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Recurrence relations for single moments of generalized order statistics from doubly truncated distributions
Journal of Statistical Planning and Inference 117 (2003) 241 – 249
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Recurrence relations for single moments of generalized order statistics from doubly truncated distributions Abd EL-Baset A. Ahmada;∗ , Mohammad A. Fawzyb a Department
b Department
of Mathematics, University of Assiut, Assiut, Egypt of Mathematics, University of South Valley, Egypt
Received 2 September 2001; accepted 4 July 2002
Abstract In this paper, we derive recurrence relations for moments of generalized order statistics within a class of doubly truncated distributions. Doubly truncated Weibull (exponential, Rayleigh), Burr type XII (Lomax) and Pareto distributions, among others, arise as special cases of this class. Recurrence relations for moments of order statistics and record values are obtained as they are special cases of the generalized order statistics. c 2002 Elsevier B.V. All rights reserved. Keywords: Generalized order statistics; Order statistics; Record values; Moments
1. Introduction Kamps (1995a) introduced the concept of generalized order statistics (gOSs). Ordinary order statistics (oOSs), k-records (ordinary record values (oRVs) when k =1), sequential order statistics, ordering via truncated distributions and censoring schemes can be discussed as they are special cases of the gOSs. Kamps’s book gave several applications in a variety of disciplines, recurrence relations for moments of oOSs and characterizations, (for a survey of the models contained and of the results obtained in the gOSs, see Kamps, 1995a, b, 1999). Ahsanullah (1996, 1997) presented the estimators of parameters of the uniform and power function distributions, respectively. Keseling (1999) characterized some continuous distributions based on conditional expectations of gOSs. Ahsanullah (2000) characterized the exponential distribution based on independence of functions of gOSs and presented the estimators of its parameters. ∗
Corresponding author.
c 2002 Elsevier B.V. All rights reserved. 0378-3758/02/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 3 8 5 - 3
242 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249
AL-Hussaini and Ahmad (2002) obtained Bayesian predictive densities and survival functions of gOSs when the underlying population is assumed to have a general class of distributions. Habibullah and Ahsanullah (2000) presented the estimators of parameters of Pareto distribution based on gOSs. Kamps and Gather (1997) characterized the exponential distributions based on the gOSs. Cramer and Kamps (2000) derived relations for expectations of functions of gOSs within a class of continuous distributions. A large number of publications are concerned with recurrence relations of moments of oOSs and oRVs, detailed surveys of the oOSs are given in Balakrishnan et al. (1988) and Balakrishnan and Sultan (1998). Kamps (1998) investigated the importance of recurrence relations of oOSs in characterization. Balakrishnan et al. (1992), Balakrishnan et al. (1993), Balakrishnan and Ahsanullah (1994) and Ahmad (1998) investigated recurrence relations of moments of oRVs, among others. In this paper, we are concerned with gOSs from a class of continuous distributions in the doubly truncated case. Section 2 gives recurrence relations for single moments of gOSs from this class. Suppose that the random variable (rv) X has a distribution function out of the following class of absolutely continuous distributions which is considered by AL-Hussaini (1999) is given by F(x) ≡ FX (x; ) = 1 − e−(x) ;
x¿0
(1.1)
and probability density function (pdf) f(x) = (x)e−(x) ;
x ¿ 0;
(1.2)
where (x) ≡ (x; ) is a nonnegative, strictly increasing and diEerentiable function of x such that (x; ) → 0 as x → 0+ and (x; ) → ∞ as x → ∞, (x) is the derivative of (x) with respect to x. The doubly truncated pdf, say fd (x); is given by fd (x) =
f(x) ; P−Q
Q1 6 x 6 P1 ;
where P = F(P1 ) and Q = F(Q1 ). The pdf of the doubly truncated case of class (1.1) is given by fd (x) =
(x)e−(x) ; P−Q
Q1 6 x 6 P1
(1.3)
and the df is given by Fd (x) = Q2 −
e−(x) ; P−Q
Q1 6 x 6 P1 ;
(1.4)
where Q2 = (1 − Q)=(P − Q). From (1.3) and (1.4), we note that fd (x) = (x)[Q2 − Fd (x)] = (x)[FJ d (x) + P2 ]; J = 1 − F(:) and P2 = (1 − P)=(P − Q). where F(:)
(1.5)
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 243
2. Recurrence relations for single moments of gOSs Let X1;n; m; k ; X2;n; m; k ; : : : ; Xn;n; m; k be n gOSs from the pdf (1.2), (n ¿ 1; m and k are real numbers and k ¿ 1). The pdf of Xr;n; m; k ; 1 6 r 6 n, is given by Kamps (1995a) as follows: cr−1 r −1 r−1 J f(x)gm (F(x)); x ∈ ; (2.1) fXr;n; m; k (x) = [F(x)] (r − 1)! where is the domain on which fXr;n; m; k (x) is positive r i ; i = k + (n − i)(m + 1) cr−1 = i=1
and for 0 ¡ z ¡ 1 [1 − (1 − z)m+1 ]=(m + 1); gm (z) = −ln(1 − z);
m = −1; m = −1:
The single moments of gOSs can be obtained, for j ¿ 1, from (2.1), as cr−1 j r −1 r−1 J xj [F(x)] f(x)gm (F(x)) d x: E[Xr;n; m; k ] = (r − 1)!
