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(1)國立高雄大學統計學研究所 碩士論文. Some Distributional Properties of Concomitants of k-Records k-紀錄值的伴隨分佈性質之研究. 研究生:李暄毅 撰 指導教授:蘇志成 博士. 中華民國 101 年 1 月.

(2) 謝辭 寒冷的冬夜,心暖暖的,終於等到這一刻!猶記兩年前,我帶著一顆既期待又惶恐的心 進入了研究所。期待著未知的研究生生活,惶恐著自己的基礎不佳。過程中,跌跌撞撞,終 於順利畢業了。 首先,最要感謝的是我的指導教授 蘇志成老師。謝謝您總是不辭辛勞的兩地奔波;謝謝 您總是耐心細心的引領教導;謝謝您總是和藹可親的接受我的粗心與無知;謝謝您這兩年 來無私的付出。從您身上,我看到了一個研究者該有的嚴謹與熱忱,也看到了一個老師對 學生的包容與關愛,若沒有您一次又一次的叮嚀與囑咐,若沒有您一次又一次的檢查與校 正,這篇論文不會如期產生。很慶幸自己在這段學習的過程中,能遇到您這麼好的老師,老 師,辛苦您了! 再來,感謝 黃文璋老師以及 蘇南誠老師於春節前夕特地撥冗前來擔任學生的口試委 員,老師們給的寶貴建議與想法,使得我對自己的研究內容有更多面向的思考,亦使這篇 論文得以更臻完善。同時,感謝 俞淑惠導師的關心與協助,讓我在課業方面能逐漸進入狀 況;感謝 黃錦輝老師告訴我「積極」與「態度」的重要;感謝 黃士峰老師帶給我們歡樂,使忙 碌的研究生活增添一分色彩;感謝 陳俞成老師不辭老遠的來為我們上課;感謝 蘭屏姐在我 剛入學那段茫然不知所措的日子裡,不斷給我鼓勵與打氣,您說「很多事情只要有心,都能 夠如期解決」,我謹記在心。 感謝慧婷一直以來的支持與陪伴,每當作業、報告兩頭燒,抑或是研究沒進度等壓力 導致心情不佳,妳總會靜靜地聽我碎念,給我鼓勵,讓我有力量繼續向前,兩年來,謝謝有 妳!另外,我親愛的同窗們,大鄭、小新、柏儒、宥文、奕童,謝謝你們常給予我課業上的協 助,有你們,枯燥的生活都變有趣了!真懷念我們曾經一同努力過的統研盃。感謝基祥學長 在論文排版上的協助,亦感謝所上所有的老師、學長姐、學弟妹們,這兩年我過得很充實也 很愉快。 最後,要感謝我摯愛的家人,謝謝老爸、老媽讓我無後顧之憂,順利完成碩士學業;謝謝 親愛的阿婆,電話中的關心,是我向前邁進的最大動力;謝謝思瑩與阿肥弟不時的來電瞎聊 解悶,家,永遠是最溫暖的。 猶記兩年前期許自己要過一個充實的研究生生活,今日似乎也沒讓自己失望。那麼,繼 續期許自己,在未來更多的兩年裡,依然要過得充實、豐富! 李暄毅 謹上 中華民國101年1月.

(3) Some Distributional Properties of Concomitants of k-Records. by Shiuan-Yi Lee Advisor Jyh-Cherng Su. Institute of Statistics National University of Kaohsiung Kaohsiung, Taiwan 811 R.O.C. January 2012.

(4) Contents. Z`Š zZ`Š. i ii. 1 Introduction. 1. 2 Multivariate pseudo-Weibull distribution. 2. 3 Concomitants of k-record values from multivariate pseudo-Weibull distributions. 5. 4 Two special cases 4.1 Bivariate pseudo-Weibull distributions . . . . . . . . . . . . . . . . . 4.2 Trivariate pseudo-Weibull distributions . . . . . . . . . . . . . . . . .. 9 10 13. 5 Concomitants of k-record values from some other distributions 5.1 Bivariate pseudo-exponential distribution with φ1 (x) = β1 x + c . . 5.2 Gumbel’s bivariate exponential distribution . . . . . . . . . . . . . .. 16 16 18. References. 21.

(5) k-紀 紀錄 值 的 伴 隨 分 佈 性 質 之 研 究 指導教授: 蘇志成 博士 陸軍軍官學校管理科學系 學生: 李暄毅 國立高雄大學統計學研究所. 摘要 在本論文中,我們首先定義一多元擬韋伯分佈為一n個韋伯隨機變數的複合分佈,並 探討其動差性質。其次,考慮由多元擬韋伯分佈所生成k-紀錄值的伴隨,探討其相關的分 佈、乘積動差、條件動差及熵。最後,針對兩個其它不同的分佈,推導其所生成k-紀錄值伴隨 之動差。 關鍵字: 伴隨、k-紀錄值、多元擬韋伯分佈。. i.

(6) Some Distributional Properties of Concomitants of k-Records Advisor: Dr. Jyh-Cherng Su Department of Management Science R.O.C. Military Academy. Student: Shiuan-Yi Lee Institute of Statistics National University of Kaohsiung. Abstract In this paper, first we define a new multivariate pseudo-Weibull distribution as a compound distribution of n Weibull random variables. The distributions of concomitants of k-records from multivariate pseudo-Weibull distribution will be derived. Some single and product moments, conditional moments and entropies have also been presented. Finally, considering two other distributions, some moments of concomitants of k-records will also be investigated. Keywords: concomitants, k-records, multivariate pseudo-Weibull distribution.. ii.

(7) 1. Introduction. Let (X1 , Y1 ), (X2 , Y2 ),... be a sequence of independent and identically distributed random vectors having some continuous bivariate distribution F (x, y). Also for n ≥ 1, let X1:n ≤ X2:n ≤ · · · ≤ Xn:n be the order statistics of X1 , X2 , ..., Xn . For some fixed integer k ≥ 1, define Uk (1) = k, Uk (n + 1) = min{j|j > Uk (n), Xj > XUk (n)−k+1:Uk (n) } and Rn = XUk (n)−k+1:Uk (n) , n ≥ 1. Then {Rn , n ≥ 1} are called the k-record values for {Xn , n ≥ 1}. For k = 1, {Rn , n ≥ 1} are called record values in general. Also the Y’s associated with the k-record values {Rn , n ≥ 1}, denoted as {Y(n) , n ≥ 1}, are called concomitants of k-record values {Rn , n ≥ 1}. Distributional properties and characterizations of record values have been widely studied and some excellent reviews can be found in books such as Ahsanullah (1995), Arnold et al. (1998) and Nevzorov (2001). However comparing with the concomitants of order statistics, the concomitants of record values have not been extensively studied. Some related results of concomitants of record values can be seen in Ahsanullah (2009), Ahsanullah et al. (2010), Mohsin et al. (2010) and Shahbaz et al. (2010). More precisely, Ahsanullah (2009) derived distributions and moments of concomitants of record values from Morgenstern family copula. Mohsin et al. (2010) and Ahsanullah et al. (2010), respectively, defined the bivariate pseudo-exponential distribution and bivariate pseudo-Weibull distribution, and investigated some distributional properties of concomitants of record values. Moreover, Shahbaz et al. (2010) considered a random sample from a trivariate distribution and studied the distributions and moments of bivariate concomitants of record values from trivariate pseudo-exponential distribution. In this paper, inspired by Shahbaz et al. (2010), we consider a more general multivariate distribution and investigate some distributional properties of concomitants of k-record values, where k ≥ 1. In Section 2, the multivariate pseudo-Weibull distribution is defined and some of its distributional properties are given. The distributional properties of concomitants of k-record values from multivariate pseudo-Weibull distribution will be studied in Section 3. In Section 4, we present some distributional properties of concomitants of record values from bivariate pseudo-Weibull distribution and trivariate pseudo-Weibull distribution, which are special cases of multivariate pseudo-Weibull distribution. Some results of the existing literature will be the corollary of our properties. Finally, in Section 5, we will investigate some moments of concomitants of k-record values from bivariate pseudo-exponential distribution and Gumbel’s bivariate exponential distribution which are not different from multivariate pseudo-Weibull distribution considered in Section 3.. 1.

