Abstract. In this paper, we obtain recurrence relations for moment and conditional moment generating functions of generalized order statistics (gos) based on ...
Metrika (2005) 61: 199–220 DOI 10.1007/s001840400332
Recurrence relations for moment and conditional moment generating functions of generalized order statistics Essam K. AL-Hussaini, Abd EL-Baset A. Ahmad and M.A. AL-Kashif Mathematics Department, Assiut University, Assiut, Egypt. Received March 2003
Abstract. In this paper, we obtain recurrence relations for moment and conditional moment generating functions of generalized order statistics (gos) based on random samples drawn from a population whose distribution is a member of a doubly truncated class of distributions denoted by =d . Members of the class =d are characterized in Section (2) based on recurrence relations for moment generating functions (moments) of gos. In Section (3), we shall characterize members of the class =d based on recurrence relations for conditional moment generating functions (conditional moments) of gos. These results are specialized to the left, right and non-truncated cases. Ordinary order statistics and ordinary record values are also obtained as special cases of the gos. Characterizations of some members of class =d such as the Weibull, compound Weibull, Pareto, power function (beta is a special case), Gompertz and compound Gompertz distributions are given as illustrative examples. Key words: Generalized order statistics, ordinary order statistics, ordinary record values. 1 Introduction The suggestion of ‘generalized order statistics’ by Kamps (1995a) as a unifying concept for ordering random variables has drawn much attention in recent years. This is due to the fact that such generalized concept includes some important concepts that have been separately treated in statistical literature such as ordinary order statistics (oos), ordinary record values (orv0 s), k th records, Pfeifer records, sequential order statistics and progressive type II censored order statistics. For details and survey see Kamps (1995a, b, 1999)], Ahsanullah and Nevzorov (2001) and AL-Hussaini (2002).
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A statistic Xr;n;m;k is said to be the rth gos, based on a random sample of size n drawn from a population whose distribution function (df) is F ðxÞ, survival function (sf) is FðxÞ ¼ 1 F ðxÞ and probability density function (pdf) is f ðxÞ, if its density function is given by Cr1 cr 1 ½F ðxÞ f ðxÞgr1 ð1:1Þ fXr;n;m;k ðxÞ ¼ m ðF ðxÞÞ; ðr 1Þ! Q where Cr1 ¼ ri¼1 ci ; r ¼ 1; 2; . . . ; n 1, ci ¼ k þ ðn iÞðm þ 1Þ, m; k are real numbers with m 1 and k 1 and for x 2 ð0; 1Þ ( ½1 ð1 xÞmþ1 =ðm þ 1Þ; m > 1, gm ðxÞ ¼ ð1:2Þ lnð1 xÞ; m¼ 1: It should be remarked that pdf (1.1) is a particular case of the more general pdf considered by Kamps (1995), in which the components of the vector ~ ¼ ðm1 ; :::; mn1P Þ are chosen such that m1 ¼ ::: ¼ mn1 ¼ m: Hence m ci ¼ k þ n i þ n1 j¼i mj ¼ k þ ðn iÞðm þ 1Þ: Recurrence relations are interesting in their own right. They are useful in reducing the number of operations necessary to obtain a general form (of order n, say) for the function under consideration. Furthermore, they are used in characterizing distributions, which is an important area, permitting the identification of population distributions from the properties of the sample. Characterizations of distributions by properties of order statistics have been considered by many authors. Details on some of which, are given in Arnold, Balakrishnan and Nagaraja (1992), and Kamps (1998). Characterizations of particular distributions based on moments and conditional moments of order statistics, were presented by some authors such as AL-Hussaini (1991), Pakes et al (1996), Wu and Ouyang (1996), Grudzien´ and Szynal (1998), Khan and Abouammoh (1999), Asadi (1999), Ahsanullah and Nevzorov (2000), Ahmad (2001), Asadi et al (2001), Govindarajulu (2001) and Marohn (2002), among others. In recent years, several characterizations have been based on properties of record values. Details on some of which, are given in Ahsanullah (1995) and Arnold, Balakrishnan and Nagaraja (1998), Ahsanullah and Wesolowski (1998), Ahmad (1998), Aliev (1998), Raqab (1998), Bairamov (2000) and Dembin´ska and Wesolowski (2000), among others. AL-Hussaini and Ahmad (2003b) obtained Bayesian predictive intervals of future records. Statistical inference and characterization based on generalized order statistics have been studied by some authors. Ahsanullah ½ð1996Þ; ð2000Þ obtained minimum variance linear unbiased and best linear invariant estimators of the parameters of a two- parameter uniform and two-parameter exponential distributions, respectively. Keseling (1999) characterized some continuous distributions based on conditional distributions of gos. Cramer and Kamps (2000) gave relations for expectations of single and product moments of gos. Pawlas and Szynal (2001) derived recurrence relations for single and product moments of gos from Pareto, generalized Pareto and Burr distributions. Ahmad and Fawzy (2002) derived recurrence relations for moments of gos within a class of doubly truncated distributions. AL-Hussaini and Ahmad (2003a) predicted gos by using Bayes method. Suppose that a random variable (rv) X has a distribution function df of the form
Recurrence relations for moment and conditional moment generating functions
F ðxÞ ¼ 1 expðhkðxÞÞ;
x 0;
201
ð1:3Þ
where kðxÞ kðx; /Þ, is a non-negative, continuous, monotone increasing, differentiable function of x such that kðxÞ ! 0 as x ! 0þ and kðxÞ ! 1 as x ! 1. The parameters h > 0 and / (could be a vector) are such that ðh; /Þ 2 H, where H is the parameter space. The probability density function (pdf) corresponding to (1.3) is given, for x 0, by f ðxÞ ¼ hk0 ðxÞ expðhkðxÞÞ ¼ hk0 ðxÞFðxÞ:
ð1:4Þ
0
Notice that hk ðxÞ is the hazard rate function and FðxÞ ¼ 1 F ðxÞ is the corresponding survival function (sf). This class of distributions shall be denoted by =, so that = ¼ fF jF ðxÞ ¼ 1 exp½hðxÞ; x 0g:
ð1:5Þ
Class = was considered by AL-Hussaini and Osman (1997), AL-Hussaini (1999), AL-Hussaini and Ahmad (2003a, 2003b). It includes, among others, some important distributions, that are used in life testing and other areas of statistics, such as Weibull (exponential, Rayleigh as special cases ), compound Weibull (or three parameter Burr type XII) (compound exponential or Lomax and compound Rayleigh as special cases), Pareto, power function (beta as a special case, Gompertz and compound Gompertz distributions. If X is a random variable with probability density function pdf f ðxÞ and cdf F ðxÞ, then a doubly truncated density fd ðxÞ (from the left at P1 and and the right at Q1 ) is given, from (1.4), for P1 x Q1 , ðP1 0 and Q1 1), by fd ðxÞ ¼ Ad f ðxÞ ¼ Ad hk0 ðxÞ expðhkðxÞÞ;
ð1:6Þ
where Ad ¼ 1=½expðhkðP1 ÞÞ expðhkðQ1 ÞÞ;
ð1:7Þ
The corresponding doubly truncated df Fd ðxÞ and survival functions are given, respectively, for P1 x Q1 , by Fd ðxÞ ¼ Ad ½expðhkðP1 ÞÞ expðhkðxÞÞ ¼ P2 Ad expðhkðxÞÞ
ð1:8Þ
and fd ðxÞ Fd ðxÞ ¼ Q2 þ 0 ; hk ðxÞ
ð1:9Þ
where P2 ¼ Ad expðhkðP1 ÞÞ; Q2 ¼ 1 P2 ¼ Ad expðhkðQ1 ÞÞ ¼ expðhkðQ1 ÞÞ=½expðhkðQ1 ÞÞ expðhkðP1 ÞÞ:
ð1:10Þ
The doubly truncated class of distributions shall be denoted by =d , so that for P1 0 and Q1 1, =d ¼ fFd jFd ðxÞ ¼ Ad ½expðhkðP1 ÞÞ expðhkðxÞÞ; P1 x Q1 g; ð1:11Þ where Ad is given by (1.7).