(2.2)
If m = 0 and k = 1 we obtain the jth moment in the oOSs case and if m = −1 and k = 1 ( j) j j = E[Xr:n ] ≡ E[Xr;n; we obtain the jth moment in the oRVs case. We shall use r:n 0; 1 ] ( j) j j and (r) = E[XU (r) ] ≡ E[Xr;n; −1; 1 ] to denote the jth moment of the rth oOS and oRV, ( j) j respectively. In general we shall write, for simplicity, r;n; m; k = E[Xr;n; m; k ]. The following general recurrence relation for single moments of gOSs can be obtained, (see Kamps, 1995a, Theorem 1.3.4, p. 98).
Lemma 2.1. Let Fd (x) be an arbitrary df de2ned on (Q1 ; P1 ), then for integers 1 6 r 6 n, real m; k with k ¿ 1 and for every absolutely continuous function , we can obtain P1 cr−2 r−1 (x)[FJ d (x)]r gm (Fd (x)) d x: (2.3) E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] = (r − 1)! Q1 Now, we shall derive an identity for single moments of gOSs from the doubly truncated class (1.4). Theorem 2.1. Let X be a rv with df Fd (x) de2ned on (Q1 ; P1 ) by (1.4), then for every absolutely continuous function , some integers satisfying 1 6 r 6 n − 1 and real m; k with k ¿ 1 and k + m ¿ 0, the following identity 1 E[ (Xr;n; m; k )] = E[ (Xr−1;n−1; m; k )] + E[ (Xr;n; m; k )] 1 − b(r)
r P2 {E[ (Xr;n−1; m; k+m )] − E[ (Xr−1;n−1; m; k+m )]} 1
(2.4)
244 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 ∗ ∗ and cr−1 = is satisMed, where (:) = (:)= (:), b(r) = cr−2 =cr−2 (n − 1 − i)(m + 1) ¿ 0.
r
i=1 i ;
i = (k + m) +
Proof. From (2.3) and (1.5), we obtain E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] P1 fd (x) cr−2 r−1 dx (Fd (x)) = (x)[FJ d (x)]r −1 gm − P 2 (x) (r − 1)! Q1 which can be written as E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] P1 cr−2 r−1 = (x)[FJ d (x)]r −1 fd (x)gm (Fd (x)) d x (r − 1)! Q1 P1 ∗ cr−2 r−1 −b(r)P2 (x)[FJ d (x)] r gm (Fd (x)) d x; (r − 1)! Q1 r ∗ ∗ where (:)= (:)= (:), cr−1 = i=1 i ; i =(k+m)+(n−1−i)(m+1) and b(r)=cr−2 =cr−2 . Making use of (2.2) and (2.3), we obtain E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] =
1 E[ (Xr;n; m; k )] − b(r)P2 {E[ (Xr;n−1; m; k+m )] − E[ (Xr−1;n−1; m; k+m )]}: r
By using of the following general recurrence relation, see Kamps (1995a, p. 71) r E[ (Xr−1;n; m; k )] + (r − 1)(m + 1)E[ (Xr;n; m; k )] =1 E[ (Xr−1;n−1; m; k )]; relation (2.4) is proved. 3. Remarks (1) It can be shown from (2.3) and (1.5), for k ¿ 1, that E[ (Xr;n; −1; k )] = E[ (Xr−1;n; −1; k )] + −
1 E[ (Xr;n; −1; k )] k
1−P E[ (Xr;n; −1; k )e(Xr;n; −1; k ) ]: k
(2.5)
(2) In the non-truncated case (P = 1; P2 = 0), (2.4) reduces to E[ (Xr;n; m; k )] = E[ (Xr−1;n−1; m; k )] +
1 E[ (Xr;n; m; k )]: 1
(2.6)
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 245
