On Bayesian interval prediction of future records - Springer Link

Sociedad de Estadistica e I~vestigaei6n Opevativa Test (2003) Vol. 12, No. 1, pp. 79-99

On Bayesian Interval Prediction of Future Records Essarn

K. AL-Hussaini* and Abd EL-Baset Department of Mathematics University of Assi~t, A. R. Egypt.

A.

Ahrnad

Abstract B a s e d on t h e o,le-sample scheme, B a y e s i a n predictio,1 b o u n d s for t h e s ~h f u t u r e record value are obtai,led. All of t h e i M o r m a t i v e a,ld f u t u r e o b s e r v a t i o n s m'e a s s u m e d to b e o b t a i n e d from a general class of d i s t r i b u t i o n s whick includes t h e Weibull, c o m p o u n d Weibull, P a r e t o , b e t a , G o m p e r t z , c o m p o u n d G o m p e r t z a m o n g o t h e r d i s t r i b u t i o n s . T h e prior belief of t h e e x p e r i m e n t e r is m e a s u r e d by a p r o p e r general c o n j u g a t e prior which was suggested b y AL HussairLi (1999b).

Key Words: AMS

1

subject

B a y e s i a n prediction, o n e - s a m p l e scheme, records.

classification:

62oo3, 62004

Introduction

Statistical prediction is the prol)lem of inferring the values of unknown future variables or functions of such variables from current available i~formative observations. As in estimation, a predictor can be either a point or an interval predictor. Parametric and nonparametric prediction have been considered in literature. Frequentist and Bayesian approaches have been used to obtain predictors and study their properties. Re~dews on parametric point and interval predictors may be found in Patel (1989), Nagaraja (1995), Kaminsky and Nelson (1998) and AL-Hussaini (2001). Nonparametric prediction w~s considered by Flinger and V~%lfe (1976, 1979a, b), euilbaud (1983) and Johnson et al. (1999), among others. The problem of prediction can be soh;ed fully within the Bayes framework (Ceisser, 1993). Several researchers have studied Bayesian prediction. Among others are Dunsmore (1974, 1976, 1983), Ceisser (1975, 1984, 1985, 1986, 1990, 1993), bingappaiah (1978, 1979, 1980, 1986, 1989), HoMader and Hossain (1995), Dunsmore and anfin (1998), Ab-Hussaini and Jaheen * C o r r e s p o n d e n c e to: D e p a r t m e n t ot M a t h e m a t i c s , U n i v e r s i t y of Assiut, Assiut 17516, A.R. E G Y P T . E - m a i h ekalh(@acc.aun.edu.eg Received: April 2001;

Accepted: J u n e 2002

E. K. AL-Hussaini and A. IF,. A. Ahrnad

80

(1995, 1996a,b), AL-Hussaini (1999a,b, 2001), Lee and Liao (1999), Corcuera and Giummolh (1999) and AL-Hussaixfi et al. (2001). A wide range of potential applications of statistical prediction includes density function estimation, calibration, classification, regulation, model comparison and model criticism. For details and references, see, for example Bernardo and Smith (1994). The two books by Aitcheson and Dunsmore (197.5) and Geisser (199a), which are primarily concerned with Bayes prediction, give illustrative examples, analysis and possible applications. Prediction is also discussed in different places of Bernardo and Smith (1994) and Johnsox~ et al. (1994, 1995). A growing interest in the estimation and prediction of records has arisen in the last two decades. For example, the best linear unbiased predictor and best linear invariant predictor of future records were obtained by Ahsanullah (1980, 1994), Nagaraja (1984) and Doganakso and Balakrishnan (1997). Interval prediction of future records was studied by Dunsmore (1983), Ahsanullah (1990), Balakrishnan and Chan (1994), Balakrishnan et al. (199.5) and Berred (1998). Let X 1 , X 2 ; . . . be an infinite sequence of independent and identically distributed (i.i.d) random variables having the same distribution as the (population) random variable X. Denote the cmnulative distribution function (cdf) of X by F. An observation X j will be called an upper record value (or simply a record) if it exceeds in value all of the preceding observations, i.e., if X j > Xi, for every i < j. The sequence of record times T,,., n ___ 0 is defined as follows: To = 1 M t h probabili~,y 1 and, for n ___ 1 ~,. rr~ir~{j 9 Xj > XT~ ~}. A sequence, {R,,}, of record values, is t h e n defined by R,, XT~, .~z 0, 1, 2, . . . . For details on prediction of records and other interesting topics related to records see, for example, the books by Ahsanullah (1994) and Arnold et al.