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Three special cases of =d are the nontruncated, left-truncated and righttruncated classes of distributions which can be obtained by appropriately choosing the end-points P1 and Q1 . For example, the nontruncated class = is obtained by setting P1 ¼ 0, Q1 ¼ 1 and taking into account that kðxÞ ! 0 as x ! 0þ , kðxÞ ! 1 as x ! 1. This leads to Ad ¼ 1 in (1.7), P2 ¼ 1 and that (1.8) reduces to (1.3). The left-truncated class, denoted by =‘ , is obtained by setting Q1 ¼ 1 and requiring only that kðxÞ ! 1 as x ! 1. The constant in (1.7) then reduces to A‘ ¼ expðhkðP1 ÞÞ. So that the left-truncated class =‘ is given, for P1 0, by =‘ ¼ fF‘ jF‘ ðxÞ ¼ 1 exp½hfkðxÞ kðP1 Þg; x P1 g:
ð1:12Þ
The right-truncated class, denoted by =r , is obtained by setting P1 ¼ 0 and requiring only that kðxÞ ! 0 as x ! 0þ . In this case, the constant in (1.7) reduces to Ar ¼ 1=½1 expðhkðQ1 ÞÞ. So that the right-truncated class =r is given, Q1 1, by =r ¼ fFr jFr ðxÞ ¼ ½1 expðhkðxÞÞ=½1 expðhkðQ1 ÞÞ; 0 x Q1 g: ð1:13Þ Notice that in the left-truncated case, P2 ¼ 1¼)Q2 ¼ 0 and in the righttruncated case, P2 ¼ Ar ¼)Q2 ¼ 1 Ar ¼ Ar expðhkðQ1 ÞÞ.
2 Characterizations of members of the class =d based on recurrence relations for moment generating functions of gos In this section, we shall characterize members of the class =d , based on recurrence relation for moment generating functions (mgf) of gos. Theorem 1 Let X1 ; :::; Xn be independently, identically distributed (iid) random variables which are copies of a random variable X having a distribution function Fd ðxÞ defined on ½P1 ; Q1 . Suppose that X1;n;m;k ; :::; Xn;n;m;k are the gos based on X1 ; :::; Xn . Let s ¼ 1; :::; n, m and k be real numbers such that m 1, k 1. Then for integers a such that a 1, the following recurrence relation is satisfied iff X has the cdf (1.8) ðaÞ
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ " # 8 a1 a Xs;n;m;k exp½tXs;n;m;k > at Q2 cs Cs1 > > E þ > > > hcs cs Cs1 k0 ðXs;n;m;k Þ > > > > > ðaÞ ðaÞ > > < fMs;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞg; a1 ¼ a Xs;n;1;k exp½tXs;n;1;k > at > > E > 0 > kh > k ðXs;n;1;k Þ > > > > h i > > > : 1 exp hfkðXs;n;1;k Þ kðQ1 Þg ;
m > 1, ð2:1Þ
m ¼ 1,
where Cs1 ¼ Psi¼1 ci , ci ¼ ci 1 ¼ ðk þ mÞ þ ðn 1 iÞðm þ 1Þ.
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Proof. Suppose that m > 1. If X has the cdf (1.8), then the mgf of the ath a power of the sth gos, Xr;n;m;k , is given, from (1.1), by ðaÞ a Ms;n;m;k ðtÞ ¼ E expðtXs;n;m;k Þ Z Q1 Cs1 cs 1 exp½txa gs1 fd ðxÞdx; ¼ m ðFd ðxÞÞ½Fd ðxÞ ðs 1Þ! P1 Z Q1 Cs1 cs ¼ exp½txa gs1 ð2:2Þ m ðFd ðxÞÞd½½Fd ðxÞ : cs ðs 1Þ! P1 Integrating by parts, we obtain Z Q1 atCs1 ðaÞ xa1 exp½txa ½Fd ðxÞcs gs1 Ms;n;m;k ðtÞ ¼ m ðFd ðxÞÞdx cs ðs 1Þ! P1 Z ðs 1ÞCs1 Q1 exp½txa ½Fd ðxÞcs1 1 fd ðxÞgs2 þ m ðFd ðxÞÞdx: cs ðs 1Þ! P1 ðaÞ
The second term in the right hand side is Ms1;n;m;k ðtÞ, so we obtain Z Q1 atCs1 ðaÞ ðaÞ xa1 exp½txa ½Fd ðxÞcs gs1 Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ ¼ m cs ðs 1Þ! P1 ðFd ðxÞÞdx: ð2:3Þ c 1 s By rewriting ½Fd ðxÞ ¼ ½Fd ðxÞ ½Fd ðxÞ, in (2.3), then making use of (1.9), we have cs
ðaÞ
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ Z Q1 atC s1 fd ðxÞ cs 1 s1 a1 a ¼ x exp½tx ½Fd ðxÞ gm ðFd ðxÞÞ Q2 þ 0 dx; cs ðs 1Þ! P1 hk ðxÞ Z atQ2 Cs1 Q1 a1 x exp½txa ½Fd ðxÞcs 1 gs1 ¼ m ðFd ðxÞÞdx cs ðs 1Þ! P1 Z Q1 a1 atCs1 x exp½txa þ ½Fd ðxÞcs 1 fd ðxÞgs1 m ðFd ðxÞÞdx; hcs ðs 1Þ! P1 k0 ðxÞ ¼
a1 a exp½tXs;n;m;k at Xs;n;m;k E 0 hcs k ðXs;n;m;k Þ Z atQ2 Cs1 Q1 a1 þ x exp½txa ½Fd ðxÞcs gs1 m ðFd ðxÞÞdx: cs ðs 1Þ! P1