(3) If we put m = 0 and k = 1 (oOSs), [r = n − r + 1; b(r) = n=(n − r + 1)], in (2.4) and (2.6), we obtain 1 (2.7) E[ (Xr:n )] = E[ (Xr:n )] + Q2 E[ (Xr−1:n−1 )] − P2 E[ (Xr:n−1 )]; n 1 (2.8) E[ (Xr:n )] = E[ (Xr−1:n−1 )] + E[ (Xr:n )]: n (4) If we put k = 1 in (2.5) and k = 1; m = −1 in (2.6) (oRVs), we obtain the recurrence relations E[ (XU (r) )] = E[ (XU (r−1) )] + E[ (XU (r) )] − (1 − P)E[ (XU (r) )e(XU (r) ) ]; E[ (XU (r) )] = E[ (XU (r−1) )] + E[ (XU (r) )]:
(2.9) (2.10)
(5) If we choose (x) = xj ; j ¿ 1, then relations (2.4)–(2.10) reduce, respectively, to j−1 X j r;n; m; k ( j) ( j) E r;n; m; k = r−1;n−1; m; k + 1 (Xr;n; m; k ) r P2 ( j) ( j) [r;n−1; m; k+m − r−1;n−1; m; k+m ]; 1 j−1 Xr;n; j −1; k ( j) = r−1;n; −1; k + E k (Xr;n; −1; k ) −b(r)
( j) r;n; −1; k
j−1 Xr;n; −1; k e(Xr;n; −1; k ) j(1 − P) − ; E k (Xr;n; −1; k ) j−1 Xr;n; j m; k ( j) ; = r−1;n−1; m; k + E 1 (Xr;n; m; k )
( j) r;n; m; k
( j) r:n
(2.11)
(2.12)
j−1 j Xr:n ( j) ( j) + Q2 r−1:n−1 = E − P2 r:n−1 n (Xr:n )
(2.13)
(Eq. (2.13) with Theorem 1 in Ahmad (2001), with diEerent parameters) j−1 j Xr:n ( j) ( j) ; r:n = r−1:n−1 + E n (Xr:n ) ( j) (r)
=
( j) (r−1)
+ jE
and
( j) (r)
=
( j) (r−1)
+ jE
XUj−1 (r)
− j(1 − P)E
(XU (r) ) XUj−1 (r) (XU (r) )
:
(XU (r) ) XUj−1 (r) e
(XU (r) )
(2.14)
246 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 j ( j) j ( j) (6) We follow the convention that i;i−1; m; k = P1 and 0;i; m; k = Q1 ; i ¿ 1. Then, from relations (2.11)–(2.14), we Mnd that j−1 X1;n; j m; k ( j) j ( j) j − b(1)P2 [1;n−1; 1;n; m; k = Q1 + E m; k+m − Q1 ]; 1 (X1;n; m; k ) ( j) 1;n; −1; k
( j) 1:n
=
Q1j
j−1 j−1 X1;n; X1;n; −1; k e(X1;n; −1; k ) j(1 − P) j −1; k − ; + E E k (X1;n; −1; k ) k (X1;n; −1; k )
j−1 X1:n j ( j) + Q2 Q1j − P2 1:n−1 = E ; (X1:n ) n X1j−1 e(X1 ) X1j−1 − j(1 − P)E : + jE (X1 ) (X1 )
( j) (1)
=
Q1j
j
(2.15)
j
(7) If we choose (x) = etx and t x , we can deduce several recurrence relations for moment and factorial moment generating functions in the gOSs case which, in the other hand, give the moments of gOSs by diEerentiating these recurrence relations with respect to t and then put t = 0 and 1, respectively. 4. Examples (1) Weibull distribution: By choosing (x) = xc ; x ¿ 0; c; ¿ 0, in (1.1), we obtain Weibull df. Recurrence relations for single moments of gOSs from doubly truncated Weibull distribution can be obtained from (2.11) and (2.12), as j ( j) ( j) r;n; ( j−c) m; k = r−1;n−1; m; k + c1 r;n; m; k −b(r)
r P2 ( j) ( j) [r;n−1; m; k+m − r−1;n−1; m; k+m ]; 1
(2.16)
( j) ( j) r;n; −1; k = r−1;n−1; −1; k c j ( j−c) j−c Xr;n; −1; k ]}; (2.17) {r;n; −1; k − (1 − P)E[Xr;n; −1; k e ck the oOSs and oRVs cases are given, from (2.16) and (2.17), as j ( j−c) ( j) ( j) ( j) r:n = + Q2 r−1:n−1 − P2 r:n−1 (2.18) nc r:n (Eq. (2.18) agrees with one of Khan et al., 1983 results) j (j−c) j(1 − P) ( j) ( j) XUc (r) (r) E[XUj−c = (r−1) + − ]: (2.19) (r) e c (r) c (Relations for single moments from doubly truncated exponential and Rayleigh distributions can be obtained from (2.16)– (2.19) when c = 1 and 2, respectively.)