(8) 2. Multivariate pseudo-Weibull distribution. Let (X, Y1 , ..., Yn ) be a random vector having the continuous multivariate probability density function (pdf) f (x, y1 , ..., yn ). Following the line of Shahbaz and Ahmad (2009) and Ahsanullah et al. (2010), the multivariate pseudo-Weibull distribution can be defined. More precisely, first let the random variable X have a Weibull distribution with parameters β0 > 0 and r0 > 0, i.e. the pdf of X is f (x; β0 , r0 ) = β0 r0 xr0 −1 exp(−β0 xr0 ),. x > 0.. (1). Next assume that for 1 ≤ i ≤ n, given that (X, Y1 , ..., Yi−1 ) = (x, y1 , ..., yi−1 ), the random variable Yi has a Weibull distribution with parameters φi (x, y1 , ..., yi−1 ) > 0 and ri > 0, i.e. f (yi ; φi (x, y1 , ..., yi−1 ), ri |x, y1 , ..., yi−1 ) = φi (x, y1 , ..., yi−1 )ri yiri −1 exp[−φi (x, y1 , ..., yi−1 )yiri ],. yi > 0.. (2). Here we let (x, y1 , ..., yi−1 ) = x for i = 1. The multivariate pseudo-Weibull distribution is defined as a compound distribution of (1) and (2), and has the pdf of form f (x, y1 , ..., yn ) = β0 r0 r1 r2 · · · rn φ1 (x)φ2 (x, y1 ) · · · φn (x, y1 , ..., yn−1 )xr0 −1 y1r1 −1 y2r2 −1 · · · ynrn −1 · exp(−β0 xr0 − φ1 (x)y1r1 − φ2 (x, y1 )y2r2 − · · · − φn (x, y1 , ..., yn−1 )ynrn ), where x > 0, yi > 0, 1 ≤ i ≤ n. If φ1 (x) = xr0 , φj+1 (x, y1 , ..., yj ) = xr0 1 ≤ j ≤ n − 1, then. (3) Qj. i=1. yiri ,. f (x, y1 , ..., yn )=β0 r0 r1 · · · rn x(n+1)r0 −1 y1nr1 −1 · · · ynrn −1 · exp[−xr0 (β0 + y1r1 + y1r1 y2r2 + · · · + y1r1 y2r2 · · · ynrn )],. (4). where x > 0, yi > 0, 1 ≤ i ≤ n. The following lemma can be found in the formula (3.241.4) of Gradshteyn and Ryzhik (2007), and will be used in the proofs of properties in Sections 2 and 3. Note that the beta function is denoted as B(·, ·). Lemma 1 For p > 0, q > 0 and 0 < Z 0. ∞. xµ−1 dx = (p + qxv )n+1. µ v. < n + 1,.   µv p 1 µ µ B( , n + 1 − ). n+1 q vp v v. Using Lemma 1, we derive the product moments of X, Y1 , ..., Yn . Here the gamma function is denoted as Γ(·). 2.

(9) Property 1 For E(X. p0. Y1p1. ···. p0 r0. + n + 1 > 0,. Ynpn ). = β0. p1 p − r0 r1 0. pj+1 rj+1. + n > j and. pj rj. pj+1 , rj+1. +1>. 0 ≤ j ≤ n − 1,. n−1 Y pj+1 p0 pj pj+1 Γ( + n + 1) B( +n − j, − + 1). r0 rj+1 rj rj+1 j=0. Proof. First, from (4), we have E(X p0 Y1p1 · · · Ynpn ) Z ∞Z = β0 r0 r1 · · · rn · · · 0. ∞. p +(n−1)r2 −1 y1p1 +nr1 −1 y2 2. ···. ynpn +rn −1. 0. Z. ∞. xp0 +(n+1)r0 −1. 0. · exp[−xr0 (β0 + y1r1 + y1r1 y2r2 + · · · + y1r1 y2r2 · · · ynrn )]dxdy1 · · · dyn Z ∞Z ∞ p0 p +(n−1)r2 −1 = β0 r1 r2 · · · rn Γ( + n + 1) · · · y2 2 · · · ynpn +rn −1 r0 0 0 Z ∞ y1p1 +nr1 −1 · dy1 · · · dyn , p0 +n+1 0 [β0 + y1r1 (1 + y2r2 + · · · + y2r2 · · · ynrn )] r0. (5). where the last equality follows from the definition of the gamma function. Now using Lemma 1, (5) implies E(X p0 Y1p1 · · · Ynpn ) = r2 · · · rn β0. p p1 − r0 r1 0. · · ·ynpn +rn −1. Z 0. p0 p1 p0 p1 Γ( + n + 1)B( + n, − + 1) r0 r1 r0 r1. ∞. [1 + y2r2 (1 +. p +(n−1)r2 −1 y2 2 y3r3 + · · · + y3r3. p1. · · · ynrn )] r1. Z. ∞Z ∞. ··· 0. +n. p +(n−2)r3 −1. y3 3. 0. dy2 · · · dyn .. Again by repeated use of Lemma 1 and after some manipulations, we obtain p1. E(X p0 Y1p1 · · · Ynpn ) = β0 r1. p. − r0 0. Γ(. n−1 Y pj+1 pj pj+1 p0 + n + 1) B( +n − j, − + 1), r0 r r r j+1 j j+1 j=0. as desired. . It can be easily seen that, for 0 ≤ m ≤ n, the joint distribution of (X, Y1 , ..., Ym ) is rm −1 f (x, y1 , ..., ym )=β0 r0 · · · rm x(m+1)r0 −1 y1mr1 −1 · · · ym rm · exp[−xr0 (β0 + y1r1 + y1r1 y2r2 + · · · + y1r1 y2r2 · · · ym )],. (6). where x > 0, yi > 0, 1 ≤ i ≤ m. From (4) and (6), we obtain that the conditional distribution of Ym+1 , ..., Yn |X = x, Y1 = y1 , ..., Ym = ym is 3.