By replacing n by n 1 and k by k þ m in (2.3), we have ðaÞ
ðaÞ
Ms;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞ Z Q1 atC ¼ s1 xa1 exp½txa ½Fd ðxÞcs gs1 m ðFd ðxÞÞdx; cs ðs 1Þ! P1
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where Cs1 ¼ Psi¼1 ci , ci ¼ ci 1 ¼ ðk þ mÞ þ ðn 1 iÞðm þ 1Þ. So, we have a1 a exp½tXs;n;m;k at Xs;n;m;k ðaÞ ðaÞ E Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ ¼ 0 hcs k ðXs;n;m;k Þ Q2 cs Cs1 ðaÞ ðaÞ þ fMs;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞg: cs Cs1
Conversely, if the characterizing condition (2.1), ðm > 1Þ is satisfied, then from (2.2) and (2.3), we have Z Q1 atCs1 xa1 exp½txa ½Fd ðxÞcs gs1 m ðFd ðxÞÞdx cs ðs 1Þ! P1 Z atQ2 Cs1 Q1 a1 ¼ x exp½txa ½Fd ðxÞcs 1 gs1 m ðFd ðxÞÞdx cs ðs 1Þ! P1 Z Q1 a1 atCs1 x exp½txa þ ½Fd ðxÞcs 1 fd ðxÞgs1 m ðFd ðxÞÞdx; hcs ðs 1Þ! P1 k0 ðxÞ which can be written as Z Q1 fd ðxÞ xa1 exp½txa ½Fd ðxÞcs 1 gs1 gdx ¼ 0: m ðFd ðxÞÞfFd ðxÞ Q2 hk0 ðxÞ P1 If t ¼ 1, the extension of Mu¨ntz-Sza´sz theorem, [see Hwang and Lin (1984)], can be applied to obtain fd ðxÞ Fd ðxÞ ¼ Q2 þ 0 ; P1 x Q1 : hk ðxÞ If m ¼ 1, then Xs ¼ Xs;n;1;k . In this case ðaÞ a Ms;n;1;k ðtÞ ¼ E expðtXs;n;1;k Þ Z Q1 ks k1 ¼ exp½txa gs1 fd ðxÞdx; 1 ðFd ðxÞÞ½Fd ðxÞ ðs 1Þ! P1 Z Q1 k s1 k ¼ exp½txa gs1 1 ðFd ðxÞÞd½½Fd ðxÞ : ðs 1Þ! P1
ð2:4Þ
Integrating by parts, we have Z Q1 atk s1 ðaÞ Ms;n;1;k ðtÞ ¼ xa1 exp½txa ½Fd ðxÞk gs1 1 ðFd ðxÞÞdx ðs 1Þ! P1 Z ðs 1Þk s1 Q1 þ exp½txa ½Fd ðxÞk1 fd ðxÞgs2 1 ðFd ðxÞÞdx: ðs 1Þ! P1 It then follows that ðaÞ
ðaÞ
Ms;n;1;k ðtÞ Ms1;n;1;k ðtÞ Z Q1 atk s1 ¼ xa1 exp½txa ½Fd ðxÞk gs1 1 ðFd ðxÞÞdx: ðs 1Þ! P1
ð2:5Þ
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Rewriting ½Fd ðxÞk ¼ ½Fd ðxÞk1 ½Fd ðxÞ in (2.5), then from (1.9), we have ðaÞ
ðaÞ
Ms;n;1;k ðtÞ Ms1;n;1;k ðtÞ Z Q1 atk s1 fd ðxÞ k1 s1 a1 a ¼ x exp½tx ½Fd ðxÞ g1 ðFd ðxÞÞ Q2 þ 0 dx ðs 1Þ! P1 hk ðxÞ Z atQ2 k s1 Q1 a1 x exp½txa ½Fd ðxÞk1 gs1 ¼ 1 ðFd ðxÞÞdx ðs 1Þ! P1 Z Q1 a1 atk s1 x exp½txa þ ½Fd ðxÞk1 fd ðxÞgs1 1 ðFd ðxÞÞdx; hðs 1Þ! P1 k0 ðxÞ " # a1 a exp½tXs;n;1;k at Xs;n;1;k ¼ E 0 hk k ðXs;n;1;k Þ Z atQ2 k s1 Q1 a1 þ x exp½txa ½Fd ðxÞk1 gs1 1 ðFd ðxÞÞdx: ðs 1Þ! P1 The second term, in the right hand side, can be written, using (1.6), as Z atQ2 k s1 Q1 xa1 exp½txa fd ðxÞ ½Fd ðxÞk1 gs1 1 ðFd ðxÞÞdx ðs 1Þ! P1 Ad hk0 ðxÞ exp½hkðxÞ " # a1 a exp½tXs;n;1;k þ hkðXs;n;1;k Þ atQ2 Xs;n;1;k E ¼ khAd k0 ðXs;n;1;k Þ " # a1 a Xs;n;1;k exp½tXs;n;1;k þ hkðXs;n;1;k Þ at ¼ exp½hkðQ1 ÞE : kh k0 ðXs;n;1;k Þ Therefore, ðaÞ
ðaÞ
Ms;n;1;k ðtÞ Ms1;n;1;k ðtÞ ( ) a1 a i h Xs;n;1;k exp½tXs;n;1;k at ¼ E 1 exp hfkðXs;n;1;k Þ kðQ1 Þg kh k0 ðXs;n;1;k Þ On the other hand, if the characterizing condition (2.1), ðm ¼ 1Þ is satisfied, then from (2.4) and (2.5), we have Z Q1 atk s1 xa1 exp½txa ½Fd ðxÞk gs1 1 ðFd ðxÞÞdx ðs 1Þ! P1 Z atQ2 k s1 Q1 a1 ¼ x exp½txa ½Fd ðxÞk1 gs1 1 ðFd ðxÞÞdx ðs 1Þ! P1 Z Q1 a1 atk s1 x exp½txa ½Fd ðxÞk1 fd ðxÞgs1 þ 1 ðFd ðxÞÞdx; hðs 1Þ! P1 k0 ðxÞ which can be written as Z Q1 fd ðxÞ k1 s1 a1 a x exp½tx ½Fd ðxÞ g1 ðFd ðxÞÞ Fd ðxÞ Q2 0 dx ¼ 0: hk ðxÞ P1
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Applying the extension of Mu¨ntz-Sza´sz theorem [see, Hwang and Lin (1984)], we obtain fd ðxÞ Fd ðxÞ ¼ Q2 þ 0 ; hk ðxÞ
P1 x Q1 :
By differentiating both sides of condition (2.1) with respect to t and then setting t ¼ 0, we obtain the following recurrence relation for moments of gos ðaÞ
ðaÞ
ls;n;m;k ls1;n;m;k h X a1 i 8 ðaÞ ðaÞ Q2 c Cs1 s;n;m;k > < hca s E k0 ðXs;n;m;k Þ þ cs Cs fls;n1;m;kþm ls1;n1;m;kþm g; s1 ¼ n a1 h io > : a E Xs;n;1;k 1 exp hfkðX Þ kðQ Þg ; 0 s;n;1;k 1 kh k ðXs;n;1;k Þ
m > 1, ð2:6Þ m ¼ 1,
ðaÞ
a where ls;n;m;k ¼ E½Xs;n;m;k .