+
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 247
Recurrence relation for single moments of gOSs from Weibull distribution can be obtained from (2.16) (when P = 1; P2 = 0), as ( j) ( j) r;n; m; k = r−1;n−1; m; k +
j ( j−c) ; c1 r;n; m; k
(2.20)
the oOSs and oRVs cases are given, from (2.20), as ( j) ( j) r:n = r−1:n−1 + ( j) ( j) (r) + = (r−1)
j ( j−c) ; nc r:n
j (j−c) : c (r)
(2.21) (2.22)
(Relations for single moments from exponential and Rayleigh distributions can be obtained from (2.20)– (2.22) when c = 1 and 2, respectively.) (2) Burr type XII distribution: By choosing (x) = ln(1 + xc ); x ¿ 0; c; ¿ 0, in (1.1), we obtain Burr type XII df. Recurrence relations for single moments of gOSs from doubly truncated Burr type XII distribution can be obtained from (2.11) and (2.12), as ( j) r;n; m; k =
c1 j + ( j) ( j−c) c1 − j r−1;n−1; m; k c1 − j r;n; m; k −
( j) r;n; −1; k =
cr ( j) ( j) b(r)P2 [r;n−1; m; k+m − r−1;n−1; m; k+m ]; c1 − j
(2.23)
ck j ( j) ( j−c) + ck − j r−1;n; −1; k ck − j r;n; −1; k −(1 − P)
j j−c c +1 ]; E[Xr;n; −1; k (1 + Xr;n; −1; k ) ck − j
(2.24)
the oOSs and oRVs cases are given, from (2.23) and (2.24), as ( j) r:n =
nc j ( j) ( j) − P2 r:n−1 ]+ [Q2 r−1:n−1 ( j−c) : nc − j nc − j r:n
(2.25)
(Eq. (2.25) agrees with one of Khan and Khan (1987) results with diEerent notation): ( j) (r) =
c j j(1 − P) ( j) ( j−c) c +1 + − ]: (r−1) (r) E[XUj−c (r) (1 + XU (r) ) c − j c − j c − j
(2.26)
(Relations for single moments from doubly truncated Lomax can be obtained from (2.23) –(2.26) when c = 1.) Recurrence relation for single moments of gOSs from Burr type XII distribution can be obtained from (2.23), as ( j) r;n; m; k =
c1 j ( j) ( j−c) ; + c1 − j r−1;n−1; m; k c1 − j r;n; m; k
(2.27)
248 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249
the oOSs and oRVs cases are given, from (2.27), as ( j) = r:n
j nc ( j−c) ; + ( j) nc − j r−1:n−1 nc − j r:n
(2.28)
( j) (r) =
j c ( j) + ( j−c) c − j (r−1) c − j (r)
(2.29)
(Eq. (2.29) agrees with Theorem 1 in Ahmad (1998) with diEerent notation). (Relations for single moments from Lomax distributions can be obtained from (2.27) – (2.29) when c = 1.) (3) Pareto distribution: By choosing (x) = ln x; 1 ¡ x ¡ ∞; ¿ 0, in (1.1), we obtain Pareto df. Recurrence relations for single moments of gOSs from doubly truncated Pareto distribution can be obtained from (2.11) and (2.12), as ( j) r;n; m; k =
1 b(r)r P2 ( j) ( j) ( j) r−1;n−1; [r;n−1; m; k+m − r−1;n−1; m; k − m; k+m ]; 1 − j 1 − j
( j) r;n; −1; k =
k j(1 − P) ( j+) ( j) ; r−1;n; −1; k − k − j k − j r;n; −1; k
(2.30) (2.31)
the oOSs and oRVs cases are given, from (2.30) and (2.31), as ( j) = r:n
n ( j) ( j) − P2 r:n−1 ] [Q2 r−1:n−1 n − j
(this result agrees with one of Balakrishnan and Joshi (1982) results with diEerent notation) ( j) = (r)
j(1 − P) ( j+) ( j) − (r) : − j (r−1) −j
Recurrence relation for single moments of gOSs from Pareto distribution can be obtained from (2.30), as ( j) r;n; m; k =
1 ; ( j) 1 − j r−1;n−1; m; k
the oOSs and oRVs cases are given, from (2.32), as ( j) = r:n
n ( j) n − j r−1:n−1
(this result agrees with one of Balakrishnan and Joshi (1982) results) and ( j) = (r)
( j) : − j (r−1)
(2.32)
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 249
Acknowledgements The authors appreciate the comments of the referees which improved the original manuscript. They also thank Professor AL-Hussaini for reading it. References Ahmad, A.A., 1998. Recurrence relations for single and product moments of record values from Burr type XII distribution and a characterization. J. Appl. Statist. Sci. 1, 7–15. Ahmad, A.A., 2001. Moments of order statistics from doubly truncated continuous distributions and characterizations. Statistics 35 (4), 479–494. Ahsanullah, M., 1996. Generalized order statistics from two parameter uniform distribution. Comm. Statist. Theory Methods 25 (10), 2311–2318. Ahsanullah, M., 1997. Generalized order statistics from power function distribution. J. Appl. Statist. Sci. 5, 283–290. Ahsanullah, M., 2000. Generalized order statistics from exponential distribution. J. Statist. Plann. Inference 85, 85–91. AL-Hussaini, E.K., 1999. Predicting observables from a general class of distributions. J. Statist. Plann. Inference 79, 79–91. AL-Hussaini, E.K., Ahmad, A.A., 2002. On Bayesian predictive distributions of generalized order statistics. Metrika, to appear. Balakrishnan, N., Ahsanullah, M., 1994. Recurrence relations for single and product moments of record values from generalized Pareto distribution. Comm. Statist. Theory Methods 23 (10), 2841–2852. Balakrishnan, N., Joshi, P.C., 1982. Moments of order statistics from doubly truncated Pareto distribution. J. Indian Statist. Assoc. 