(10) f (ym+1 , ..., yn |x, y1 , ..., ym ) (n−m)r1. = rm+1 · · · rn x(n−m)r0 y1. (n−m)rm+1 −1 (n−m−1)rm+2 −1 ym+2 rm+2 ym+2 + · · · + y1r1 y2r2 · · · ynrn )],. (n−m)rm ym+1 · · · ym. r. m+1 + y1r1 y2r2 · · · · exp[−xr0 (y1r1 y2r2 · · · ym+1. · · · ynrn −1 (7). where x > 0, yi > 0, 1 ≤ i ≤ n. On the other hand, for 0 ≤ m < n, the joint distribution of (Ym+1 , ..., Yn ) is f (ym+1 , ..., yn ) Z ∞Z ∞ = ··· f (x, y1 , ..., yn )dxdy1 · · · dym 0 0 Z ∞Z ∞ (n−m)rm+1 −1 (n−m+1)rm −1 rn −1 y1nr1 −1 · · · ym = β0 r0 r1 · · · rn ym+1 · · · yn ··· 0 0 Z ∞ x(n+1)r0 −1 exp[−xr0 (β0 + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )]dxdy1 · · · dym · 0 Z ∞Z ∞ (n−m)rm+1 −1 (n−1)r2 −1 rn −1 (n−m+1)rm −1 = β0 r1 · · · rn ym+1 · · · yn Γ(n + 1) · · · y2 · · · ym 0 0 Z ∞ nr1 −1 y1 dy1 · · · dym · , yi > 0, m + 1 ≤ i ≤ n. r1 r2 [β0 + y1 (1 + y2 + · · · + y2r2 · · · ynrn )]n+1 0 By repeated use of Lemma 1, and the fact B(k, 1) = Γ(k)/Γ(k + 1), it can be shown that f (ym+1 , ..., yn ) = rm+1 · · · rn Γ(n + 1)B(n, 1)B(n − 1, 1) · · · B(n − m + 1, 1) (n−m)r. ·. −1. ym+1 m+1 · · · ynrn −1 rm+1 rm+1 (1 + ym+1 + · · · + ym+1 · · · ynrn )n−m+1 (n−m)r. −1. ym+1 m+1 · · · ynrn −1 = rm+1 · · · rn Γ(n − m + 1) , rm+1 rm+1 (1 + ym+1 + · · · + ym+1 · · · ynrn )n−m+1. (8). where yi > 0, m + 1 ≤ i ≤ n. In view of (4) and (8), the conditional distribution of X, Y1 , ..., Ym |Ym+1 = ym+1 , ..., Yn = yn is obtained as f (x, y1 , ..., ym |ym+1 , ..., yn ) β0 r0 · · · rm (n+1)r0 −1 nr1 −1 rm+1 rm+1 (n−m+1)rm −1 + · · · + ym+1 · · · ynrn )n−m+1 x y1 · · · ym (1 + ym+1 Γ(n − m + 1) · exp[−xr0 (β0 + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )], (9). =. where x > 0, yi > 0, 1 ≤ i ≤ n. Now from (7) and (9), using the definition of the gamma function and Lemma 1, the following two conditional moments can also be derived. 4.

(11) Property 2 For. pj+1 rj+1. + n > j, m ≤ j ≤ n − 1 and. pj rj. +1 >. pj+1 , rj+1. m + 1 ≤ j ≤ n − 1,. p. m+1 E(Ym+1 · · · Ynpn |X = x, Y1 = y1 , ..., Ym = ym ) n−1 pm+1 Y pm+1 pj pj+1 pj+1 rm − rm+1 ) = Γ( + n − m) + n − j, − + 1)(xr0 y1r1 · · · ym , B( rm+1 rj+1 rj rj+1 j=m+1. where x > 0, yi > 0, 1 ≤ i ≤ m.. Property 3 For. p0 r0. + n + 1 > 0,. pj+1 rj+1. + n > j and. pj rj. +1>. pj+1 , rj+1. 0 ≤ j ≤ m − 1,. E(X p0 Y1p1 · · · Ympm |Ym+1 = ym+1 , ..., Yn = yn ) Y Γ( pr00 + n + 1) p1 − p0 m−1 pj+1 pj pj+1 = β0 r1 r0 B( + n − j, − + 1) Γ(n − m + 1) r r r j+1 j j+1 j=0 r. pm. r. m+1 m+1 ·(1 + ym+1 + · · · + ym+1 · · · ynrn )− rm ,. where yi > 0, m + 1 ≤ i ≤ n.. 3. Concomitants of k-record values from multivariate pseudo-Weibull distributions. In this section, we investigate the moments of concomitants of k-record values from multivariate pseudo-Weibull distributions. Let (X1 , Y11 , ..., Y1n ), (X2 , Y21 , ..., Y2n ), · · · be a sequence of independent and identically distributed random vectors having the multivariate pseudo-Weibull distribution as in (4). Assume that {Ri , i ≥ 1} are the k-record values for {Xi , i ≥ 1} and {Y1(i) , Y2(i) , ..., Yn(i) , i ≥ 1} are the concomitants of the k-record values {Ri , i ≥ 1}. For i ≥ 1, Dziubdziela and Kopoci´ nski (1976) had shown that the pdf of the ith k-record values Ri is of form ki fRi (x) = [− ln(1 − F (x))]i−1 [1 − F (x)]k−1 f (x), Γ(i). x > 0,. (10). where F (x) and f (x) are, respectively, the cumulative distribution function and the pdf of the random variable X1 . It is known that f (x) = β0 r0 xr0 −1 exp(−β0 xr0 ), x > 0, and F (x) = 1 − exp(−β0 xr0 ), x > 0. Hence the pdf of Ri is 5.

(12) r0 (β0 k)i r0 i−1 fRi (x) = x exp(−β0 kxr0 ), Γ(i). x > 0.. (11). As the conditional distribution of Y1(i) , ..., Yn(i) |Ri = x is the same as that of Yj1 , ..., Yjn |Xj = x, j = 1, 2, ..., and for x > 0, yi > 0, 1 ≤ i ≤ n, (n−1)r2 −1. f (y1 , ..., yn |x)=r1 · · · rn xnr0 y1nr1 −1 y2. · · · ynrn −1. · exp[−xr0 (y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )], the joint distribution of (Ri , Y1(i) , ..., Yn(i) ) is given by g(x, y1 , ..., yn )=f (y1 , ..., yn |x)fRi (x) r0 r1 · · · rn (β0 k)i (n+i)r0 −1 nr1 −1 (n−1)r2 −1 = y2 · · · ynrn −1 x y1 Γ(i) · exp[−xr0 (β0 k + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )],. (12). where x > 0, yi > 0, 1 ≤ i ≤ n, and the joint distribution of the ith concomitants Y1(i) , ..., Yn(i) is Z ∞ g(y1 , y2 , ..., yn ) = g(x, y1 , y2 , ..., yn )dx 0 (n−1)r −1. 2 r1 · · · rn (β0 k) Γ(n + i)y1nr1 −1 y2 · · · ynrn −1 , yi ≥ 0, 1 ≤ i ≤ n. Γ(i)(β0 k + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )n+i. i. =. (13). The product moments of Ri , Y1(i) , ..., Yn(i) is derived in Property 4. Property 4 For −n <. p1 r1. <. p0 r0. + i,. pj+1 rj+1. + n > j and. pj rj. +1>. pj+1 , rj+1. 1 ≤ j ≤ n − 1,. p1 pn E(Rip0 Y1(i) · · · Yn(i) ). =. Γ( pr11 + n)Γ( pr00 − p0. Γ(i)(β0 k) r0. p1 r1. Y + i) n−1. p. − r1 1.  B. j=1.  pj+1 pj pj+1 + n − j, − +1 . rj+1 rj rj+1. Proof. First, from (12), we have p1 pn E(Rip0 Y1(i) · · · Yn(i) ) Z Z Z ∞ r0 r1 · · · rn (β0 k)i ∞ ∞ p1 +nr1 −1 p2 +(n−1)r2 −1 pn +rn −1 ··· y1 y2 · · · yn xp0 +(n+i)r0 −1 = Γ(i) 0 0 0 · exp[−xr0 (β0 k + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )]dxdy1 · · · dyn Z Z r1 · · · rn (β0 k)i Γ( pr00 + n + i) ∞ ∞ p2 +(n−1)r2 −1 = ··· y2 · · · ynpn +rn −1 Γ(i) 0 0 Z ∞ y1p1 +nr1 −1 · dy1 dy2 · · · dyn , (14) p0 +n+i 0 [β0 k + y1r1 (1 + y2r2 + y2r2 y3r3 + · · · + y2r2 · · · ynrn )] r0. 6.