Special cases (1) If we put m ¼ 0 and k ¼ 1 in (2.1) and (2.6), ðoosÞ, [cs ¼ n s þ 1 and Xs;n;m;k Xs:n ], we have " # a1 a at Xs:n exp½tXs:n ðaÞ ðaÞ E Ms:n ðtÞ Ms1:n ðtÞ ¼ hðn s þ 1Þ k0 ðXs:n Þ þ
nQ2 ðaÞ ðaÞ fMs:n1 ðtÞ Ms1:n1 ðtÞg; nsþ1
ð2:7Þ
and ðaÞ
lðaÞ s:n ls1:n ¼
h X a1 i a nQ2 ðaÞ ðaÞ E 0 s:n fl ls1:n1 g: ð2:8Þ þ hðn s þ 1Þ k ðXs:n Þ n s þ 1 s:n1
Expressions (2.7) and (2.8) agree with the two expressions, in the oos case, given by (2.12) in AL-Hussaini et al (2004a), when h ¼ 1. (2) If we put k ¼ 1 and m ¼ 1 in (2.1) and (2.6) ðorv0 sÞ [cs ¼ k and Xs;n;m;k XU ðsÞ ], we have ðaÞ
ðaÞ
MU ðsÞ ðtÞ MU ðs1Þ ðtÞ ( a1 ) i h XU ðsÞ exp½tXUa ðsÞ at ¼ E 1 exp hfkðXU ðsÞ Þ kðQ1 Þg h k0 ðXU ðsÞ Þ and ðaÞ lðsÞ
ðaÞ lðs1Þ
( ) i h XUa1 a ðsÞ ¼ E 0 : 1 exp hfkðXU ðsÞ Þ kðQ1 Þg h k ðXU ðsÞ Þ
ð2:9
ð2:10Þ
Expressions (2.9) and (2.10) agree with the two expressions, in the orv0 s case, given by (2.12) in AL-Hussaini et al (2004b), when h ¼ 1.
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2.1. Left, right and non-truncated cases for class = Special doubly truncated cases are the left, right and non-truncated distributions. Recurrence relations of moment generating functions and moments of gos; oos and orv0 s corresponding to each one of such cases characterizes the class =. The following corollary characterizes the left truncated and (non-truncated) distributions of class = by recurrence relations for the moment generating functions and moments. Corollary 2.1 Let X1 ; . . . ; Xn be independently, identically distributed (iid) random variables which are copies of a random variable X having a distribution function FX ðxÞ defined on ½P1 ; Q1 . Suppose that X1;n;m;k ; . . . ; Xn;n;m;k are the gos based on X1 ; . . . ; Xn . Let s ¼ 1; . . . ; n, m and k be real numbers such that m 1, k 1. Then for integer a such that a 1, the necessary and sufficient condition for X to be distributed as (1.12) is obtained by setting Q2 ¼ 0 in (2.1) and (2.6) as follows " # a1 a Xs;n;m;k exp½tXs;n;m;k at ðaÞ ðaÞ Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ ¼ E ; m 1; ð2:11Þ hcs k0 ðXs;n;m;k Þ or ðaÞ
ðaÞ
ls;n;m;k ls1;n;m;k ¼
a1 i a h Xs;n;m;k E 0 ; hcs k ðXs;n;m;k Þ
m 1:
ð2:12Þ
Special cases The left truncated case is obtained from the doubly truncated case by setting Q2 ¼ 0. So that the relations corresponding to (2.7), (2.8), (2.9) and (2.10) are given by h X a1 exp½tX a i at ðaÞ ðaÞ s:n E s:n 0 ðtÞ Ms1:n ðtÞ ¼ ; ð2:13Þ Ms:n hðn s þ 1Þ k ðXs:n Þ h X a1 i a ðaÞ E 0 s:n lðaÞ ; ð2:14Þ s:n ls1:n ¼ hðn s þ 1Þ k ðXs:n Þ ðaÞ
ðaÞ
MU ðsÞ ðtÞ MU ðs1Þ ðtÞ ¼
a1 a at h XU ðsÞ exp½tXU ðsÞ i E ; h k0 ðXU ðsÞ Þ
ð2:15Þ
and ðaÞ lðsÞ
ðaÞ lðs1Þ
a1 a h XU ðsÞ i ¼ E 0 : h k ðXU ðsÞ Þ
ð2:16Þ
In the non-truncated case (also Q2 ¼ 0) the relations are obtained as in the left truncated case. The next corollary characterizes the right truncated distributions of class = by recurrence relations for the moment generating functions and moments. Corollary 2.2 Let X1 ; :::; Xn be independently, identically distributed (iid) random variables which are copies of a random variable X having a distribution function FX ðxÞ defined on ½P1 ; Q1 . Suppose that X1;n;m;k ; :::; Xn;n;m;k are the gos
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based on X1 ; :::; Xn . Let s ¼ 1; :::; n, m and k be real numbers such that m 1, k 1. Then for integer a such that a 1, the necessary and sufficient condition for X to be distributed as (1.13) is that, ðaÞ
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ 8 h X a1 exp½tX a i s;n;m;k s;n;m;k at > E > > hcs k0 ðXs;n;m;k Þ > > < ðaÞ ðaÞ c AC ¼ þ scs Cr s1 fMs;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞg; s1 > > > n a1 io h > a > : at E Xs;n;1;k0 exp½tXs;n;1;k 1 exp hfkðXs;n;1;k Þ kðQ1 Þg ; kh k ðXs;n;1;k Þ
m > 1, m ¼ 1, ð2:17Þ
or ðaÞ
ðaÞ
ls;n;m;k ls1;n;m;k 8 h X a1 i ðaÞ ðaÞ c A C > þ sc Cr s1 fls;n1;m;kþm ls1;n1;m;kþm g; < hca s E k0 ðXs;n;m;k s;n;m;k Þ s s1 ¼ io h a1 > a n Xs;n;1;k : ; kh E k0 ðXs;n;1;k Þ 1 exp hfkðXs;n;1;k Þ kðQ1 Þg
m > 1, m ¼ 1, ð2:18Þ
Special cases In the right truncated case, relations (2.7), (2.8), (2.9) and (2.10) reduce, respectively to, a1 a at X exp½tXs:n ðaÞ ðaÞ Ms:n E s:n 0 ðtÞ Ms1:n ðtÞ ¼ hðn s þ 1Þ k ðXs:n Þ n o nAr ðaÞ ðaÞ þ Ms:n1 ðtÞ Ms1:n1 ðtÞ ; ð2:19Þ ðn s þ 1Þ ðaÞ
lðaÞ s:n ls1:n ¼
ðaÞ
h X a1 i a E 0 s:n hðn s þ 1Þ k ðXs:n Þ n o nAr ðaÞ ðaÞ þ ls:n1 ls1:n1 ; ðn s þ 1Þ
ð2:20Þ
ðaÞ
MU ðsÞ ðtÞ MU ðs1Þ ðtÞ
( a1 ) h i XU ðsÞ exp½tXUa ðsÞ at ¼ E 1 exp hfkðXU ðsÞ Þ kðQ1 Þg h k0 ðXU ðsÞ Þ ð2:21Þ
and ðaÞ lðsÞ
ðaÞ lðs1Þ
( ) i h XUa1 a ðsÞ ¼ E 0 : 1 exp hfkðXU ðsÞ Þ kðQ1 Þg h k ðXU ðsÞ Þ
ð2:22Þ
Recurrence relations for moment and conditional moment generating functions
209
2.2. Examples (1) Doubly truncated Weibull distribution ða; hÞ kðxÞ ¼ xa , and k0 ðxÞ ¼ axa1 . From (2.1), ðaÞ
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ h i 8 at aa a > E X exp½tX > s;n;m;k s;n;m;k ahcs > > > < n o ðaÞ ðaÞ Q c C ¼ þ c2 Cs s1 Ms;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞ ; m > 1, s s1 > > > n h io > > a : at E X aa exp½tX a a ; m ¼ 1. s;n;1;k s;n;1;k 1 exp hfXs;n;1;k ðQ1 Þ g ahk From (2.6), ðaÞ
ðaÞ
ls;n;m;k ls1;n;m;k n o 8 ðaaÞ ðaÞ ðaÞ Q c C a > ls;n;m;k þ c2 Cs s1 ls;n1;m;kþm ls1;n1;m;kþm ; < ahc s s s1 ¼ n io h > a a : a E X aa ðQ Þ g ; 1 exp hfX 1 s;n;m;k s;n;1;k ahk
m > 1, m ¼ 1.