20, 109–117. Balakrishnan, N., Sultan, K.S., 1998. Recurrence relations and identities for moments of order statistics. In: Balakrishnan, N., Rao, C.R. (Eds.), Handbook of Statistics, 16, Order Statistics: Theory and Methods. North-Holland, Amsterdam, pp. 149–228. Balakrishnan, N., Malik, H.J., Ahmed, S.E., 1988. Recurrence relations and identities for moments of order statistics, II: speciMc continuous distributions. Comm. Statist. Theory Methods 17, 2657–2694. Cramer, E., Kamps, U., 2000. Relations for expectations of functions of generalized order statistics. J. Statist. Plann. Inference 89 (1–2), 79–89. Habibullah, M., Ahsanullah, M., 2000. Estimation of parameters of a Pareto distribution by generalized order statistics. Comm. Statist. Theory Methods 29 (7), 1597–1609. Kamps, U., 1995a. A Concept of Generalized Order Statistics. Teubner, Stuttgart. Kamps, U., 1995b. A concept of generalized order statistics. J. Statist. Plann. Inference 48, 1–23. Kamps, U., 1998. Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: Balakrishnan, N., Rao, C.R. (Eds.), Handbook of Statistics, 16, Order Statistics: Theory and Methods. North-Holland, Amsterdam, pp. 291–311. Kamps, U., 1999. Order statistics, generalized. In: Kotz, S., Read, C.B., Banks, D.L. (Eds.), Encyclopedia of Statistical Sciences, Update Vol. 3. Wiley, New York, pp. 553–557. Kamps, U., Gather, U., 1997. Characteristic property of generalized order statistics for exponential distributions. Appl. Math. (Warsaw) 24 (4), 383–391. Keseling, C., 1999. Conditional distributions of generalized order statistics and some characterizations. Metrika 49 (1), 27–40. Khan, A.H., Khan, I.A., 1987. Moments of order statistics from Burr distribution and its characterizations. Metron 45, 21–29. Khan, A.H., Yaqub, M., Parvez, S., 1983. Recurrence relations between moments of order statistics. Naval Res. Logist. Quart. 30, 419–461.
www.elsevier.com/locate/jspi
Recurrence relations for single moments of generalized order statistics from doubly truncated distributions Abd EL-Baset A. Ahmada;∗ , Mohammad A. Fawzyb a Department
b Department
of Mathematics, University of Assiut, Assiut, Egypt of Mathematics, University of South Valley, Egypt
Received 2 September 2001; accepted 4 July 2002
Abstract In this paper, we derive recurrence relations for moments of generalized order statistics within a class of doubly truncated distributions. Doubly truncated Weibull (exponential, Rayleigh), Burr type XII (Lomax) and Pareto distributions, among others, arise as special cases of this class. Recurrence relations for moments of order statistics and record values are obtained as they are special cases of the generalized order statistics. c 2002 Elsevier B.V. All rights reserved. Keywords: Generalized order statistics; Order statistics; Record values; Moments
1. Introduction Kamps (1995a) introduced the concept of generalized order statistics (gOSs). Ordinary order statistics (oOSs), k-records (ordinary record values (oRVs) when k =1), sequential order statistics, ordering via truncated distributions and censoring schemes can be discussed as they are special cases of the gOSs. Kamps’s book gave several applications in a variety of disciplines, recurrence relations for moments of oOSs and characterizations, (for a survey of the models contained and of the results obtained in the gOSs, see Kamps, 1995a, b, 1999). Ahsanullah (1996, 1997) presented the estimators of parameters of the uniform and power function distributions, respectively. Keseling (1999) characterized some continuous distributions based on conditional expectations of gOSs. Ahsanullah (2000) characterized the exponential distribution based on independence of functions of gOSs and presented the estimators of its parameters. ∗
Corresponding author.
c 2002 Elsevier B.V. All rights reserved. 0378-3758/02/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 2 ) 0 0 3 8 5 - 3
