such as ordinary order statistics [David (1981), Arnold, Balakrishnan and. Nagaraja (1992) and Balakrishnan and Rao (1998)], records [Ahsanullah. (1994) and ...
Metrika (2003) 57: 165–176
> Springer-Verlag 2003
On Bayesian predictive distributions of generalized order statistics Essam K. AL-Hussaini and Abd EL-Baset A. Ahmad Department of Mathematics, University of Assiut, Assiut 71516 Egypt
Abstract. Bayesian predictive densities and survival functions of generalized order statistics are obtained when the underlying population is assumed to have a general class which includes several important distributions. The prior belief of the experimenter is measured by a general class of distributions, suggested by AL-Hussaini (1999) b , which covers most prior distributions used in literature. Specializations to predictive densities and survival functions of ordinary order statistics and records are obtained and compared with existing results. Applications to the Weibullða; bÞ model are illustrated when a is the only unknown parameter and when both ða; bÞ are unknown. Key words: Bayesian prediction, one-sample scheme, order statistics, records, generalized order statistics
1. Introduction A concept of generalized order statistics (gos) was introduced by Kamps (1995) to unify several important concepts that have been used in statistics such as ordinary order statistics [David (1981), Arnold, Balakrishnan and Nagaraja (1992) and Balakrishnan and Rao (1998)], records [Ahsanullah (1994) and Arnold, Balakrishnan and Nagaraja (1998)], sequential order statistics [Cramer and Kamps (1996), (2001)] and ordering via truncated distributions and censoring schemes [Kamps (1995)]. Applications in a variety of disciplines, recurrence relations for moments of gos and characterizations of distributions are among the topics of Kamps’s book. Ahsanullah (1996, 2000) and Habibullah and Ahsanullah (2000) studied some distributional properties of gos and obtained minimum variance linear unbiased estimates of the parameters of a two-parameter uniform, exponential
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and Pareto type II distributions based on gos, respectively. For the uniform model, Ahsanullah (1996) also obtained the best linear invariant estimates of the parameters and in the exponential case, Ahsanullah (2000), furthermore, characterized such distribution based on gos. Kamps and Gather (1997) developed a characteristic property of gos for exponential distributions and Keseling (1999) used conditional distributions of gos in characterizing some continuous distributions. Cramer and Kamps (2000) established relations for expectations of functions of gos. Kamps and Cramer (2001) developed marginal density and distribution functions in a wide subclass of gos and applied their results to obtain recurrence relations and moments of gos from Pareto, power function and Weibull distributions. They characterized the exponential distribution by using a distributional identity involving subrange gos or an identity using the expectations of gos defining the subrange. Nasri-Roudsari and Cramer (1999) derived results for the convergence rates of the n th and ðn r þ 1Þ st gos and showed that such rates are highly influenced by the choice of the normalizing sequence. Cramer and Kamps (2001) proposed the use of sequential order statistics to model a change of