i y j [ F ] m fgm r F [h mf y h mf ] s r [ F y] γs dyd the constant of integration anishes since the integral in I is a definite integral. On sing the relation.3, we obtain E[X i r, n, m, kx j s, n, m, k] E[X i r, n, m, kx j s, n, m, k] jσc s αβγ sr!s r! i y j [ F ] m fgm r F [h mf y h mf ] s r [ F y] γs fydyd jσ β+ C s + αβγ sr!s r! i y j β [ F ] m fgm r F [h mf y h mf ] s r [ F y] γs fydyd and hence the reslt gien in 3.4. Remark 3.2 Setting m, k in 3.4, we obtain recrrence relations for prodct moments of order statistics of the type II eponentiated log-logistic distribtion in the form σj E[Xr,s:n] i,j E[X i,j αβn s + r,s :n] + jσ β+ αβn s + E[Xi,j β r,s:n ]. Remark 3.3 Ptting m, k in 3.5, we get the recrrence relations for prodct moments of pper k records of the type II eponentiated log-logistic distribtion in the form σj E[X k Ur αβk i X k Us j ] E[X k Ur i X k Us j ] + jσβ+ αβk E[Xk Ratio moments of gos from type II eponentiated log-logistic distribtion can be obtain by the following Theorem. Ur i X k Us j β ]. 3.5. Theorem. For type II eponentiated log-logistic distribtion as gien in.2 E[X i r, n, m, kx j β s, n, m, k] r a + p+q i+j/β αγ a Proof From.6, we hae p q s br+ j p+q σ i+j β i Γ Γ + β β p!q!γ j Γ p i + p β β, β > j. 3.6 + p+ j/β αγ b C s E[X i r, n, m, kx j β s, n, m, k] r!s r!m + s 2 r s r + r s r i [ F ] s r+ m+ fjd, 3.7
727 where J y j β [ F y] γ s fydy. 3.8 By setting z [ F y] /α in 3.8, we find that p Γ j [ F ] γ s + p+ j/β α J σ j β β [ ]. p p!γ j p γ β s + p+ j/β α On sbstitting the aboe epression of J in 3.7, we get E[X i r, n, m, kx j β s, n, m, k] s r ++p r [ ] γ s + p+ j/β α σ j β C s r!s r!m + s 2 s r Γ j p!γ j β β p r p i [ F ] γ r + p+ j/β α d. 3.9 Again by setting t [ F ] /α in 3.9, we get E[X i r, n, m, kx j β s, n, m, k] σi+j β C s p+q m + s j β Γ p!q! Γ j Γ i p q [ ] α{k+n rm+}+p+q {i+j/β} αm+ [ α{k+nm+}+p+q {i+j/β} αm+ [ ] α{k+n sm+}+p+ j/β Γ + β Γ p i Γ + q β β Γ Γ αm+ [ α{k+n rm+}+p+ j/β αm+ ] ] 3.2 and hence the reslt gien in 3.6. Special cases iii Ptting m, k in 3.2, the eplicit formla for the ratio moments of order statistics of the type II eponentiated log-logistic distribtion can be obtained as E[Xr:nX i s:n j β ] n!σi+j β n s! p q p+q j β Γ Γ p!q! Γ j Γ p β i + β i + q β Γ[αn r + + p + q {i + j/β}]γ[αn s + + p + j/β]. Γ[αn + + p + q {i + j/β}]γ[αn r + + p + j/β] i Ptting m in 3.6, the eplicit epression for the ratio moments of pper k record ales for the type II eponentiated log-logistic distribtion can be obtained as E[X k Ur i X k Us j β ] σ i+j β p+q Γ j i Γ β β + p!q! p q Γ j Γ p i + q β β r s r. + p+q {i+j/β} + p+ j/β αk αk Making se of.6, we can derie recrrence relations for ratio moments of gos from.2.