242 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249
AL-Hussaini and Ahmad (2002) obtained Bayesian predictive densities and survival functions of gOSs when the underlying population is assumed to have a general class of distributions. Habibullah and Ahsanullah (2000) presented the estimators of parameters of Pareto distribution based on gOSs. Kamps and Gather (1997) characterized the exponential distributions based on the gOSs. Cramer and Kamps (2000) derived relations for expectations of functions of gOSs within a class of continuous distributions. A large number of publications are concerned with recurrence relations of moments of oOSs and oRVs, detailed surveys of the oOSs are given in Balakrishnan et al. (1988) and Balakrishnan and Sultan (1998). Kamps (1998) investigated the importance of recurrence relations of oOSs in characterization. Balakrishnan et al. (1992), Balakrishnan et al. (1993), Balakrishnan and Ahsanullah (1994) and Ahmad (1998) investigated recurrence relations of moments of oRVs, among others. In this paper, we are concerned with gOSs from a class of continuous distributions in the doubly truncated case. Section 2 gives recurrence relations for single moments of gOSs from this class. Suppose that the random variable (rv) X has a distribution function out of the following class of absolutely continuous distributions which is considered by AL-Hussaini (1999) is given by F(x) ≡ FX (x; ) = 1 − e−(x) ;
x¿0
(1.1)
and probability density function (pdf) f(x) = (x)e−(x) ;
x ¿ 0;
(1.2)
where (x) ≡ (x; ) is a nonnegative, strictly increasing and diEerentiable function of x such that (x; ) → 0 as x → 0+ and (x; ) → ∞ as x → ∞, (x) is the derivative of (x) with respect to x. The doubly truncated pdf, say fd (x); is given by fd (x) =
f(x) ; P−Q
Q1 6 x 6 P1 ;
where P = F(P1 ) and Q = F(Q1 ). The pdf of the doubly truncated case of class (1.1) is given by fd (x) =
(x)e−(x) ; P−Q
Q1 6 x 6 P1
(1.3)
and the df is given by Fd (x) = Q2 −
e−(x) ; P−Q
Q1 6 x 6 P1 ;
(1.4)
where Q2 = (1 − Q)=(P − Q). From (1.3) and (1.4), we note that fd (x) = (x)[Q2 − Fd (x)] = (x)[FJ d (x) + P2 ]; J = 1 − F(:) and P2 = (1 − P)=(P − Q). where F(:)
(1.5)
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 243
2. Recurrence relations for single moments of gOSs Let X1;n; m; k ; X2;n; m; k ; : : : ; Xn;n; m; k be n gOSs from the pdf (1.2), (n ¿ 1; m and k are real numbers and k ¿ 1). The pdf of Xr;n; m; k ; 1 6 r 6 n, is given by Kamps (1995a) as follows: cr−1 r −1 r−1 J f(x)gm (F(x)); x ∈ ; (2.1) fXr;n; m; k (x) = [F(x)] (r − 1)! where is the domain on which fXr;n; m; k (x) is positive r i ; i = k + (n − i)(m + 1) cr−1 = i=1
and for 0 ¡ z ¡ 1 [1 − (1 − z)m+1 ]=(m + 1); gm (z) = −ln(1 − z);
m = −1; m = −1:
The single moments of gOSs can be obtained, for j ¿ 1, from (2.1), as cr−1 j r −1 r−1 J xj [F(x)] f(x)gm (F(x)) d x: E[Xr;n; m; k ] = (r − 1)!
(2.2)
If m = 0 and k = 1 we obtain the jth moment in the oOSs case and if m = −1 and k = 1 ( j) j j = E[Xr:n ] ≡ E[Xr;n; we obtain the jth moment in the oRVs case. We shall use r:n 0; 1 ] ( j) j j and (r) = E[XU (r) ] ≡ E[Xr;n; −1; 1 ] to denote the jth moment of the rth oOS and oRV, ( j) j respectively. In general we shall write, for simplicity, r;n; m; k = E[Xr;n; m; k ]. The following general recurrence relation for single moments of gOSs can be obtained, (see Kamps, 1995a, Theorem 1.3.4, p. 98).
Lemma 2.1. Let Fd (x) be an arbitrary df de2ned on (Q1 ; P1 ), then for integers 1 6 r 6 n, real m; k with k ¿ 1 and for every absolutely continuous function , we can obtain P1 cr−2 r−1 (x)[FJ d (x)]r gm (Fd (x)) d x: (2.3) E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] = (r − 1)! Q1 Now, we shall derive an identity for single moments of gOSs from the doubly truncated class (1.4). Theorem 2.1. Let X be a rv with df Fd (x) de2ned on (Q1 ; P1 ) by (1.4), then for every absolutely continuous function , some integers satisfying 1 6 r 6 n − 1 and real m; k with k ¿ 1 and k + m ¿ 0, the following identity 1 E[ (Xr;n; m; k )] = E[ (Xr−1;n−1; m; k )] + E[ (Xr;n; m; k )] 1 − b(r)
r P2 {E[ (Xr;n−1; m; k+m )] − E[ (Xr−1;n−1; m; k+m )]} 1
(2.4)
244 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 ∗ ∗ and cr−1 = is satisMed, where (:) = (:)= (:), b(r) = cr−2 =cr−2 (n − 1 − i)(m + 1) ¿ 0.