the residual liftime distribution after the breakdown of some component in a k-out-of-n system based on a one – and two-parameter exponential distributions. Furthermore, they compared two sequential k-out-of-n systems by using PðX < Y Þ. Several authors have predicted future order statistics and records from homogeneous and heterogeneous populations that can be represented by single or finite mixtures of distributions. Among others are Dunsmore [(1974), (1976), (1983)], Geisser [(1975), (1984), (1985), (1986), (1990), (1993)], Lingappaiah [(1978), (1979), (1980), (1986), (1989)], Howlader and Hossain (1995), Dunsmore and Amin (1998), AL-Hussaini and Jaheen [(1995), (1996), (1999)], AL-Hussaini [(1999) a , (1999) b , (2001) a , (2001) b ], Lee and Liao (1999), Corcuera and Giummole` (1999), AL-Hussaini, Nigm and Jaheen (2001) and AL-Hussaini and Ahmad (2001). In this paper, Bayesian predictive densities and survival functions of gos are obtained when the underlying population is assumed to have a general class which includes several important distributions such as the Weibull, compound Weibull (or three-parameter Burr type XII), Pareto, beta, Gompertz and compound Gompertz distributions, among others. The prior belief of the expermenter is measured by a general class of distributions, suggested by AL-Hussaini (1999) b , which covers most of the prior distributions used in literature. Specializations to predictive densities and survival functions of ordinary order statistics and records are obtained and compared with existing results. The Weibull model is used as an example of the underlying population distribution. In this case, Bayesian predicitive survival functions for gos are obtained when only one or both parameters are unknown. 2. Bayesian predictive densities and survival functions Let X1 ; X2 ; . . . ; Xr be r gos based on an absolutely continuous distribution function Fy , where Xi 1 X ði; n; m; kÞ, i ¼ 1; 2; . . . ; n, m and k are real numbers and k b 1. The likelihood function is given by Kamps (1995) as follows
On Bayesian predictive distributions of generalized order statistics
) r1 Y m Lðy; xÞ z ½Fy ðxi Þ fy ðxi Þ ½Fy ðxr Þ gr 1 fy ðxr Þ;
167
(
y < x1 < < xr < y;
i¼1
ð2:1Þ Fy ð:Þ and fy ð:Þ are the population reliability and density functions, respectively, Fy ð:Þ ¼ 1 Fy ð:Þ is the corresponding cumulative distribution function, gr ¼ k þ ðn rÞðm þ 1Þ > 0, x ¼ ðx1 ; . . . ; xr Þ and y is the parameter (could be a vector) indexing the population distribution such that y A Y, where Y is the parameter space. Suppose that Fy ðxÞ ¼ 1 exp½ly ðxÞ;
ð2:2Þ
x > 0;
ly ðxÞ is a nonnegative continuous di¤erentiable function of x such that ly ðxÞ ! 0 as x ! 0þ and ly ðxÞ ! y as x ! y. Then the corresponding density, reliability and hazard rate functions are given, respectively, for x > 0, by fy ðxÞ ¼ ly0 ðxÞ exp½ly ðxÞ;
Fy ðxÞ ¼ exp½ly ðxÞ and ly0 ðxÞ:
By substituting Fy ð:Þ and fy ð:Þ in (2.1), the likelihood function Lðy; xÞ is given, for xi > 0, i ¼ 1; . . . ; n, by ð2:3Þ
Lðy; xÞ z Aðy; xÞ exp½Bðy; xÞ; where Aðy; xÞ ¼
r Y
ly0 ðxi Þ;
Bðy; xÞ ¼ ðm þ 1Þ
i¼1
r1 X
ly ðxi Þ þ gr ly ðxr Þ:
ð2:4Þ
i¼1
Consider a conjugate prior density, suggested by AL-Hussaini (1999) b , to be given by pðy; dÞ z Cðy; dÞ exp½Dðy; dÞ;
y A Y; d A W;
ð2:5Þ
where W is the hyperparameter space. The posterior density function, p ðyjxÞ, is then given by p ðyjxÞ z Lðy; xÞpðy; dÞ ¼ hðy; xÞ exp½zðy; xÞ;