(13) where the last equality follows from the definition of the gamma function. Using Lemma 1, (14) in turn implies pn p1 ) · · · Yn(i) E(Rip0 Y1(i). =. r2 · · · rn Γ( pr00 + n + i) Γ(i)(β0 k) Z. · 0. p0 p − r1 r0 1. p1 p0 p1 B( + n, − + i) r1 r0 r1. ∞. [1 + y2r2 (1 + y3r3 +. p +(n−1)r2 −1 y2 2 y3r3 y4r4 + · · · +. ∞Z ∞. Z. ··· 0. p +(n−2)r3 −1. y3 3. · · · ynpn +rn −1. 0. p1 +n. y3r3 · · · ynrn )] r1. dy2 · · · dyn .. By repeated use of Lemma 1 and after some manipulations, it yields pn p1 )= · · · Yn(i) E(Rip0 Y1(i). Γ( pr00 + n + i). p1 p0 p 1 + n, − + i) r1 r0 r1  pj pj+1 + n − j, − +1 . rj rj+1. p0 p − r1 r0 1. Γ(i)(β0 k) n−1 Y  pj+1 · B rj+1 j=1. B(. As B(. Γ( p1 + n)Γ( pr00 − pr11 + i) p1 p0 p1 + n, − + i) = r1 , r1 r0 r1 Γ( pr00 + n + i). the assertion follows immediately.. . From Property 4, letting p0 = 0, the product moments of the ith concomitants Y1(i) · · · Yn(i) is given by p1 pn E(Y1(i) · · · Yn(i) ) p1. =. (β0 k) r1 Γ( pr11 + n)Γ(i − Γ(i) p. p1 ) r1. ·. n−1 Y.  B. j=1 p.  pj+1 pj pj+1 + n − j, − +1 , rj+1 rj rj+1. (15). p. j+1 j+1 + n > j and rjj + 1 > rj+1 , 1 ≤ j ≤ n − 1. Note that (15) can also where i > pr11 , rj+1 be derived from (13). In the following, we will give the conditional product moments Ym+1(i) , ..., Yn(i) given Y1(i) , ..., Ym(i) . It is known that for 1 ≤ m < n, from (13), the conditional distribution of Ym+1(i) , ..., Yn(i) |Y1(i) = y1 , ..., Ym(i) = ym is,. g(ym+1 , ..., yn |y1 , ..., ym )=. rm+1 · · · rn Γ(n + i) (n−m)r1 (n−m)rm (n−m)rm+1 −1 y1 · · · ym ym+1 · · · ynrn −1 Γ(m + i) rm m+i (β0 k + y1r1 + y1r1 y2r2 + · · · + y1r1 y2r2 · · · ym ) · , (16) r1 r1 r2 r1 r2 r n (β0 k + y1 + y1 y2 + · · · + y1 y2 · · · yn )n+i. where yj > 0, 1 ≤ j ≤ n. 7.

(14) On the other hand, for 1 ≤ m < n and yi > 0, m + 1 ≤ i ≤ n, the joint distribution of Ym+1(i) , ..., Yn(i) is, g(ym+1 , ..., yn ) Z ∞Z ∞ g(y1 , ..., yn )dy1 · · · dym = ··· 0. 0. r1 · · · rn (β0 k)i Γ(n + i) (n−m)rm+1 −1 ym+1 · · · ynrn −1 Γ(i) Z ∞Z ∞ Z ∞ y1nr1 −1 dy1 · · · dym (n−1)r2 −1 (n−m+1)rm −1 . · ··· y2 · · · ym [β0 k + y1r1 (1 + y2r2 + · · · + y2r2 · · · ynrn )]n+i 0 0 0. =. By repeated use of Lemma 1 and after some manipulations, we have g(ym+1 , ..., yn ) (n−m)rm+1 −1. rm+1 · · · rn Γ(n + i)B(n, i)B(n − 1, 1) · · · B(n − m + 1, 1)ym+1 = rm+1 rm+1 Γ(i)(1 + ym+1 + · · · + ym+1 · · · ynrn )n−m+1. · · · ynrn −1. .. (17) As B(a, b) = Γ(a)Γ(b)/Γ(a + b), it is easy to see that B(n, i)B(n − 1, 1) · · · B(n − m + 1, 1)=. Γ(i)Γ(n − m + 1) . Γ(n + i). Thus (17) implies (n−m)r. −1. rm+1 · · · rn Γ(n − m + 1)ym+1 m+1 · · · ynrn −1 g(ym+1 , ..., yn )= . rm+1 rm+1 (1 + ym+1 + · · · + ym+1 · · · ynrn )n−m+1. (18). In view of (13) and (18), the conditional distribution of Y1(i) , ..., Ym(i) |Ym+1(i) = ym+1 , ..., Yn(i) = yn is (n−m+1)r −1. m r1 · · · rm (β0 k)i Γ(n + i)y1nr1 −1 · · · ym g(y1 , ..., ym |ym+1 , ..., yn )= Γ(i)Γ(n − m + 1) rm+1 rm+1 (1 + ym+1 + · · · + ym+1 · · · ynrn )n−m+1 · , (βk + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )n+i. (19). where yi > 0, 1 ≤ i ≤ n. Now from (16), (19) and Lemma 1, the following two conditional moments of concomitants can be derived. Property 5 For m − n < 1 ≤ j ≤ n − m − 1,. pm+1 rm+1. pm+j+1 rm+j+1. < m + i,. + n > m + j and. pm+j rm+j. +1 >. pm+j+1 , rm+j+1. p. pn m+1 E(Ym+1(i) · · · Yn(i) |Y1(i) = y1 , Y2(i) = y2 , ..., Ym(i) = ym ). =. m+1 Γ( prm+1 + n − m)Γ(i + m −. Γ(m + i) . ·. pm+1 n−m−1 ) Y rm+1. . B. j=1. rm β0 k + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ym rm y1r1 · · · ym. pm+j+1 pm+j pm+j+1 + n − m − j, − +1 rm+j+1 rm+j rm+j+1.  prm+1 m+1. 8. ,. .

(15) where yi > 0, 1 ≤ i ≤ m. Property 6 For −n <. p1 r1. < i,. pj+1 rj+1. + n > j and. pj rj. +1>. pj+1 , rj+1. 1 ≤ j ≤ m − 1,. pm p2 p1 |Ym+1(i) = ym+1 , ..., Yn(i) = yn ) · · · Ym(i) Y2(i) E(Y1(i) p1  Y  pj+1 Γ( pr11 + n)Γ(i − pr11 )(β0 k) r1 m−1 pj pj+1 = B + n − j, − +1 Γ(i)Γ(n − m + 1) rj+1 rj rj+1 j=1 r. pm. r. m+1 m+1 ·(1 + ym+1 + · · · + ym+1 · · · ynrn )− rm ,. where yi > 0, m + 1 ≤ i ≤ n. It is know that the conditional distribution of Y1(i) , ..., Yn(i) |Ri = x is the same as pj+1 that of Yj1 , ..., Yjn |Xj = x, j = 1, 2, ... From Property 2, we know that for rj+1 +n > j, pj pj+1 0 ≤ j ≤ n − 1 and rj + 1 > rj+1 , 1 ≤ j ≤ n − 1, p1 E(Y1(i). ···. pn |Ri Yn(i). n−1 Y pj+1 −r0 p1 p1 pj pj+1 = x)=Γ( + n) B( + n − j, − + 1)x r1 . r1 rj+1 rj rj+1 j=1. In the following, we will give the conditional moments of Ri |Y1(i) , ..., Yn(i) . In view of (12) and (13), it can be seen that the conditional distribution of Ri |Y1(i) , ..., Yn(i) is g(x|y1 , ..., yn )=. r0 (β0 k + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )n+i x(n+i)r0 −1 Γ(n + i) · exp[−xr0 (β0 k + y1r1 + y1r1 y2r2 + · · · + y1r1 · · · ynrn )].. Using the definition of the gamma function, the following result can be proved. Property 7 For. p0 r0. + n + i ≥ 0,. E(Rip0 |Y1(i) = y1 , Y2(i) = y2 , ..., Yn(i) = yn ) Γ( pr00 + n + i) = p0 . Γ(n + i)(β0 k + y1r1 + y1r1 y2r2 + .. + y1r1 y2r2 ...ynrn ) r0. 4. Two special cases. In this section we will present some distributional properties of concomitants of k-record values from bivariate pseudo-Weibull distribution and trivariate pseudoWeibull distribution. It will be shown that some properties in Section 3 are extensions of results known in the literature.. 9.