r
i=1 i ;
i = (k + m) +
Proof. From (2.3) and (1.5), we obtain E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] P1 fd (x) cr−2 r−1 dx (Fd (x)) = (x)[FJ d (x)]r −1 gm − P 2 (x) (r − 1)! Q1 which can be written as E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] P1 cr−2 r−1 = (x)[FJ d (x)]r −1 fd (x)gm (Fd (x)) d x (r − 1)! Q1 P1 ∗ cr−2 r−1 −b(r)P2 (x)[FJ d (x)] r gm (Fd (x)) d x; (r − 1)! Q1 r ∗ ∗ where (:)= (:)= (:), cr−1 = i=1 i ; i =(k+m)+(n−1−i)(m+1) and b(r)=cr−2 =cr−2 . Making use of (2.2) and (2.3), we obtain E[ (Xr;n; m; k ) − (Xr−1;n; m; k )] =
1 E[ (Xr;n; m; k )] − b(r)P2 {E[ (Xr;n−1; m; k+m )] − E[ (Xr−1;n−1; m; k+m )]}: r
By using of the following general recurrence relation, see Kamps (1995a, p. 71) r E[ (Xr−1;n; m; k )] + (r − 1)(m + 1)E[ (Xr;n; m; k )] =1 E[ (Xr−1;n−1; m; k )]; relation (2.4) is proved. 3. Remarks (1) It can be shown from (2.3) and (1.5), for k ¿ 1, that E[ (Xr;n; −1; k )] = E[ (Xr−1;n; −1; k )] + −
1 E[ (Xr;n; −1; k )] k
1−P E[ (Xr;n; −1; k )e(Xr;n; −1; k ) ]: k
(2.5)
(2) In the non-truncated case (P = 1; P2 = 0), (2.4) reduces to E[ (Xr;n; m; k )] = E[ (Xr−1;n−1; m; k )] +
1 E[ (Xr;n; m; k )]: 1
(2.6)
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 245
(3) If we put m = 0 and k = 1 (oOSs), [r = n − r + 1; b(r) = n=(n − r + 1)], in (2.4) and (2.6), we obtain 1 (2.7) E[ (Xr:n )] = E[ (Xr:n )] + Q2 E[ (Xr−1:n−1 )] − P2 E[ (Xr:n−1 )]; n 1 (2.8) E[ (Xr:n )] = E[ (Xr−1:n−1 )] + E[ (Xr:n )]: n (4) If we put k = 1 in (2.5) and k = 1; m = −1 in (2.6) (oRVs), we obtain the recurrence relations E[ (XU (r) )] = E[ (XU (r−1) )] + E[ (XU (r) )] − (1 − P)E[ (XU (r) )e(XU (r) ) ]; E[ (XU (r) )] = E[ (XU (r−1) )] + E[ (XU (r) )]:
(2.9) (2.10)
(5) If we choose (x) = xj ; j ¿ 1, then relations (2.4)–(2.10) reduce, respectively, to j−1 X j r;n; m; k ( j) ( j) E r;n; m; k = r−1;n−1; m; k + 1 (Xr;n; m; k ) r P2 ( j) ( j) [r;n−1; m; k+m − r−1;n−1; m; k+m ]; 1 j−1 Xr;n; j −1; k ( j) = r−1;n; −1; k + E k (Xr;n; −1; k ) −b(r)
( j) r;n; −1; k
j−1 Xr;n; −1; k e(Xr;n; −1; k ) j(1 − P) − ; E k (Xr;n; −1; k ) j−1 Xr;n; j m; k ( j) ; = r−1;n−1; m; k + E 1 (Xr;n; m; k )
( j) r;n; m; k
( j) r:n
(2.11)
(2.12)
j−1 j Xr:n ( j) ( j) + Q2 r−1:n−1 = E − P2 r:n−1 n (Xr:n )
(2.13)
(Eq. (2.13) with Theorem 1 in Ahmad (2001), with diEerent parameters) j−1 j Xr:n ( j) ( j) ; r:n = r−1:n−1 + E n (Xr:n ) ( j) (r)
=
( j) (r−1)
+ jE
and
( j) (r)
=
( j) (r−1)
+ jE
XUj−1 (r)
− j(1 − P)E
(XU (r) ) XUj−1 (r) (XU (r) )
:
(XU (r) ) XUj−1 (r) e
(XU (r) )
(2.14)
246 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 j ( j) j ( j) (6) We follow the convention that i;i−1; m; k = P1 and 0;i; m; k = Q1 ; i ¿ 1. Then, from relations (2.11)–(2.14), we Mnd that j−1 X1;n; j m; k ( j) j ( j) j − b(1)P2 [1;n−1; 1;n; m; k = Q1 + E m; k+m − Q1 ]; 1 (X1;n; m; k ) ( j) 1;n; −1; k
( j) 1:n
=
Q1j
j−1 j−1 X1;n; X1;n; −1; k e(X1;n; −1; k ) j(1 − P) j −1; k − ; + E E k (X1;n; −1; k ) k (X1;n; −1; k )
j−1 X1:n j ( j) + Q2 Q1j − P2 1:n−1 = E ; (X1:n ) n X1j−1 e(X1 ) X1j−1 − j(1 − P)E : + jE (X1 ) (X1 )
( j) (1)
=
Q1j
j
(2.15)
j
(7) If we choose (x) = etx and t x , we can deduce several recurrence relations for moment and factorial moment generating functions in the gOSs case which, in the other hand, give the moments of gOSs by diEerentiating these recurrence relations with respect to t and then put t = 0 and 1, respectively. 4. Examples (1) Weibull distribution: By choosing (x) = xc ; x ¿ 0; c; ¿ 0, in (1.1), we obtain Weibull df. Recurrence relations for single moments of gOSs from doubly truncated Weibull distribution can be obtained from (2.11) and (2.12), as j ( j) ( j) r;n; ( j−c) m; k = r−1;n−1; m; k + c1 r;n; m; k −b(r)
r P2 ( j) ( j) [r;n−1; m; k+m − r−1;n−1; m; k+m ]; 1
(2.16)
( j) ( j) r;n; −1; k = r−1;n−1; −1; k c j ( j−c) j−c Xr;n; −1; k ]}; (2.17) {r;n; −1; k − (1 − P)E[Xr;n; −1; k e ck the oOSs and oRVs cases are given, from (2.16) and (2.17), as j ( j−c) ( j) ( j) ( j) r:n = + Q2 r−1:n−1 − P2 r:n−1 (2.18) nc r:n (Eq. (2.18) agrees with one of Khan et al., 1983 results) j (j−c) j(1 − P) ( j) ( j) XUc (r) (r) E[XUj−c = (r−1) + − ]: (2.19) (r) e c (r) c (Relations for single moments from doubly truncated exponential and Rayleigh distributions can be obtained from (2.16)– (2.19) when c = 1 and 2, respectively.)