ð2:6Þ
where hðy; xÞ ¼ Aðy; xÞCðy; dÞ;
zðy; xÞ ¼ Bðy; xÞ þ Dðy; dÞ:
ð2:7Þ
It is assumed that only the first r gos, X1 ; X2 ; . . . ; Xr have been observed and we wish to predict the future Xrþ1 ; . . . ; Xn gos, (one-sample scheme). Let Xs ¼ Xrþs , s ¼ 1; 2; . . . ; n r. Write gr ðxs jy; xÞ to denote the conditional density function of the s th future gos given that the first r gos had been observed. Then
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E. K. AL-Hussaini, A. EL-B. A. Ahmad
gr ðxs jy; xÞ z ½hm ðFy ðxs ÞÞ hm ðFy ðxr ÞÞ s1 ½Fy ðxs Þ grþs 1 ½Fy ðxr Þgrþ1 fy ðxs Þ s1 X s1 ð1Þ sj1 ½hm ðFy ðxs ÞÞ j ½hm ðFy ðxr ÞÞ sj1 j j¼0
z
½Fy ðxs Þ grþs 1 ½Fy ðxr Þgrþ1 fy ðxs Þ;
xs > xr ;
where, for 0 a z < 1, ( ð1 zÞ mþ1 =ðm þ 1Þ; m 0 1 hm ðzÞ ¼ lnð1 zÞ; m ¼ 1:
ð2:8Þ
ð2:9Þ
The predictive density function is given by ð gr ðxs jy; xÞp ðyjxÞ dy; xs > xr : f ðxs jxÞ ¼
ð2:10Þ
Y
Substitution of Eqs. (2.8) and (2.6) in Eq. (2.10), yields the predictive density of future Xs , ðs ¼ 1; 2; . . .Þ given the first r gos. Such a density function is given by f ðxs jxÞ ¼ K
s1 X
aj Ij ðxs Þ;
xs > xr ;
ð2:11Þ
j¼0
where K¼
ð y xr
f ðxs jxÞ dxs
1 ;
aj ¼ ð1Þ sj1
s1 ; j
and Ij ðxs Þ
¼
ð Y
Gj ðxs ; yÞ exp½Hj ðxs ; yÞ dy;
ð2:12Þ
Gj ðxs ; yÞ ¼ hðy; xÞly0 ðxs Þ½hm ðFy ðxs ÞÞ j ½hm ðFy ðxr ÞÞ sj1 Hj ðxs ; yÞ ¼ Dðy; dÞ þ ðm þ 1Þ
r X
ly ðxi Þ þ grþs ly ðxs Þ:
ð2:13Þ
i¼1
It then follows that the predictive survival function is given, for the s th future gos, by ð y ðy f ðxs jxÞ dxs f ðxs jxÞ dxs P½Xs > n j x ¼ n
xr
( P s1 ¼
P s1
m 0 1 j¼0 Qj ðnÞ= j¼0 Qj ðxr Þ; P s1 P j P s1 P j j¼0 j¼0 l¼0 Qjl ðnÞ= l¼0 Qjl ðxr Þ; m ¼ 1;
ð2:14Þ
On Bayesian predictive distributions of generalized order statistics
169
where Qj ðnÞ ¼ ð1Þ
Jj ðnÞ ¼
j
s1 Jj ðnÞ=½grþsj ðm þ 1Þ s1 ; j
ð
hðy; xÞ exp Dðy; dÞ ðm þ 1Þ
Y
r X
ð2:15Þ
ly ðxi Þ
i¼1
ðm þ 1Þðs j 1Þly ðxr Þ grþsj ly ðnÞ dy;
ð2:16Þ
and s1 Qjl ðnÞ ¼ ð1Þ j! Jjl ðnÞ=k jlþ1 l!; j j
Jjl ðnÞ ¼
ð
½ly ðxr Þ sj1 hðy; xÞ½ly ðnÞl expf½kly ðnÞ þ Dðy; dÞg dy:
ð2:17Þ
ð2:18Þ
Y
The proof of (2.14) is given in APPENDIX A. Remark. It may be noted that although our interest is in the predictive survival function of the ðr þ sÞ th gos given the first r gos, yet, in our derivation of such a function, only the r th gos appears. This is due to the fact that gos based on a continuous distribution form a Markov chain. Special cases. (1) Ordinary order statistics. This is the case in which k ¼ 1 and m ¼ 0. The informative and future order statistics are then given by X1:n < < Xr:n and Xrþ1:n < < Xn:n , respectively. It can be shown that the predictive survival function coincides with that obtained by AL-Hussaini (1999) b in the case of one-sample prediction, where Xs is replaced by Ys . (2) Records. This is the case in which k ¼ 1 and m ¼ 1. The informative and future record values are then given by R0 < < Rp and Rpþ1 < Rpþ2 < : It can be shown that the predictive survival function coincides with that obtained by AL-Hussaini and Ahmad (2001) in the case of one-sample prediction. 3. Example In this section, a member of class (2.2) is used as an example to illustrate the use of the survival function in obtaining predictive bounds of future gos. Such a member is the Weibull distribution, which is used as the population distribution in two cases: when a is the only unknown parameter and when ða; bÞ are both unknown.