(1998). Suppose that the population cdf is of the form F(~)-F,-(~I0)=I

exp[ a(x)],

~>0,

(1.1)

%(.z)~ %(.z;0) is a nonnegative continuous function of z such t h a t %(.~;0) 0 as z --+ 0 + and k(z; 0) -+ oc as z --+ oo. The reliability, density and hazard

On Bayesian Interval Prediction of Future Records

81

rate functions are given, respectively, for x > 0 by F(~:) = 1 -

f(~:) and

p(~:)

e•

;~'(~:)exp[- ~,(~:)],

(1.2)

h(~) = a'(~). Let R 0 , R I , . . . , R , ,

(1.3)

be the first upper record values. Their likelihood

function (L~) is given by Arnold et al. (199S) in the form, m 1

L(0, r)

h(r.,),

f(r,,) H i

(1.4)

o

where f(r)

F(~-)'

dr

m

H(~)=

lnr(G (1.5)

By substituting Equations (1.2) and (1.3) in Equation (1.4), we obtain

L(O,r) = A(0, r) exp[ A(r,,,)],

oo < r0 < rl < ...
,,,, and H(.) is given by (1.~). The predictive density function of the s th record given the first (m + 1) records is defined by f*(r~ It_) = f 9,,,(r; [ 0)Tr*(0 [ r_)d0, Jo

r; > r,,,.

(2.2)

Application to class (i.i) by substituting Equations (2.i) and (i.9) in Equation (2.2), yields the predictive density of r*, (s ---- 1, 2 , . . . ) given the (m + 1) records. Such a density function is given by

f*(,; I,)

~

.~ A'(,-~FI(O I,-)[A(,-~)- A(,,,,)F lexp[-r

l,~)] dO

s--]

A

~.Ik(,-;), k=0

,-~* >

,-....

(2.3)

On Bayesian Interval Prediction of Future Records

83

where

f*(r~ I c)dr~,

k

and

Ik(,-~)

L

/VCp*~.r/~,',p*~.lkr/~,',p ~.ls--k--i

t~)t

t.)J

t t .,;J

~lte 0 I , ) e x p [ - r

Ir~)]dO.

(2.4)

It then follows t h a t the predictive surx,ival function for the s th record is given by

F f*(,-; I r)dr; /F if(r;

PER; > ,~ I,-]

I r)dr;

~a

s

1

k

1

k=o j=o

~

1

k

~.=0 j=o

where

(2.6)

.~ [AO-,,,)]~-k-~,ffO l ,-)[,',(,.,)]j exp[-r

J~(,.')

l ,.,)]dO~

(2.7)

where (9 [ 1., is given by (1.10). This result follows by observing t h a t the

n~me,'~tor of the ~,,,,~,,~1 f~n~-tion (2.S) i~ gi,r~n~ from ~q~tion~ (2.3) ~nd (2.~), by =

f (r~ [E)dr*

A

ck k=O s-]

I~.(r~)dr~ t~

= A ~ ~k [,[A(,,,,)] ~ k b ( o I~_) k=O

exp [-D (o; ~,)]Io (,.,)dO~

(2.8)

where

•

c~

=

L

*

k !

*

?,0-,)1 ~, O-~)~xp[ aO-2)ld,-2 ~(~)

(.) k

~.! ~ j=O

[a(j~)? exp[-a(,J/].