(16) 4.1. Bivariate pseudo-Weibull distributions. Let n = 1 in (4), the bivariate pseudo-Weibull distribution can be given as f (x, y) = β0 r0 r1 x2r0 −1 y r1 −1 exp(−xr0 (β0 + y r1 )), x, y > 0, where β0 , r0 , r1 > 0. Note that if r0 = r1 = 1, f (x, y) is the pdf of bivariate pseudoexponential distribution, and if r0 = r1 = 2, f (x, y) is the pdf of bivariate pseudoRayleigh distribution. Assume that Y(i) is the concomitant of the ith k-record values Ri , i ≥ 1. Letting (12) with n = 1, the joint distribution of (Ri , Y(i) ) is g(x, y) =. r0 r1 (β0 k)i r0 (i+1)−1 r1 −1 x y exp(−xr0 (β0 k + y r1 )), x, y > 0. Γ(i). (20). Also letting (13) with n = 1, the marginal distribution of the ith concomitant Y(i) is g(y) =. r1 i(β0 k)i y r1 −1 , y > 0. (y r1 + β0 k)i+1. (21). It can be easily seen that the survival function S(y) and hazard function h(y) of Y(i) are  S(y) =. β0 k y r1 + β0 k. i. and h(y) =. r1 iy r1 −1 , (y r1 + β0 k). respectively. Now from Property 4, we have p1. p1 E(Y(i) ). =. (β0 k) r1 Γ( pr11 + 1)Γ(i − Γ(i). p1 ) r1. , p1 < ir1 .. (22). Letting r0 = r1 = 1, the moments of the concomitant Y(i) of the k-record value Ri from bivariate pseudo-exponential distribution can also be obtained, and the mean µ, variance σ 2 , skewness γ1 and kurtosis γ2 of Y(i) , can be expressed as follows:. µ=. β0 k , i−1. σ2 =. i > 1;. i(β0 k)2 , (i − 1)2 (i − 2) 10. i > 2;.

(17) √ 2(i + 1) i − 2 √ γ1 = , (i − 3) i. i > 3;. and γ2 =. 3(i − 2)(3i2 + i + 2) , i(i − 3)(i − 4). i > 4.. For k = 1, µ, σ 2 , γ1 and γ2 are also derived by Mohsin et al. (2010), however the expressions of γ1 and γ2 in Mohsin et al. (2010) are wrong. Moreover, Ahsanullah et al.(2010) derived (22) with k = 1. Also letting r0 = r1 = 2 in (22), the moment of the concomitant Y(i) of the k-record value Ri from bivariate pseudo-Rayleigh distribution can be obtained. The other corresponding moments derived in Sections 2 and 3 can also be obtained. For example, from Property 7 with n = 1, we have, for pr00 + i + 1 > 0, E(Rip0 |Y(i). Γ( pr00 + i + 1). = y)=. p0. ,. (23). Γ(i + 1)(β0 k + y r1 ) r0. and the corresponding result for bivariate pseudo-exponential distribution can be obtained from (23) with r0 = r1 = 1. In the following, we will investigate the entropies of Y(i) and (Ri , Y(i) ). The proofs need the following integral formulas which can be seen in formulas (4.352.1), (4.253.3) and (4.253.6) of Gradshteyn and Ryzhik (2007). Z ∞ 1 xν−1 exp(−µx) ln xdx = ν Γ(ν)[ψ(ν) − ln µ], µ > 0, ν > 0, (24) µ 0 ∞. Z u. and Z 0. (x − u)µ−1 ln x dx = uµ−λ B(λ − µ, µ)[ln u + ψ(λ) − ψ(λ − µ)], 0 < µ < λ, (25) xλ. ∞. ln x 1 dx = µ (ln a + ψ(1) − ψ(µ)), µ > 0, a 6= 0, µ+1 (a + x) µa. where ψ(x) =. d dx. ln Γ(x) is the psi function.. 11. (26).

(18) Property 8 The joint entropy of Ri and Y(i) is     r0 r1 (β0 k)i 1 1 H(Ri , Y(i) )=− ln − i− + [ψ(i) − ln(β0 k)] Γ(i) r0 r1   1 ψ(1) + i + 1. − 1− r1 Proof. First, from (20), the joint entropy of Ri and Y(i) is H(Ri , Y(i) )=E[− ln g(Ri , Y(i) )]   r0 r1 (β0 k)i =− ln − [r0 (i + 1) − 1]E(ln Ri ) Γ(i) r1 + β0 k)]. −(r1 − 1)E(ln Y(i) ) + E[Rir0 (Y(i). (27). Next, from (11), we know that Z r0 (β0 k)i ∞ r0 i−1 E(ln Ri )= x exp(−β0 kxr0 ) ln x dx Γ(i) Z 0 (β0 k)i ∞ i−1 = u exp(−β0 ku) ln u du. r0 Γ(i) 0. (28). Applying the formula (24), it can be seen that (28) implies E(ln Ri )=. ψ(i) − ln(β0 k) . r0. (29). From (21), it can be seen that i. Z. ∞. E(ln Y(i) )=r1 i(β0 k). i = (β0 k)i r1. Z 0∞ 0. y r1 −1 ln y dy (β0 k + y r1 )i+1 ln u du. (β0 k + u)i+1. Applying the formula (26), it yields E(ln Y(i) )=. ln(β0 k) + ψ(1) − ψ(i) . r1. (30). Also from (20), we have r1 E[Rir0 (Y(i). Z Z ∞ r1 (β0 k)i ∞ r1 r1 −1 + β0 k)]= (y + β0 k)y r0 xr0 (i+2)−1 exp(−xr0 (y r1 + β0 k)) dxdy Γ(i) 0 Z0 Γ(i + 2) r1 (β0 k)i ∞ r1 (y + β0 k)y r1 −1 r1 dy = Γ(i) (y + β0 k)i+2 0 Z r1 (β0 k)i Γ(i + 2) ∞ y r1 −1 = dy. Γ(i) (y r1 + β0 k)i+1 0. Using Lemma 1, after some manipulations, we have r1 E[Rir0 (Y(i) + β0 k)]=i + 1.. (31). Finally, from (27) and (29)-(31), the joint entropy of Ri and Y(i) can be obtained. 12.