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3.1 Weibull(a, b) model, a is the only unknown parameter. In this model, ly ðxÞ ¼ ax b , ly0 ðxÞ ¼ abx b1 , x > 0 where y ¼ a and a; b > 0. It follows, from (2.4), that Aða; xÞ ¼ AðbÞa r and Bða; xÞ ¼ BðbÞa, where Qr P r1 b AðbÞ ¼ b r ð i¼1 xi Þ b1 and BðbÞ ¼ ðm þ 1Þ i¼1 xi þ gr xrb . Set Cða; dÞ ¼ a1 a , Dða; dÞ ¼ ba, d ¼ ða; bÞ, a; b > 0, so that the prior density function pða; dÞ, given by (2.5), is gamma with parameters ða; bÞ. From (2.7), we then have hða; xÞ ¼ AðbÞa aþr1 and zða; xÞ ¼ ½b þ BðbÞa. So that (2.16) yields Jj ðnÞ ¼ AðbÞ
( "
ðy a
aþr1
exp b þ ðm þ 1Þ
0
r X
! xib
þ ðs j
1Þxrb
i¼1
# ) þ grþsj n
¼
b
a da
AðbÞGða þ rÞ : Pr ½b þ ðm þ 1Þð i¼1 xib þ ðs j 1Þxrb Þ þ grþsj n b aþr
ð3:1Þ
For m 0 1, the survival function of the s th future gos from the Weibullða; bÞ model, (b known), can then be obtained by substituting (3.1) in (2.15) to get (2.14). The first future gos is of particular interest. This is the case when s ¼ 1, ð j ¼ 0Þ. In this case, the survival function (2.14) then becomes P½X1 > n j x ¼ Q0 ðnÞ=Q0 ðxr Þ " ¼
b þ ðm þ
Pr
b i¼1 xi Pr 1Þ i¼1 xib
b þ ðm þ 1Þ
þ grþ1 xrb
#aþr ;
þ grþ1 n b
n > xr :
ð3:2Þ
To obtain the survival function when m ¼ 1, we observe, from (2.18), that Jjl ðnÞ ¼ AðbÞxðsj1Þb nlb r
ðy
a ejl ðsÞ1 exp½ðb þ kn b Þa da
0
ðsj1Þb lb
¼
AðbÞG½ejl ðsÞxr ½b þ
kn b ejl ðsÞ
n
ejl ðsÞ ¼ a þ r þ s þ l j 1:
;
ð3:3Þ
For m ¼ 1, the survival function of the s th future gos from the Weibullða; bÞ model, (b known), can then be obtained by substituting (3.3) in (2.17) to get (2.14). When s ¼ 1, ð j ¼ l ¼ 0Þ; the survival function (2.14) then becomes P½X1
b þ kxrb > n j x ¼ Q00 ðnÞ=Q00 ðxr Þ ¼ b þ kn b
aþr :
ð3:4Þ
On Bayesian predictive distributions of generalized order statistics
171
The predictive bounds of a two-sided interval with cover t for the Xs , ðs ¼ 1; 2; . . .Þ, may thus be obtained by solving the following two equations for the lower (L) and upper (U) bounds: t1 1
1þt ¼ P½Xs > L j x ; 2
t2 1
1t ¼ P½Xs > U j x : 2
ð3:5Þ
Substituting (3.2) and (3.4) in (3.5), (s ¼ 1), explicit lower and upper bounds of the predictive interval with cover t for the first future gos can thus be shown to be of the following forms: when m 0 1: " # ( r X 1 1=b ðt1 Þ1=ðaþrÞ b þ ðm þ 1Þ xib þ grþ1 xrb L¼ grþ1 i¼1 " b þ ðm þ 1Þ
r X
#)1=b xib
i¼1
U¼
1
grþ1
1=b (
" ðt2 Þ1=ðaþrÞ b þ ðm þ 1Þ
r X
#
ð3:6Þ
xib þ grþ1 xrb
i¼1
" b þ ðm þ 1Þ
r X
#)1=b xib
;
i¼1
when m ¼ 1 1=b 1 fðt1 Þ1=ðaþrÞ ½b þ kxrb bg 1=b L¼ k 1=b 1 U¼ fðt2 Þ1=ðaþrÞ ½b þ kxrb bg 1=b : k
ð3:7Þ
3.2 Weibull(a, b) model, a; b are both unknown. In this model, ly ðxÞ and ly0 ðxÞ with y ¼ ða; bÞ are the same as in Sec. (3.1). So that (2.4) yields Aðy; xÞ ¼ a r AðbÞ and Bðy; xÞ ¼ BðbÞa, where AðbÞ and BðbÞ are as given in Sec. (3.1). Suppose that the prior density function for a and b is such that bja @ gammaða; baÞ and a @ gammaðc; dÞ. Then pðy; dÞ ¼ p1 ðbjaÞp2 ðaÞ z ða a b a1 exp½ðbaÞbÞða c1 exp½dbÞ ¼ Cðy; dÞ exp½Dðy; dÞ; where Cðy; dÞ ¼ a aþc1 b a1 , Dðy; dÞ ¼ ðd þ bbÞa, d ¼ ða; b; c; dÞ. It follows, from (2.7), that hðy; xÞ ¼ a rþaþc1 b a1 AðbÞ and zðy; xÞ ¼ ½d þ bb þ BðbÞa. If m 0 1, substituting in (2.16) and integrating, we obtain