From (2.7), ðaÞ
ðaÞ Ms:n ðtÞ Ms1:n ðtÞ ¼
h i at aa a E Xs:n exp½tXs:n ahðn s þ 1Þ o nQ2 n ðaÞ ðaÞ þ Ms:n1 ðtÞ Ms1:n1 ðtÞ : nsþ1
From (2.8), ðaÞ
lðaÞ s:n ls1:n ¼
o a nQ2 n ðaÞ ðaÞ lðaaÞ ls:n1 ls1:n1 : þ s:n ahðn s þ 1Þ nsþ1
Recurrence relations (2.9) and (2.10) reduce, respectively, to ðaÞ
ðaÞ
MU ðsÞ ðtÞ MU ðs1Þ ðtÞ ¼
h io at n aa E XU ðsÞ exp½tXUa ðsÞ 1 exp hfXUa ðsÞ ðQ1 Þa g ah
and ðaÞ
ðaÞ
lðsÞ lðs1Þ ¼
io h a n aa E XU ðsÞ 1 exp hfXUa ðsÞ ðQ1 Þa g : ah
(2) Doubly truncated compound Weibull distribution (three-parameter Burr type XII distribution) ðh; b; dÞ
kðxÞ ¼ ln 1 þ ðxd =bÞ , then k0 ðxÞ ¼ dxd1 =ðb þ xd Þ. From (2.1) ,
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E.K. AL-Hussaini et al. ðaÞ
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ h i 8 at ad d a > E Xs;n;m;k ðb þ Xs;n;m;k Þ exp½tXs;n;m;k > hdc > s > > o > > Q2 cs Cs1 n ðaÞ ðaÞ > > Ms;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞ ; < þ cs C s1 ¼ n > at ad d a > > > khd E Xs;n;1;k ðb þ Xs;n;1;k Þ exp½tXs;n;1;k > > > h io > > d : 1 exp hfðb þ X d Þ=ðb þ ðQ Þ Þg ; 1 s;n;1;k
m > 1,
m ¼ 1.
From (2.6), ðaÞ
ðaÞ
ls;n;m;k ls1;n;m;k n o 8 Q c C a ad a > bl þ l þ c2 Cs s1 > s;n;m;k s;n;m;k hdc s s s1 > > > n o > > ðaÞ ðaÞ > > < ls;n1;m;kþm ls1;n1;m;kþm ; ¼ n > a ad d > > E Xs;n;1;k ðb þ Xs;n;1;k Þk0 ðXs;n;1;k Þ > khd > > > h io > > d : 1 exp hfðb þ X d ; s;n;m;k Þ=ðb þ ðQ1 Þ Þg From (2.7) and (2.8), ðaÞ
ðaÞ Ms:n ðtÞ Ms1:n ðtÞ ¼
m > 1,
m ¼ 1.
h i at ad d a E Xs:n ðb þ Xs:n Þ exp½tXs:n hdðn s þ 1Þ o nQ2 n ðaÞ ðaÞ þ Ms:n1 ðtÞ Ms1:n1 ðtÞ nsþ1
and ðaÞ
½hdðn s þ 1Þ alðaÞ s:n hdðn s þ 1Þls1:n n o ðaÞ ðaÞ ¼ ablðadÞ þ nhdQ2 ls:n1 ls1:n1 : s:n From (2.9) and (2.10), ðaÞ
ðaÞ
at n ad E XU ðsÞ ðb þ XUd ðsÞ Þ exp½tXUa ðsÞ hd io h 1 exp hfðb þ XUd ðsÞ Þ=ðb þ ðQ1 Þd Þg
MU ðsÞ ðtÞ MU ðs1Þ ðtÞ ¼
and ðaÞ
ðaÞ
lðsÞ lðs1Þ ¼
h io a n ad E XU ðsÞ ðb þ XUd ðsÞ Þ 1 exp hfðb þ XUd ðsÞ Þ=ðb þ ðQ1 Þd Þg : hd
(3) Doubly truncated Pareto distribution ða; hÞ kðxÞ ¼ lnða=xÞ, and k0 ðxÞ ¼ 1=x. From (2.1),
Recurrence relations for moment and conditional moment generating functions ðaÞ
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ 8 h i at a a > E X exp½tX > s;n;m;k s;n;m;k hcs > > > < n o ðaÞ ðaÞ Q c C ¼ þ c2 Cs s1 Ms;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞ ; s s1 > > > n h io > > a : at E X a exp½tX 1 exp hflnðX =Q Þg ; s;n;1;k 1 s;n;1;k s;n;1;k kh From (2.6),
ðaÞ
211
ðaÞ
ls;n;m;k ls1;n;m;k
8 Q2 cs Cs1 a ðaÞ > > hcs ls;n;m;k þ cs Cs1 > > > n o > > ðaÞ > lðaÞ < s;n1;m;kþm ls1;n1;m;kþm ; n ¼ > a a > E Xs;n;1;k > kh > > > h io > > : 1 exp hflnðXs;n;1;k =Q1 Þg ;
m > 1, m ¼ 1.
m > 1,
m ¼ 1.
From (2.7), ðaÞ
ðaÞ Ms:n ðtÞ Ms1:n ðtÞ h i o at nQ2 n ðaÞ ðaÞ a a E Xs:n Ms:n ðtÞ Ms1:n ðtÞ : ¼ expðtXs:n Þ þ hðn s þ 1Þ nsþ1
From (2.8),
n o ðaÞ ðaÞ ðaÞ ½hðn s þ 1Þ alðaÞ s:n hðn s þ 1Þls1:n ¼ nhQ2 ls:n1 ls1:n1 :
Recurrence relations (2.9) and (2.10) reduce, respectively, to h io at n ðaÞ ðaÞ MU ðsÞ ðtÞ MU ðs1Þ ðtÞ ¼ E XUa ðsÞ exp½tXUa ðsÞ 1 exp hflnðXU ðsÞ =Q1 Þg h and io h a n ðaÞ ðaÞ lðsÞ ðtÞ lðs1Þ ðtÞ ¼ E XUa ðsÞ 1 exp hflnðXU ðsÞ =Q1 Þg : h (4) Doubly truncated beta distribution ðaÞ kðxÞ ¼ lnð1=ð1 xÞÞ; From (2.1), ðaÞ
h ¼ 1=a, and k0 ðxÞ ¼ 1=ð1 xÞ.