(19) Property 9 The entropy of Y(i) is     1 1 1 i H(Y(i) )=− ln[r1 i(β0 k) ] + 1 − [ψ(i) − ψ(1)] + i + ln(β0 k) + + 1. r1 r1 i Proof. From (21), the entropy of Y(i) is r1 + β0 k)], H(Y(i) )=− ln[r1 i(β0 k)i ] − (r1 − 1)E(ln Y(i) ) + (i + 1)E[ln(Y(i). (32). and r1 E[ln(Y(i). ∞. r1 i(β0 k)i y r1 −1 ln(y r1 + β0 k) dy r1 + β k)i+1 (y 0 Z ∞ 0 ln u =i(β0 k)i du. i+1 β0 k u Z. + β0 k)]=. (33). Applying the formula (25) and using the equality ψ(i + 1) = ψ(i) + 1i , (33) implies that 1 r1 E[ln(Y(i) + β0 k)]= + ln(β0 k). i. (34). Now from (30), (32) and (34), the assertion is verified.. 4.2. . Trivariate pseudo-Weibull distributions. Let n = 2 in (4), the trivariate pseudo-Weibull distribution can be given as f (x, y1 , y2 ) = β0 r0 r1 r2 x3r0 −1 y12r1 −1 y2r2 −1 exp(−xr0 (β0 + y1r1 + y1r1 y2r2 )), x, y1 , y2 > 0. where β0 , r0 , r1 , r2 > 0. Again if r0 = r1 = r2 = 1, f (x, y1 , y2 ) is the pdf of trivariate pseudo-exponential distribution, and if r0 = r1 = r2 = 2, f (x, y1 , y2 ) is the pdf of trivariate pseudo-Rayleigh distribution. Letting (13) with n = 2, the pdf of the ith concomitants (Y1(i) , Y2(i) ) is g(y1 , y2 ) = r1 r2 i(i + 1)(β0 k)i. y12r1 −1 y2r2 −1 . (β0 k + y1r1 + y1r1 y2r2 )i+2. (35). From Properties 4-7 in Section 3, we have p1. p1 p2 E(Y1(i) Y2(i) )=i(i + 1)(β0 k) r1 B(. p2 E(Y2(i) |Y1(i) = y1 )=. p1 p1 p2 p1 p2 + 2, i − )B( + 1, − + 1), r1 r1 r2 r1 r2 p1 p2 p1 −2 < < i, −1 < < + 1, r1 r2 r1. Γ( pr22 + 1)Γ(i −. p2 r2. + 1). Γ(i + 1) 13. . β0 k 1 + r1 y1.  pr2 2. , −1 <. (36). p2 < i + 1, (37) r2.

(20) p1 |Y2(i) E(Y1(i). = y2 )=. Γ( pr11 + 2)Γ(i −. p1 ) r1. . Γ(i). β0 k 1 + y2r2.  pr1 1. −2 <. ,. p1 < i, (38) r1. and E(Rip0 |Y1(i) = y1 , Y2(i) = y2 )=. Γ( pr00 + i + 2) Γ(i + 2)(β0 k + y1r1 + y1r1 y2r2 ). p0 r0. ,. p0 + i + 2 > 0.(39) r0. Consider the record values from trivariate pseudo-exponential distribution, i.e. k = 1 and r0 = r1 = r2 = 1, then (36)-(39) imply that p2 p1 )=i(i + 1)β0 p1 B(p1 + 2, i − p1 )B(p2 + 1, p1 − p2 + 1), Y2(i) E(Y1(i). −2 < p1 < i, −1 < p2 < p1 + 1, (40). p2 |Y1(i) E(Y2(i). Γ(p2 + 1)Γ(i − p2 + 1) = y1 )= Γ(i + 1). p1 E(Y1(i) |Y2(i). Γ(p1 + 2)Γ(i − p1 ) = y2 )= Γ(i). . . β0 1+ y1. β0 1 + y2.  p2 ,. −1 < p2 < i + 1, (41). p1 ,. −2 < p1 < i, (42). and E(Rip0 |Y1(i) = y1 , Y2(i) = y2 )=. Γ(p0 + i + 2) , Γ(i + 2)(β0 + y1 + y1 y2 )p0. p0 + i + 2 > 0. (43). Note that (40) and (42) are also given in Shahbaz et al. (2010). In the following we will investigate the entropies of (Y1(i) , Y2(i) ) and (Ri , Y1(i) , Y2(i) ). Property 10 The joint entropy of Y1(i) and Y2(i) is   1 i H(Y1(i) , Y2(i) )=− ln[r1 r2 i(i + 1)(β0 k) ] + 2 − [ψ(i) − ψ(1)] r1   (i + 2)(2i + 1) 1 . + i+ ln(β0 k) + r1 i(i + 1) Proof. From (35), the entropy of Y1(i) and Y2(i) is H(Y1(i) , Y2(i) )=− ln[r1 r2 i(i + 1)(β0 k)i ] − (2r1 − 1)E(ln Y1(i) ) − (r2 − 1)E(ln Y2(i) ) r1 r1 r2 +(i + 2)E[ln(β0 k + Y1(i) + Y1(i) Y2(i) )].. (44). It is easy to see that E(ln Y1(i) ) is given as in (30) and Z ∞ Z ∞ y12r1 −1 r2 −1 i E(ln Y2(i) )=r1 r2 i(i + 1)(β0 k) y2 ln y2 dy1 dy2 . [β0 k + y1r1 (1 + y2r2 )]i+2 0 0 (45) 14.

(21) Using Lemma 1, it can be seen that Z 0. ∞. 1 y12r1 −1 r2 i+2 dy1 = r1 [β0 k + y1 (1 + y2 )] r1 (β0 k)i+2. . β0 k 1 + y2r2. 2 B(2, i).. (46). In view of (45) and (46), we have y2r2 −1 E(ln Y2(i) )=r2 ln y2 dy2 (1 + y2r2 )2 0 Z 1 ∞ ln u = du. r2 0 (1 + u)2 Z. ∞. (47). Applying the formula (26), (47) implies E(ln Y2(i) )=0.. (48). Also from (35), we have r2 r1 r1 )] Y2(i) + Y1(i) E[ln(β0 k + Y1(i) Z ∞ 2r1 −1 Z ∞ y1 ln[β0 k + y1r1 (1 + y2r2 )] r2 −1 i dy1 dy2 y2 = r1 r2 i(i + 1)(β0 k) [β0 k + y1r1 (1 + y2r2 )]i+2 0 0 Z ∞ Z ∞ y2r2 −1 (u − β0 k) ln u i = r1 r2 i(i + 1)(β0 k) dudy2 . r2 2 r1 (1 + y2 ) β0 k ui+2 0. Applying the formula (25), it can be shown that r1 r1 r2 E[ln(β0 k + Y1(i) + Y1(i) Y2(i) )]. Z = r2 i(i + 1)B(i, 2)[ψ(i + 2) − ψ(i) + ln(β0 k)] 0. ∞. y2r2 −1 dy2 . (1 + y2r2 )2. Now using Lemma 1 and the equality ψ(i + 2) = ψ(i) + 1i + r1 r1 r2 E[ln(β0 k + Y1(i) + Y1(i) Y2(i) )] =. 1 , i+1. it yields. 1 1 + + ln(β0 k). i i+1. Finally, from (30), (44), (48) and (49), the result follows immediately. Property 11 The joint entropy of (Ri , Y1(i) , Y2(i) ) is     r0 r1 r2 (β0 k)i 1 1 H(Ri , Y1(i) , Y2(i) )=− ln − i− + [ψ(i) − ln(β0 k)] Γ(i) r0 r1   1 − 2− ψ(1) + i + 2. r1 Proof. From (12) with n = 2, the entropy of (Ri , Y1(i) , Y2(i) ) is. 15. (49) .