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E. K. AL-Hussaini, A. EL-B. A. Ahmad
Jj ðnÞ ¼
ðy 0
Gða þ r þ cÞb a1 AðbÞ db: Pr ½d þ bb þ ðm þ 1Þð i¼1 xib þ ðs j 1Þxrb Þ þ grþsj n b aþrþc ð3:8Þ
For m 0 1, the survival function of the s th future gos from the Weibullða; bÞ model, (both parameters unknown), can then be obtained by substituting (3.8) in (2.17) to get (2.14). The survival function of the first future gos (s ¼ 1; j ¼ 0) is then given by: P½X1 > n j x ¼ I0 ðnÞ=I0 ðxr Þ;
n > xr ;
where I0 ðnÞ ¼
ðy 0
b a1 AðbÞ db: Pr ½d þ bb þ ðm þ 1Þ i¼1 xib þ grþ1 n b aþrþc
When m ¼ 1 Jjl ðnÞ ¼
ðy 0
ðsj1Þb lb
AðbÞG½ejl ðsÞxr ½b þ
kn b ejl ðsÞ
n
db;
where ejl ðsÞ is given in (3.3). The survival function of the first gos (s ¼ 1; j ¼ l ¼ 0), is then given by ðnÞ=I00 ðxr Þ; P½X1 > n j x ¼ I00
n > xr ;
where ðnÞ ¼ I00
ðy 0
AðbÞ db: ½b þ kn b aþr
It may be noticed that, even in this simple case of the first gos, if a; b are both unknown, we will have to resort to numerical methods [solution of the two equations, given by Eq. (3.5)] in order to be able to obtain L and U for X1 . Remarks and conclusions (1) Weibullða; bÞ model when a is the only unknown parameter (i) If m ¼ 0, k ¼ 1, (ordinary order statistics), then gr ¼ n r þ 1 and the predictive bounds (3.6) reduce to those obtained for the first future order statistic. (ii) If m ¼ 1, k ¼ 1, (records), then gr ¼ 1 and the predictive bounds (3.7) reduce to those obtained by AL-Hussaini and Ahmad (2001) for the first future record. (iii) The exponential(a) and Rayleigh(a) models are special cases of the Weibull(a; b) model when b ¼ 1 and b ¼ 2, respectively. So, all of the
On Bayesian predictive distributions of generalized order statistics
173
previous results specialize to the exponential(a) and Rayleigh(a) models by setting b ¼ 1 and b ¼ 2, respectively. (iv) Predictive bounds for other gos rather than the first require numerical computations as they could not be obtained in closed form. (2) Weibull(a; b) model, when both (a; b) are unknown (i) If m ¼ 0, k ¼ 1, (ordinary order statistics), the predictive survival distribution can be shown to agree with AL-Hussaini (1999) b . (Notice that the symbols di¤er and the roles of a; b in the prior density are reversed, where b @ gamma and ajb @ gamma). (ii) If m ¼ 1, k ¼ 1, (records), the predictive survival distribution coincides with that obtained by AL-Hussaini and Ahmad (2001). (iii) Predictive bounds in all cases require numerical computations. (3) We have obtained densities and survival functions of any future gos based on a general class of population distibutions and a general class of priors. Similar results, to those obtained in the Weibull case, could