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ h i 8 ata a1 a > E X ð1 X Þ exp½tX > s;n;m;k s;n;m;k s;n;m;k > cs > > > o n > > ðaÞ ðaÞ Q c C > < þ c2s Cs s1 Ms;n1;m;kþm ðtÞ Ms1;n1;m;kþm ðtÞ ; s1 ¼ n > ata a1 a > > > k E Xs;n;1;k ð1 Xs;n;1;k Þexp½tXs;n;1;k > > > h io > > : 1 exp hfln½ð1 Q1 Þ=ð1 Xs;n;1;k Þg ;
m > 1,
m ¼ 1.
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From (2.6), o 8 n aa > la1 las;n;m;k > s;n;m;k c s > > n o > > ðaÞ > < þ Qc2 cCs C s1 lðaÞ ls1;n1;m;kþm ; s;n1;m;kþm ðaÞ ðaÞ ns s1 ls;n;m;k ls1;n;m;k ¼ > aa a1 > > k E Xs;n;1;k ð1 Xs;n;1;k Þ > > io h > > : ; 1 exp hfln½ð1 Q1 Þ=ð1 Xs;n;1;k Þg
m > 1,
m ¼ 1.
From (2.7) and (2.8), ðaÞ
ðaÞ ðtÞ Ms1:n ðtÞ ¼ Ms:n
h i ata a1 a E Xs:n ð1 Xs:n Þ exp½tXs:n ðn s þ 1Þ o nQ2 n ðaÞ ðaÞ Ms:n1 ðtÞ Ms1:n1 ðtÞ þ nsþ1
and ðaÞ
½ðn s þ 1Þ þ aalðaÞ s:n ðn s þ 1Þls1:n o aa nQ2 n ðaÞ ðaÞ la1 l ¼ þ l s1:n1 : ðn s þ 1Þ s:n n s þ 1 s:n1 From (2.9) and (2.10),
n ðaÞ ðaÞ a MU ðsÞ ðtÞ MU ðs1Þ ðtÞ ¼ ataE XUa1 ðsÞ ð1 XU ðsÞ Þexp½tXU ðsÞ h io 1 exp hfln½ð1 Q1 Þ=ð1 XU ðsÞ Þg
and n h io ðaÞ ðaÞ lðsÞ lðs1Þ ¼ aaE XUa1 ð1 X Þ 1 exp hfln½ð1 Q Þ=ð1 X Þg : 1 U ðsÞ U ðsÞ ðsÞ (5) Doubly truncated Gompertz distribution ða; bÞ kðxÞ ¼ ½exp½bx 1=b; From (2.1), ðaÞ
ðaÞ
h ¼ 1=a, then k0 ðxÞ ¼ exp½bx.
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ h i 8 ata a1 a > E X exp½tX bX s;n;m;k > s;n;m;k s;n;m;k cs > > > n o > > ðaÞ > þ Q2 cs Cs1 M ðaÞ > ðtÞ M ðtÞ ; m > 1, < s;n1;m;kþm s1;n1;m;kþm cs Cs1 ¼ n > ata a1 a > > > k E Xs;n;1;k exp½tXs;n;1;k bXs;n;1;k > > > io h > > : 1 exp hfbðX ; m ¼ 1. s;n;1;k Q1 Þg From (2.6),
Recurrence relations for moment and conditional moment generating functions ðaÞ
213
ðaÞ
ls;n;m;k ls1;n;m;k h i 8 aa a1 > E X exp½bX > s;n;m;k s;n;m;k cs > > > > n o > > ðaÞ ðaÞ Q c C > < þ c2s Cs s1 ls;n1;m;kþm ls1;n1;m;kþm ; s1 ¼ n > aa a1 > > > k E Xs;n;1;k exp½bXs;n;1;k > > > h io > > : 1 exp hfbðXs;n;1;k Q1 Þg ;
m > 1,
m ¼ 1.
From (2.7) and (2.8), ðaÞ
ðaÞ Ms:n ðtÞ Ms1:n ðtÞ ¼
h i ata a1 a E Xs:n exp½tXs:n bXs:n ðn s þ 1Þ o nQ2 n ðaÞ ðaÞ þ Ms:n1 ðtÞ Ms1:n1 ðtÞ nsþ1
and ðaÞ
lðaÞ s:n ls1:n ¼
h i o aa nQ2 n ðaÞ ðaÞ a1 E Xs:n ls:n1 ls1:n1 : exp½bXs:n þ ðn s þ 1Þ nsþ1
From (2.9) and (2.10), ðaÞ
ðaÞ
MU ðsÞ ðtÞ MU ðs1Þ ðtÞ h io n a ¼ ataE XUa1 ðsÞ exp½tXU ðsÞ bXU ðsÞ 1 exp hfbðXU ðsÞ Q1 Þg and n h io ðaÞ ðaÞ lðsÞ lðs1Þ ¼ aaE XUa1 exp½bX 1 exp hfbðX Q Þg : 1 U ðsÞ U ðsÞ ðsÞ (6) Doubly truncated Compound Gompertz distribution ða; b; hÞ .h i kðxÞ ¼ ln½1 þ ððexpðaxÞ 1Þ=abÞ, then k0 ðxÞ ¼ a 1 þ ðab 1Þ exp½ax . From (2.1), ðaÞ
ðaÞ
Ms;n;m;k ðtÞ Ms1;n;m;k ðtÞ 8 h i i h at a1 a > E Xs;n;m;k 1 þ ðab 1Þexp½aXs;n;m;k exp½tXs;n;m;k > > ahc s > > > o n > > ðaÞ Q > þ 2 cs Cs1 M ðaÞ > ðtÞ M ðtÞ ; m > 1, > s;n1;m;kþm s1;n1;m;kþm c C > s s1 > > < n i h at a1 a ¼ ahk E Xs;n;1;k 1 þ ðab 1Þ exp½aXs;n;1;k exp½tXs;n;1;k > > > h n > > > > 1 exp h ln ½ðab 1Þ > > > > > oio > > : þ expðaXs;n;1;k Þ ½ðab 1Þ þ expðaQ1 Þ ; m ¼ 1.
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From (2.6), ðaÞ
ðaÞ
ls;n;m;k ls1;n;m;k h ii h 8 a a1 > E X 1 þ ðab 1Þ exp½aX > s;n;m;k s;n;m;k ahcs > > > n o > > ðaÞ > Q2 cs Cs1 ðaÞ > þ l l > s;n1;m;kþm s1;n1;m;kþm ; cs Cs1 > > > < n i h a a1 ¼ ahk E Xs;n;1;k 1 þ ðab 1Þ exp½aXs;n;1;k > > > h n > > > 1 exp h ln ½ðab 1Þ > > > > > . oio > > : þ expðaXs;n;1;k Þ ½ðab 1Þ þ expðaQ1 Þ ;
m > 1,
m ¼ 1.