728 3.6. Theorem. For type II eponentiated log-logistic distribtion σj β E[X i r, n, m, kx j β s, n, m, k] αβγ s E[X i r, n, m, kx j β s, n, m, k] j βσβ+ + E[X i r, n, m, kx j 2β s, n, m, k], αβγ s β > j. 3.2 Proof The proof is easy. Remark 3.4 Setting m, k in 3.2, we obtain a recrrence relation for Ratio moments of order statistics for type II eponentiated log-logistic distribtion in the form σj β E[Xr:nX i s:n j β ] E[Xr:nX i j β j βσβ+ αβn s + s :n] + αβn s + E[Xi r:nxs:n j 2β ]. Remark 3.5 Ptting m, in Theorem 3.6, we get a recrrence relation for ratio moments of pper k record ales from type II eponentiated log-logistic distribtion in the form σj β E[X k Ur αβk i X k Us j β ] E[X k Ur i X k Us j β ] + j βσβ+ E[X k Ur αβk i X k Us j 2β ]. Remark 3.6 At γ r n r + + j ir mi, r j n, mi N, k mn + in 3.6 the prodct moment of progressie type II censored order statistics of type II eponentiated log-logistic distribtion can be obtained. Remark 3.7 The reslt is more general in the sense that by simply adjsting j β in 3.6, we can get interesting reslts. For eample if j β then E [ ] i Xr,n,m,k Xs,n,m,k gies the moments of qotient. For j β >, E[X i r, n, m, k X j β s, n, m, k] represent prodct moments, whereas for j < β, it is moment of the ratio of two generalized order statistics of different powers. 4. Characterization This Section contains characterization of type II eponentiated log-logistic distribtion by sing the conditional epectation of gos. Let Xr, n, m, k, r, 2,..., n be gos, then from a continos poplation with cdf F and pdf f, then the conditional pdf of Xs, n, m, k gien Xr, n, m, k, r < s n, in iew of.5 and.6, is C s f Xs,n,m,k Xr,n,m,k y s r!c r [hmf y hmf ]s r [F y] γs [ F fy. < y 4. ] γ r+ 4.. Theorem. Let X be a non-negatie random ariable haing an absoltely continos distribtion fnction F with F and < F < for all >, then
729 if and only if E[Xs, n, m, k Xr, n, m, k ] σ F Proof From 4., we hae /β p [ + /σ β ] p p s r γ r+j γ r+j p/α j { } β α, +, α, σ >, β >. σ 4.2 C s E[Xs, n, m, k Xr, n, m, k ] s r!c r m + s r [ F y m+ ] s r F y γs fy y dy. 4.3 F F F By setting F y F +/σ β +y/σ β α from.2 in 4.3, we obtain E[Xs, n, m, k Xr, n, m, k ] σc s s r!c r m + s r [{ + /σ β } /α ] /β γs m+ s r d σc s /β s r!c r m + s r p [ + /σ β ] p Again by setting t m+ in 4.4, we get p γs p/α m+ s r d 4.4 E[Xs, n, m, k Xr, n, m, k ] σc s /β s r!c r m + s r p [ + /σ β ] p p t k p/α m+ +n s t s r dt σc s /β s r!c r m + s r p [ + /σ β ] p p Γ k p/α + n s Γs r m+ Γ k p/α + n r m+ σc s /β s r!c r m + s r p [ + /σ β ] p p m + s r Γs r γr+j p/α s r j and hence the relation in 4.2. To proe sfficient part, we hae from 4. and 4.2 C s s r!c r m + s r y[ F m+ F y m+ ] s r
73 where H r σ [ F y] γs fydy [ F ] γ r+ H r, 4.8 s r /β p [ + /σ β ] p p j γ r+j γ r+j p/α Differentiating 4.5 both sides with respect to and rearranging the terms, we get or C s [ F ] m f s r 2!C r m + s r 2. y[ F m+ F y m+ ] s r 2 [ F y] γs fydy H r[ F ] γ r+ γ r+h r[ F ] γ r+ f γ r+h r+[ F ] γ r+2+m f H r[ F ] γ r+ + γ r+h r[ F ] γ r+ f. Therefore, f F H r γ r+[h r+ H αβ/σβ r] σ[ + /σ β ] which proes that { } β α, F +, α, σ >, β >. σ Remark For m, k and m, k, we obtain the characterization reslts of the type II eponentiated log-logistic distribtion based on order statistics and record ales respectiely. 