(22)  r0 r1 r2 (β0 k)i H(Ri , Y1(i) , Y2(i) )=− ln − (2r0 + ir0 − 1)E(ln Ri ) − (2r1 − 1)E(ln Y1(i) ) Γ(i) r2 r1 r1 )]. (50) Y2(i) + Y1(i) −(r2 − 1)E(ln Y2(i) ) + E[Rir0 (β0 k + Y1(i) . It can be seen that E(ln Ri ), E(ln Y1(i) ) and E(ln Y2(i) ) are given as in (29), (30) and (48) respectively. Also using the definition of the gamma function, we have r2 r1 r1 )] Y2(i) + Y1(i) E[Rir0 (β0 k + Y1(i) Z Z r0 r1 r2 (β0 k)i ∞ ∞ = (β0 k + y1 r1 + y1 r1 y2 r2 )y12r1 −1 y2r2 −1 Γ(i) 0 0 Z ∞ x(3+i)r0 −1 exp[−xr0 (β0 k + y1 r1 + y1 r1 y2 r2 )] dxdy1 dy2 · 0 Z Z r1 r2 (β0 k)i ∞ ∞ 2r1 −1 r2 −1 Γ(3 + i) = y2 y1 dy1 dy2 . · Γ(i) (β0 k + y1 r1 + y1 r1 y2 r2 )2+i 0 0. (51). By repeated use of Lemma 1 and after some manipulations, (51) implies that r2 r1 r1 )] = i + 2. Y2(i) + Y1(i) E[Rir0 (β0 k + Y1(i). (52). Now from (29), (30), (48), (50) and (52), the result is immediately obtained.. 5. . Concomitants of k-record values from some other distributions. In this section, we will consider the k-record values from bivariate pseudo-exponential distribution with φ1 (x) = β1 x + c, β1 , c > 0, and Gumbel’s bivariate exponential distribution, and investigate the moments of concomitants of k-record values.. 5.1. Bivariate pseudo-exponential distribution with φ1 (x) = β1 x + c. In (3), if we let n = 1, r0 = r1 = 1, and φ1 (x) = β1 x + c, then the joint distribution of bivariate pseudo-exponential distribution is given as f (x, y) = β0 (β1 x + c) exp[−x(β0 + β1 y) − cy], where β0 , β1 , c > 0. Assume that Y(i) is the concomitant of the k-record values Ri , i ≥ 1. It can be seen that the joint distribution of (Ri , Y(i) ) is g(x, y) =. (β0 k)i (β1 x + c)xi−1 exp(−β1 xy − β0 kx − cy), Γ(i) 16.

(23) and the marginal distribution of Y(i) is   (β0 k)i e−cy β1 i g(y) = +c . (β1 y + β0 k)i β1 y + β0 k Hence the pth moment of Y(i) is Z ∞ Z ∞ y p e−cy y p e−cy p i i E(Y(i) ) = β1 i(β0 k) dy + c(β k) dy. 0 (β1 y + β0 k)i+1 (β1 y + β0 k)i 0 0. (53). From the formulas (3.351.2) and (3.351.4) in Gradshteyn and Ryzhik (2007), for an integer u and real numbers v, k > 0, we have. Z. ∞. xu e−vx dx =. k. P  ul=0. e−kv Γ(u+1)ku−l , v l+1 Γ(u−l+1). if u ≥ 0,.  (−1)−u v−u−1 Ei(−kv) − P−u−2 l=0 Γ(−u). v l e−kv Γ(−u−l−1)ku+l+1 , (−1)l+1 Γ(−u). (54) if u < 0,. Rz where Ei(z)= −∞ x−1 ex dx is an exponential integral function. Also for some integral p, and real numbers c, k, θ, letting z = θy + k, and using the binomial theorem, it can be shown that ∞. Z 0. Z ∞ ck p   y p e−cy eθ X p m−i − θc z p−m z e dz. dy = (−k) (θy + k)i θp+1 m=0 m k. (55). From (54), (55) implies that Z ∞ p −cy y e dy (θy + k)i 0     (−1)p+i ci−m−1 e ckθ kp−m Ei(− ck ) Pi−m−2 (−1)p−m−l cl kp−i+l+1 Γ(i−m−l−1) Pp  p  θ  + l=0 ,  m=0 m θi−m+p Γ(i−m) θp+l+1 Γ(i−m)      if p < i.    ck = Pi−1 p  (−1)p+i ci−m−1 e θ kp−m Ei(− ckθ ) Pi−m−2 (−1)p−m−l cl kp−i+l+1 Γ(i−m−l−1)  + l=0  m=0 m θi−m+p Γ(i−m) θp+l+1 Γ(i−m)         c−l−1 kp−i−l Γ(m−i+1) + Pp Pm−i p (−1)p−mp−l , if p ≥ i. (56) m=i l=0 m θ Γ(m−i−l+1) In view of (53) and (56), if p ≤ i − 1, then p E(Y(i) ). " =. Pp.  p. +. Pp.  p. β0 ck β1. Ei(−. β0 ck ) β1. β1i−m+p Γ(i−m+1). m=0 m. " m=0 m. (−1)i+p+1 ici−m (β0 k)p−m+i e. (−1)i+p ci−m (β0 k)p−m+i e. β0 ck β1. Ei(−. β0 ck ) β1. β1i−m+p Γ(i−m). 17. # +. Pi−m−1 l=0. (−1)p−m−l icl (β. 0. k)l+p Γ(i−m−l). β1p+l Γ(i−m+1). # P + i−m−2 l=0. (−1)p−m−l cl+1 (β. 0. k)l+p+1 Γ(i−m−l−1). β1p+l+1 Γ(i−m). ;.

(24) if p = i, then p E(Y(i) ). " =. Pp.  p. +. Pp−1.  p. +. . ". cp−m (β0 k)2p−m e. β0 ck β1. β0 ck ) β1. Ei(−. Ei(−. β12p−m Γ(p−m). m=0 m. p. β0 ck β1. β12p−m Γ(p−m+1). m=0 m. β0 k β1. −pcp−m (β0 k)2p−m e. β0 ck ) β1. # +. Pp−m−1 l=0. (−1)p−m−l pcl (β0 k)l+p Γ(p−m−l) β1p+l Γ(p−m+1). # +. Pp−m−2(−1)p−m−l cl+1 (β0 k)l+p+1 Γ(p−m−l−1) β1p+l+1 Γ(p−m). l=0. ;. and if p ≥ i + 1, then p E(Y1(i) ). " =. Pi.  p. m=0. P + i−1 m=0. (−1)i+p+1 ici−m (β0 k)p−m+i e. β0 ck β1. Ei(−. β0 ck ) β1. β1i−m+p Γ(i−m+1). m. # P + i−m−1 l=0. (−1)p−m−l icl (β. 0. k)l+p Γ(i−m−l). β1p+l Γ(i−m+1). # " β0 ck  (−1)i+p ci−m (β0 k)p−m+i e β1 Ei(− ββ0 ck ) Pi−m−2(−1)p−m−l cl+1 (β0 k)l+p+1 Γ(i−m−l−1) 1 + l=0 β i−m+p Γ(i−m) β p+l+1 Γ(i−m). p m. 1. 1.  h (−1)p−m ic−l−1 (β0 k)p−l−1 Γ(m−i) i p. P P + pm=i+1 m−i−1 l=0 m β1p−l−1 Γ(m−i−l) h  (−1)p−m c−l (β0 k)p−l Γ(m−i+1) i P P p + pm=i m−i . l=0 m β p−l Γ(m−i−l+1) 1. 5.2. Gumbel’s bivariate exponential distribution. Consider the Gumbel’s bivariate exponential distribution (see Kotz, Balakrishnan and Johnson (2000)), its pdf is f (x, y) = {(1 + θx)(1 + θy) − θ} exp(−x − y − θxy),. x, y > 0,. 0 6 θ 6 1.. It is known that X has exponential distribution with parameter 1. From (10), it can be seen that the ith k-record value Ri arised by X’s has the pdf fRi (x) =. k i i−1 −kx x e , Γ(i). x > 0.. The joint distribution of (Ri , Y(i) ) is g(x, y) =. k i xi−1 [(1 + θx)(1 + θy) − θ] exp(−y − θxy − kx), Γ(i) 18.