+
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 247
Recurrence relation for single moments of gOSs from Weibull distribution can be obtained from (2.16) (when P = 1; P2 = 0), as ( j) ( j) r;n; m; k = r−1;n−1; m; k +
j ( j−c) ; c1 r;n; m; k
(2.20)
the oOSs and oRVs cases are given, from (2.20), as ( j) ( j) r:n = r−1:n−1 + ( j) ( j) (r) + = (r−1)
j ( j−c) ; nc r:n
j (j−c) : c (r)
(2.21) (2.22)
(Relations for single moments from exponential and Rayleigh distributions can be obtained from (2.20)– (2.22) when c = 1 and 2, respectively.) (2) Burr type XII distribution: By choosing (x) = ln(1 + xc ); x ¿ 0; c; ¿ 0, in (1.1), we obtain Burr type XII df. Recurrence relations for single moments of gOSs from doubly truncated Burr type XII distribution can be obtained from (2.11) and (2.12), as ( j) r;n; m; k =
c1 j + ( j) ( j−c) c1 − j r−1;n−1; m; k c1 − j r;n; m; k −
( j) r;n; −1; k =
cr ( j) ( j) b(r)P2 [r;n−1; m; k+m − r−1;n−1; m; k+m ]; c1 − j
(2.23)
ck j ( j) ( j−c) + ck − j r−1;n; −1; k ck − j r;n; −1; k −(1 − P)
j j−c c +1 ]; E[Xr;n; −1; k (1 + Xr;n; −1; k ) ck − j
(2.24)
the oOSs and oRVs cases are given, from (2.23) and (2.24), as ( j) r:n =
nc j ( j) ( j) − P2 r:n−1 ]+ [Q2 r−1:n−1 ( j−c) : nc − j nc − j r:n
(2.25)
(Eq. (2.25) agrees with one of Khan and Khan (1987) results with diEerent notation): ( j) (r) =
c j j(1 − P) ( j) ( j−c) c +1 + − ]: (r−1) (r) E[XUj−c (r) (1 + XU (r) ) c − j c − j c − j
(2.26)
(Relations for single moments from doubly truncated Lomax can be obtained from (2.23) –(2.26) when c = 1.) Recurrence relation for single moments of gOSs from Burr type XII distribution can be obtained from (2.23), as ( j) r;n; m; k =
c1 j ( j) ( j−c) ; + c1 − j r−1;n−1; m; k c1 − j r;n; m; k
(2.27)
248 A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249
the oOSs and oRVs cases are given, from (2.27), as ( j) = r:n
j nc ( j−c) ; + ( j) nc − j r−1:n−1 nc − j r:n
(2.28)
( j) (r) =
j c ( j) + ( j−c) c − j (r−1) c − j (r)
(2.29)
(Eq. (2.29) agrees with Theorem 1 in Ahmad (1998) with diEerent notation). (Relations for single moments from Lomax distributions can be obtained from (2.27) – (2.29) when c = 1.) (3) Pareto distribution: By choosing (x) = ln x; 1 ¡ x ¡ ∞; ¿ 0, in (1.1), we obtain Pareto df. Recurrence relations for single moments of gOSs from doubly truncated Pareto distribution can be obtained from (2.11) and (2.12), as ( j) r;n; m; k =
1 b(r)r P2 ( j) ( j) ( j) r−1;n−1; [r;n−1; m; k+m − r−1;n−1; m; k − m; k+m ]; 1 − j 1 − j
( j) r;n; −1; k =
k j(1 − P) ( j+) ( j) ; r−1;n; −1; k − k − j k − j r;n; −1; k
(2.30) (2.31)
the oOSs and oRVs cases are given, from (2.30) and (2.31), as ( j) = r:n
n ( j) ( j) − P2 r:n−1 ] [Q2 r−1:n−1 n − j
(this result agrees with one of Balakrishnan and Joshi (1982) results with diEerent notation) ( j) = (r)
j(1 − P) ( j+) ( j) − (r) : − j (r−1) −j
Recurrence relation for single moments of gOSs from Pareto distribution can be obtained from (2.30), as ( j) r;n; m; k =
1 ; ( j) 1 − j r−1;n−1; m; k
the oOSs and oRVs cases are given, from (2.32), as ( j) = r:n
n ( j) n − j r−1:n−1
(this result agrees with one of Balakrishnan and Joshi (1982) results) and ( j) = (r)
( j) : − j (r−1)
(2.32)
A.A. Ahmad, M.A. Fawzy / Journal of Statistical Planning and Inference 117 (2003) 241 – 249 249
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