be obtained for any other member of the class by defining ly ðxÞ; ly0 ðxÞ; Cðy; dÞ; Dðy; dÞ and then applying the expressions obtained for the predictive survival function. (4) Predictive bounds of gos could be obtained in closed forms in some special cases. However, in most cases we have to resort to numerical computations to obtain such predictive bounds. (5) In the one-sided case, a predictive interval of the form (0; U) with cover t has the same U value as in the two-sided case with ð1 tÞ=2 being replaced by ð1 tÞ. Similarly, a one-sided interval of the form (L; y) has the same L as in the two-sided case with ð1 þ tÞ=2 being replaced by t. (6) If the prior parameters are unknown, the empirical Bayes approach may be used in their estimation by using past samples [see, for example Maritz and Lwin (1989)]. Alternatively, one could use the hierarchical Bayes approach in which a suitable prior for the vector of hyperparameters d is used [see, Bernardo and Smith (1994)]. Appendix A Survival function The proof of expression (2.15) for the survival function of the s th future gos follows by observing that, from its definition and the use of (2.11), we have P½Xs > n j x ¼
ðy n
f ðxs jxÞ dxs ¼ K
s1 X
ðy aj
j¼0
n
Ij ðxs Þ dxs :
By using (2.12) and (2.13), we obtain P½Xs > n j x ¼ K
s1 X
ð
½hm ðFy ðxr ÞÞ sj1 hðy; xÞ
aj Y
j¼0
" exp Dðy; dÞ ðm þ 1Þ
r X i¼1
where
# ly ðxi Þ oy ðnÞ dy;
ðA:1Þ
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E. K. AL-Hussaini, A. EL-B. A. Ahmad
oy ðnÞ ¼
ðy n
ly0 ðxs Þ½hm ðFy ðxs ÞÞ j exp½grþs ly ðxs Þ dxs :
It follows from (A.1) and (2.9) that if m 0 1, then P½Xs
s1 X
> n j x ¼ K
aj
j¼0 r X
ð
ð1Þ sj1 ðm þ 1Þ sj1
" hðy; xÞ exp Dðy; dÞ ðm þ 1Þ
Y
!# ly ðxi Þ ðs j 1Þly ðxr Þ
oy ðnÞ dy;
ðA:2Þ
i¼1
where oy ðnÞ ¼ ð1Þ j
ðy n
ly0 ðxs Þ exp½grþsj ly ðxs Þ dxs =ðm þ 1Þ j
¼ ð1Þ j exp½grþsj ly ðnÞ=½grþsj ðm þ 1Þ j :
ðA:3Þ By substituting Eq. (A.3) in Eq. (A.2) and writing aj ¼ ð1Þ sj1 s1 , we j obtain s1 X
P½Xs > n j x ¼ K
Qj ðnÞ;
n > xr ;
j¼0
where Qj ðnÞ is given by Eq. (2.15). Expression (2.14) follows (when m 0 1), by using the identity P½Xs > xr j x ¼ 1. It follows from (A.1) and (2.9) that if m ¼ 1, then P½Xs > n j x ¼ K
s1 X
aj ð1Þ sj1
ð
½ly ðxr Þ sj1 hðy; xÞ exp½Dðy; dÞoy ðnÞ dy; Y
j¼0
where oy ðnÞ ¼ ð1Þ
j
ðy n
½ly ðxs Þ j ly0 ðxs Þ exp½kly ðxs Þ dxs :
By using the substitution z ¼ kly ðxs Þ and the relation between the incomplete gamma integral and Poisson sum, we obtain oy ðnÞ ¼ ð1Þ j j!
j X
f½ly ðnÞl exp½kly ðnÞ=l!k jlþ1 g:
l¼0
Notice that when m ¼ 1, then gr ¼ k. Therefore, P½Xs > n j x ¼ K
j s1 X X j¼0 l¼0
Qjl ðnÞ;
n > xr ;
On Bayesian predictive distributions of generalized order statistics
175
where Qjl ðnÞ is given by Eq. (2.17). Expression (2.14) follows (when m ¼ 1), by using the identity P½Xs > xr j x ¼ 1. Acknowledgement. The authors appreciate the comments of the referees which improved the original manuscript.
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