(25) and the marginal distribution of Y(i) is given by g(y) =. k i e−y (1 − θ + θy) iθk i (1 + θy)e−y + , (θy + k)i (θy + k)i+1. y > 0, 0 ≤ θ ≤ 1.. Hence the moment of Y(i) is given by  Z ∞ Z ∞ p+1 −y  y p e−y y e p i E(Y(i) ) = k (1 − θ) dy + θ dy i (θy + k) (θy + k)i 0 0 +iθk. i. Z 0. ∞. y p e−y dy + θ (θy + k)i+1. Z 0. ∞.  y p+1 e−y dy . (θy + k)i+1. Using (56), we know that, if p < i − 1, then p E(Y(i) ) " # k p   i−m−2 X X Γ(i − m − l − 1)k p+l+1 (−1)p+i e θ k p−m+i Ei(− kθ ) p (1 − θ) = + p+i−m m Γ(i − m)θ (−1)p−m−l Γ(i − m)θp+l+1 m=0 l=0 # " k p+1  i−m−2 X X (−1)p+i+1 e θ k p−m+i+1 Ei(− kθ ) Γ(i − m − l − 1)k p+l+2 p+1 + + p+i−m m Γ(i − m)θ (−1)p−m−l+1 Γ(i − m)θp+l+1 m=0 l=0 " # k p   i−m−1 X X (−1)p+i+1 ie θ k p−m+i Ei(− kθ ) p iΓ(i − m − l)k p+l + + m Γ(i − m + 1)θp+i−m (−1)p−m−l Γ(i − m + 1)θp+l m=0 l=0 # " k p+1  i−m−1 X X (−1)p+i ie θ k p−m+i+1 Ei(− kθ ) iΓ(i − m − l)k p+l+1 p+1 + + ; m Γ(i − m + 1)θp+i−m (−1)p−m−l+1 Γ(i − m + 1)θp+l m=0 l=0. if p = i − 1, then p E(Y(i) ) " # k p   p−m−1 X X −e θ k 2p−m+1 Ei(− kθ ) p Γ(p − m − l)k p+l+1 = (1 − θ) + m Γ(p − m + 1)θ2p−m+1 (−1)p−m−l Γ(p − m + 1)θp+l+1 m=0 l=0 #  " k 2p−m+2 p  p−m−1 X X eθ k Ei(− kθ ) p+1 Γ(p − m − l)k p+l+2 + + 2p−m+1 m Γ(p − m + 1)θ (−1)p−m−l+1 Γ(p − m + 1)θp+l+1 m=0 l=0 " # k p   p−m X (p + 1)e θ k 2p−m+1 Ei(− kθ ) X (p + 1)Γ(p − m − l + 1)k p+l p + + 2p−m+1 m Γ(p − m + 2)θ (−1)p−m−l Γ(p − m + 2)θp+l m=0 l=0 # " k p−m p+1  k 2p−m+2 p+l+1 X X θ −(p + 1)e k Ei(− ) p+1 (p + 1)Γ(p − m − l + 1)k θ + + 2p−m+1 m Γ(p − m + 2)θ (−1)p−m−l+1 Γ(p − m + 2)θp+l m=0 l=0. +. k p+1 ; θp. 19.

(26) if p = i, then p E(Y(i) ) " k # p−1   p−m−2 X X Γ(p − m − l − 1)k p+l+1 e θ k 2p−m Ei(− kθ ) p (1 − θ) = + m Γ(p − m)θ2p−m (−1)p−m−l Γ(p − m)θp+l+1 m=0 l=0 #  " k 2p−m+1 p−1  p−m−2 X X Γ(p − m − l − 1)k p+l+2 −e θ k Ei(− kθ ) p+1 + + 2p−m m Γ(p − m)θ (−1)p−m−l+1 Γ(p − m)θp+l m=0 l=0 " # k p   p−m−1 X X −pe θ k 2p−m Ei(− kθ ) p pΓ(p − m − l)k p+l + + 2p−m m Γ(p − m + 1)θ (−1)p−m−l Γ(p − m + 1)θp+l m=0 l=0 #  " k 2p−m+1 p  p−m−1 X X pe θ k Ei(− kθ ) p+1 pΓ(p − m − l)k p+l+1 + + m Γ(p − m + 1)θ2p−m (−1)p−m−l+1 Γ(p − m + 1)θp+l m=0 l=0  p k + (1 + pθ − pk); θ. and if p ≥ i + 1, then p E(Y(i) ) " # k i−1   i−m−2 X X Γ(i − m − l − 1)k p+l+1 (−1)p+i e θ k p−m+i Ei(− kθ ) p = (1 − θ) + p+i−m m Γ(i − m)θ (−1)p−m−l Γ(i − m)θp+l+1 m=0 l=0 " #  i−1  i−m−2 k p+i+1 kθ p−m+i+1 p+l+2 X X (−1) e k Ei(− ) p+1 Γ(i − m − l − 1)k θ + + p+i−m m Γ(i − m)θ (−1)p−m−l+1 Γ(i − m)θp+l+1 m=0 l=0 " # k i−m−1 i   X X (−1)p+i+1 ie θ k p−m+i Ei(− kθ ) iΓ(i − m − l)k p+l p + + Γ(i − m + 1)θp+i−m (−1)p−m−l Γ(i − m + 1)θp+l m m=0 l=0 # " k i−m−1 i  X X (−1)p+i ie θ k p−m+i+1 Ei(− kθ ) p+1 iΓ(i − m − l)k p+l+1 + + m Γ(i − m + 1)θp+i−m (−1)p−m−l+1 Γ(i − m + 1)θp+l m=0 l=0   p m−i   X X p Γ(m − i + 1)k p−l p−m + (1 − θ)(−1) m θp−l Γ(m − i − l + 1) m=i l=0   p+1 m−i  X X p + 1 Γ(m − i + 1)k p−l+1 p−m+1 (−1) + m θp−l Γ(m − i − l + 1) m=i l=0   p m−i−1 X X p iΓ(m − i)k p−l−1 p−m + (−1) m θp−l−1 Γ(m − i − l) m=i+1 l=0   p+1 m−i−1  X X p + 1 iΓ(m − i)k p−l p−m+1 (−1) . + m θp−l−1 Γ(m − i − l) m=i+1 l=0. 20.

(27) References [1] Ahsanullah, M. (1995). Record statistics. Nova Science Publishers, Commack, New York. [2] Ahsanullah, M. (2009). Records and concomitants. Bull. Malays. Math. Sci. Soc. 32, 101-117. [3] Ahsanullah, M., Shahbaz, S., Shahbaz, M. Q. and Mohsin, M. (2010). Concomitants of upper record statistics for bivariate pseudo-Weibull distribution. Appl. Appl. Math. 5, 1379-1388. [4] Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1998). Records. Wiley, New York. [5] Dziubdziela, W. and Kopoci´ nski, B. (1976). Limiting properties of the k-th record value. Appl. Math. 15, 187-190. [6] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series and Products. Academic, New York. [7] Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, Models and Applications, second edition. John Wiley, New York. [8] Mohsin, M., Pilz, J., Gunter, S., Shahbaz, S. and Shahbaz, M. Q. (2010). Some distributional properties of the concomitants of record statistics for bivariate pseudo-exponential distribution and characterization. J. Prime Res. Math. 6, 32-37. [9] Nevzorov, V. B. (2001). Records: Mathematical Theory, Translations of Mathematical Monographs. 194, American Mathematical Society. [10] Shahbaz, S. and Ahmad, M. (2009). Concomitants of order statistics for bivariate pseudo-Weibull distribution. World Appl. Sci. J. 6, 1409-1412. [11] Shahbaz, M. Q., Shahbaz, S., Mohsin, M. and Rafiq, A. (2010). On distribution of bivariate concomitants of records. Appl. Math. Lett. 23, 567-570.. 21.

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