From (2.7) and (2.8), ðaÞ
h i i h at a1 a E Xs:n 1þðab1Þexp½aXs:n exp½tXs:n ahðnsþ1Þ o nQ2 n ðaÞ ðaÞ Ms:n1 ðtÞMs1:n1 ðtÞ þ nsþ1
ðaÞ Ms:n ðtÞMs1:n ðtÞ ¼
and h ii h a a1 E Xs:n 1 þ ðab 1Þ exp½aXs:n ahðn s þ 1Þ o nQ2 n ðaÞ ðaÞ þ ls:n1 ls1:n1 : nsþ1 From (2.9) and (2.10), i h at n ðaÞ ðaÞ a 1 þ ðab 1Þ exp½aX MU ðsÞ ðtÞ MU ðs1Þ ðtÞ ¼ E XUa1 U ðsÞ exp½tXU ðsÞ ðsÞ ah h n . oio 1 exp h ln ½ðab 1Þ þ expðaXU ðsÞ Þ ½ðab 1Þ þ expðaQ1 Þ ðaÞ
lðaÞ s:n ls1:n ¼
and
i h a n ðaÞ ðaÞ lðsÞ lðs1Þ ¼ E XUa1 ðsÞ 1 þ ðab 1Þ exp½aXU ðsÞ ah h n . oio 1 exp h ln ½ðab 1Þ þ expðaXU ðsÞ Þ ½ðab 1Þ þ expðaQ1 Þ :
3 Characterizations of Members of the Class =d Based on Recurrence Relations for Conditional Moment Generating Functions of gos By following similar steps as used in Theorem 1, recurrence relation for conditional moment generating functions of gos based on random sample drawn from a population whose df is a member of =d can be shown to be of the same form as (2.1), after imposing the condition. Such recurrence relation may be used in characterizing members of =d . The conditional version of Theorem 1 shall be stated, without proof, as follows:
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Theorem 2 Let X1 ; . . . ; Xn be independently, identically distributed ðiidÞ random variables which are copies of a random variable X having a distribution function Fd ðxÞ defined on ½P1 ; Q1 . Suppose that X1;n;m;k ; . . . ; Xn;n;m;k are the gos based on X1 ; . . . ; Xn . Let r, s be two integers such that 1 r < s n, m and k are real numbers such that m 1, k 1. Then for integer a such that a 1, the following recurrence relation is satisfied iff X has the cdf ð1:8Þ. a a MXs;n;m;k jXr;n;m;k ðtjyÞ MXs1;n;m;k jXr;n;m;k ðtjyÞ 8 h X a1 expðtX a Þ i s;n;m;k s;n;m;k at > > E ¼ y X 0 r;n;m;k > k ðXs;n;m;k Þ > hcs > n o > > > < þBQ2 MX a a ðtjyÞ M ðtjyÞ ; m > 1, jX X jX r;n1;m;kþm r;n1;m;kþm s;n1;m;kþm s1;n1;m;kþm ¼ n X a1 exp½tX a > s;n;1;k s;n;1;k at > > kh E > k0 ðXs;n;1;k Þ > > h i > > : 1 exp hfkðXs;n;1;k Þ kðQ1 Þg Xr;n;1;k ¼ yg; m ¼ 1,
ð3:1Þ Cs1 Cr1 cs =ðCs1 Cr1 cs Fd ðyÞÞ.
where B ¼ By differentiating both sides of condition (3.1) with respect to t and then setting t ¼ 0, we have the following recurrence relation for conditional moments of gos i h i h a Xr;n;m;k ¼ y E X a Xr;n;m;k ¼ y E Xs;n;m;k s1;n;m;k h X a1 i 8 s;n;m;k a > > hcs E k0 ðXs;n;m;k Þ Xr;n;m;k ¼ y þ BQ2 > > > n h i > > > a Xr;n1;m;kþm ¼ y > E X > s;n1;m;kþm > > > < h io a Xr;n1;m;kþm ¼ y ; m > 1, ð3:2Þ ¼ E Xs1;n1;m;kþm > > > h > a1 > a n Xs;n;1;k > > E 1 exp hfkðXs;n;1;k Þ 0 > kh k ðX Þ > s;n;1;k > > i o > > : kðQ1 Þg Xr;n;1;k ¼ y ; m ¼ 1. Remark Similar recurrence relations if the more general
could be obtained a a expectations E exp twðX Þ or E exp twðX s;n;m;k Þ Xr;n;m;k a
as;n;m;k ¼ y are used instead of E exp tXs;n;m;k and E exp tXs;n;m;k Xr;n;m;k ¼ y . Special cases (1) If we put m ¼ 0 and k ¼ 1 (case of oos) in (3.1) and (3.2), [cs ¼ n s þ 1 and Xs;n;m;k Xs:n ], we have h X a1 exp½tX a i at s:n s:n a E MXs:na jXr:n ðtjyÞ MXs1:n ðtjyÞ ¼ ¼ y X r:n jXr:n hðn s þ 1Þ k0 ðXs:n Þ n o ðn rÞQ2 a a þ ðtjyÞ M ðtjyÞ ; ð3:3Þ M X jX X jX r:n1 r:n1 s:n1 s1:n1 ðn s þ 1ÞFd ðyÞ
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and h X a1 i a E 0 s:n Xr:n ¼ y hðn s þ 1Þ k ðXs:n Þ n h i h io a Xr:n1 ¼ y E X a E Xs:n1 : ð3:4Þ s1:n1 Xr:n1 ¼ y
h i h i a a Xr:n ¼ y ¼ E Xs:n Xr:n ¼ y E Xs1:n þ
ðn rÞQ2 ðn s þ 1ÞFd ðyÞ
Expressions (3.3) and (3.4) agree with the corresponding results given in Theorem 1 of AL-Hussaini et al (2004c). (2) If we put k ¼ 1 and m ¼ 1 (case of orv0 s) in (3.1) and (3.2) [cs ¼ k and Xs;n;m;k XU ðsÞ , we have ( a1 XU ðsÞ exp½tXUa ðsÞ at MXUa ðsÞ jXU ðrÞ ðtjyÞ MXUa ðs1Þ jXU ðrÞ ðtjyÞ ¼ E h k0 ðXU ðsÞ Þ h i o 1 exp hfkðXU ðsÞ Þ kðQ1 Þg XU ðrÞ ¼ y ð3:5Þ and
( h i h i a XUa1 ðsÞ a a E XU ðsÞ XU ðrÞ ¼ y E XU ðs1Þ XU ðrÞ ¼ y ¼ E 0 h k ðXU ðsÞ Þ h i o 1 exp hfkðXU ðsÞ Þ kðQ1 Þg XU ðrÞ ¼ y :
ð3:6Þ