5. Applications In this Section, we sggest some applications based on moments discssed in Section 2. Order statistics, record ales and their moments are widely sed in statistical inference [see for eample Balakrishnan and Sandh [], Sltan and Moshref [38] and Mahmod et al. [3], among seeral others]. i Estimation: The moments of order statistics and record ales gien in Section 2 can be sed to obtain the best linear nbiased estimate of the parameters of the type II eponentiated log-logistic distribtion. Some works of this natre based on gos hae been done by Ahsanllah and habibllah [6], Malinowska et al. [32] and Brkchat et al. [6]. ii Characterization: The type II eponentiated log-logistic distribtion gien in.2 can be characterized by sing recrrence of single moment of gos as follows: Let La, b stand for the space of all integrable fnctions on a, b. A seqence f n La, b is called complete on La, b if for all fnctions g La, b the condition b a gf nd, n N, implies g a.e. on a, b. We start with the following reslt of Lin [3]. Proposition 5. Let n be any fied non-negatie integer, a < b and g an absoltely continos fnction with g a.e. on a, b. Then the seqence of fnctions {g n e g, n n } is complete in La, b iff g is strictly monotone on a, b. Using the aboe Proposition we get a stronger ersion of Theorem 2.4.
73 5.. Theorem. A necessary and sfficient conditions for a random ariable X to be distribted with pdf gien by. is that σj αβγ r E[X j r, n, m, k] E[X j r, n, m, k] + jσβ+ E[X j β r, n, m, k]. 5. αβγ r Proof The necessary part follows immediately from 2.4 on the other hand if the recrrence relation 5. is satisfied then on sing.5, we hae C r r! Cr γ rr 2! + σjcr αβγ rr! j [ F ] γr fgm r F d j [ F ] γr+m fgm r 2 F d j [ F ] γr fgm r F d + jσβ+ C r j β [ αβγ rr! F ] γr fgm r F d. 5.2 Integrating the first integral on the right-hand side of the aboe eqation by parts and simplifying the reslting epression, we get jc r γ rr! j [ F ] γr gm r F { σ } F αβ f σβ+ αβ f d. β It now follows from Proposition 5., we get which proes that f has the form.. 6. Conclding Remarks αβ F σ[ + /σ β ]f, In the stdy presented aboe, we established some new eplicit epressions and recrrence relations between the single and prodct moments of gos from the type II eponentiated log-logistic distribtion. In addition ratio and inerse moments of type II eponentiated log-logistic distribtion are also established. Frther, the conditional epectation of gos is sed to characterize the distribtion. Acknowledgements The athor appreciates the comments and remarks of the referees which improed the original form of the paper. References [] Aggarwala, R. and Balakrishnan, N. Recrrence relations for progressie type II right censored order statistics from eponential trncated eponential distribtions, Ann. Instit. Statist. Math., 48, 757-77, 996. [2] Ahmad, A.A. and Fawzy, M. Recrrence relations for single moments of generalized order statistics from dobly trncated distribtion, J. Statist. Plann. Inference, 77, 24-249, 23. [3] Ahmad, A.A. Relations for single and prodct moments of generalized order statistics from dobly trncated Brr type XII distribtion, J. Egypt. Math. Soc., 5, 7-28, 27. [4] Ahsanllah, M. Record Statistics, Noa Science Pblishers, New York, 995.
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