Expressions (3.5) and (3.6) agree with the corresponding results given in Theorem 2 of AL-Hussaini et al (2004c). 3.1. Left, right and non-truncated cases for class = Recurrence relations of conditional moment generating functions and conditional moments of gos; oos and orv0 s corresponding to each one of such cases characterize class =. The following corollary characterizes the left truncated and non-truncated distributions for class = by recurrence relations for the conditional moment generating functions and conditional moments. Corollary 3.1 Let X1 ; . . . ; Xn be independently, identically distributed ðiidÞ random variables which are copies of a random variable X having a distribution function FX ðxÞ defined on ½P1 ; Q1 . Suppose that X1;n;m;k ; . . . ; Xn;n;m;k are the gos based on X1 ; . . . ; Xn . Let s ¼ 1; . . . ; n, m and k be real numbers such that m 1, k 1. Then for integer a such that a 1, the necessary and sufficient condition for a random variable X to be distributed as (1.12) is that, a a MXs;n;m;k jXr;n;m;k ðtjyÞ MXs1;n;m;k jXr;n;m;k ðtjyÞ a1 a i at h Xs;n;m;k exp½tXs;n;m;k ¼ E jXr;n;m;k ¼ y ; 0 hcs k ðXs;n;m;k Þ
m 1;
ð3:7Þ
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or
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h i h i a a E Xs;n;m;k jXr;n;m;k ¼ y E Xs1;n;m;k jXr;n;m;k ¼ y ¼
a1 i a h Xs;n;m;k E 0 jXr;n;m;k ¼ y ; hcs k ðXs;n;m;k Þ
m 1:
ð3:8Þ
Special cases In the left truncated case, relations (3.3), (3.4), (3.5) and (3.6) reduce, respectively to, h X a1 expðtX a Þ i at sþ1:n a a jX ðtjyÞ ¼ E sþ1:n0 MXsþ1:n jXr:n ¼ y ; ð3:9Þ jXr:n ðtjyÞ MXs:n r:n hðn sÞ k ðXsþ1:n Þ i h i h a a jXr:n ¼ y E Xs:n jXr:n ¼ y ¼ E Xsþ1:n
MXUa ðsÞ jXU ðrÞ ðtjyÞ MXUa ðs1Þ jXU ðrÞ ðtjyÞ ¼
h X a1 i a E 0 sþ1:n jXr:n ¼ y ; ð3:10Þ hðn sÞ k ðXsþ1:n Þ
a1 a i at h XU ðsÞ expðtXU ðsÞ Þ E ¼ y ; ð3:11Þ jX U ðrÞ h k0 ðXU ðsÞ Þ
and h i h i a h X a1 i U ðsÞ E XUa ðsÞ jXU ðrÞ ¼ y E XUa ðs1Þ jXU ðrÞ ¼ y ¼ E 0 jXU ðrÞ ¼ y : ð3:12Þ h k ðXU ðsÞ Þ In the non-truncated case the relations are obtained as in the left truncated case. The next corollary characterizes the right truncated distribution for class = by using recurrence relations for the conditional moment generating functions and conditional moments. Corollary 3.2 Let X1 ; . . . ; Xn be independently, identically distributed ðiidÞ random variables which are copies of a random variable X having a distribution function FX ðxÞ defined on ½P1 ; Q1 . Suppose that X1;n;m;k ; . . . ; Xn;n;m;k are the gos based on X1 ; . . . ; Xn . Let s ¼ 1; . . . ; n, m and k be real numbers such that m 1, k 1. Then for integer a such that a 1, the necessary and sufficient condition for X to be distributed as (1.13) is that, a a MXs;n;m;k jXr;n;m;k ðtjyÞ MXs1;n;m;k jXr;n;m;k ðtjyÞ 8 h X a1 expðtX a Þ i at > E s;n;m;kk0 ðXs;n;m;ks;n;m;k ¼ y > X r;n;m;k > hc Þ s > > n > > > > a þBAr MXs;n1;m;kþm > jXr;n1;m;kþm ðtjyÞ > > < o a ¼ m > 1, MXs1;n1;m;kþm jXr;n1;m;kþm ðtjyÞ ; > > n h > a > X a1 exp½tXs;n;1;k > at > E s;n;1;k 1 exp hfkðXs;n;1;k Þ 0 > kh > k ðX Þ s;n;1;k > > i o > > : kðQ1 Þg Xr;n;1;k ¼ y ; m ¼ 1,
ð3:13Þ
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h i h i a a E Xs;n;m;k Xr;n;m;k ¼ y E Xs1;n;m;k Xr;n;m;k ¼ y 8 h X a1 i s;n;m;k a > E ¼ y X > 0 r;n;m;k > hcs k ðXs;n;m;k Þ > > i n h > > > a > þBA ¼ y E X X r r;n1;m;kþm > s;n1;m;kþm > > < h io a ¼ E Xs1;n1;m;kþm m > 1, Xr;n1;m;kþm ¼ y ; > > n h > a1 >a Xs;n;1;k > > > > kh E k0 ðXs;n;1;k Þ 1 exp hfkðXs;n;1;k Þ > > i o > > : kðQ Þg X m ¼ 1. 1 r;n;1;k ¼ y ;
ð3:14Þ
Special cases In the right truncated truncated case, relations (3.3), (3.4), (3.5) and (3.6) reduce, respectively to, h X a1 exp½tX a i at s:n a MXs:na jXr:n ðtjyÞ MXs1:n E s:n 0 jXr:n ¼ y jXr:n ðtjyÞ ¼ hðn s þ 1Þ k ðXs:n Þ n o ðn rÞAr a a þ ðtjyÞ M ðtjyÞ ; ð3:15Þ M X jX X jX r:n1 r:n1 s:n1 s1:n1 ðn s þ 1ÞFd ðyÞ h X a1 i a E 0 s:n jXr:n ¼ y hðn s þ 1Þ k ðXs:n Þ n h i h io a a jXr:n1 ¼ y E Xs1:n1 jXr:n1 ¼ y ; ð3:16Þ E Xs:n1
i h i h a a jXr:n ¼ y E Xs1:n jXr:n ¼ y ¼ E Xs:n þ
ðn rÞAr ðn s þ 1ÞFd ðyÞ
MXUa ðsÞ jXU ðrÞ ðtjyÞ MXUa ðs1Þ jXU ðrÞ ðtjyÞ ( a1 ) i h XU ðsÞ exp½tXUa ðsÞ at ¼ E 1 exp hfkðXU ðsÞ Þ kðQ1 Þg XU ðrÞ ¼ y ð3:17Þ h k0 ðXU ðsÞ Þ and
i h i h E XUa ðsÞ jXU ðrÞ ¼ y E XUa ðs1Þ jXU ðrÞ ¼ y ( ) i h XUa1 a ðsÞ ¼ E 0 1 exp hfkðXU ðsÞ Þ kðQ1 Þg XU ðrÞ ¼ y : h k ðXU ðsÞ Þ
ð3:18Þ
Applications to members of the class =d can be made by the appropriate choices of kðxÞ, as was done in the examples given in Sec. 2 for different choices of kðxÞ. References Ahmad AA (1998) Recurrence relations for single and product moments of record values from Burr type XII distribution and a characterization. J Appl Statist Sci 7(1):7–15 Ahmad AA (2001) Moments of order statistics from doubly truncated continuous distributions and characterizations. Statistics 35(4):479–494
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