(2.9)

E. K. AL-Hussaini and A. E. A. Ahraad

84

By substituting Equation (2.9) in Equation (2.8) and writing cA,

( 1 ) s - k - ' ( s ; 0 , we, obtain fc~

s-1 k f* (r* I r_)dr; = A ~ ~ Qj~.(r,), k 0 j=0

where Qj~,(u) is given by Equation (2.6). The expression for the predictive density then follows by forming the r~tio given in (2.5). The predictive bounds of a two-sided interval ~qth cover r, for the B; record value (s = 1, 2 , . . . ), may thus be obtained by solving the following two equations for lower bound (L) and upper bound (U): l+r

2

i

-RIB* >LIt],

r

2

-RIB* >Ulr],

(2.10)

where P [ ~ ; > 1, I,-] is gi,r~n by (2.5). In the special case, s and (2.6), that

l(j

k

0), it follows, from Equations (2.5)

P[R~ > i, I~1 = Jo(4/Jo(,-,,,),

i, > ,-,,,,

where, from Equation (2.7), ,)TOO/) is given by 9J 0 ( 4 = f , ( 0 1 Jo 3

~)expL 0,0 > O,,S > O. ~t fol-

A(O,,')

S;,C'..~""+~O""+~,S,,. I].;20"~.-

On Bayesian Interval Prediction of Future Records

85

Set C(0;~') 0 ~ 1,D(0;q') bO, 7 (a,b), a and b are positive. It follows, from Equation (1.10), t h a t ~l(OIr) 6~ 1..d'"+10'"+~and r I r,,, ) .Y O(b + r;,,.). Application to (2.7) t h e n yields 4(,4

= c,,, ( , , ~ ) , Y r b~-(,~)]/~,~J(b + , / ) ~ ( ~ ) ,

where =,-,,

o . . . . . ~,

,

ejk(s)

= m + a + s + j

k.

It t h e n follows t h a t the survival function (2.5) is such t h a t

%k(,,)=(1) When ~ l(j (3.1), of the form

k (s k 1) k,uJ.~F.[ej~.(s)]/j,r,~:~(b+ u.~)~k(~). ~-

(3.1)

0), t h e surviv,1 f u n c t i o n (2.5) t h e n becon~e~, u~ing

P[R~ > ,~ I ~-,,,] (~o0(4/r

[b + r~,] "+a+~

[b + ,/J

(3.2)

Substituting ]Equation (3.2) in Equation (2.10) and solving, explicit lower and upper bounds of a predictive interval ~ i t h cover ~- for the first future record R~ can be shown to be of the forms: L

-

b + ~-;,~) - b

U

-

b + ,-;,~) - b

In a sequence of generated V~.%ibull(3,4) variates, the following 10 records h~ve been reported (ra = 9): 0.67825, 0.87021, 0.91960, 0.94436, 0.95554, 0.98119, 0.99798, 1.01388, 1.03495, 1.17963. It can be seen, with prior parameters a = 10 and b = 2, t h a t 95% predictive limits of the first future record (11 th in the sequence) are given by L 1.18039 and U 1.28384.

86

E. K. AL-Hussaini and A. E. A. Abroad

R e m a r k 3.1. The exponential(0)model is a V~'eibull(0,.3 1) model and the R~yleigh(0) model is V~'eibull(0,.3 = 2) model. So, all of the previous results regarding the U:eibull(0, .3)model, reduce to the exponential(0) and Rayleigh(0) models by setting .3 = 1 and .3 = 2, respectively. R e m a r k 3.2. Lower and upper bounds for the s th future record R; can lye obtained by numerically solving (2.10) for L and U, where P[R* > L ] r] and P[R* > U ]r] are given by (2.5) after replacing l/by L and U, respectively. 3.1.2

0,

B u r r t y p e XII(0,.3) m o d e l

~ > 0. ~t follows, fron~ (1.S), that A(0,,-)

4,, 3'"+~0'"+~, a,,, (.3)

I];" 0 [ r / - ~ / ( l + r / ) ] . Set C(0;?0 =O~-~,D(O;?O=bO, ?~=(a,b), aand b are positive. It follows, from (1.10), that rl(O I t ) 5,,,.3"+10 ~ and 3 C(O]r,,,) = O[b + ln(1 + v~,,.)]. Application to (2.7) then yields a k Ib+ ln(1 + , / ) 1 ~ + , .JJ(,d = c,,, (a) Iln( 1 + ,/)l~r [~k('~)l/Iln( 1 + ~;,,)1 where G,,, (..3) ---- [ln(1 + r;,,.)] ~ 15(..3)..3,,,-+1 and

ejt.(s)=rn+a+s+j

k.

It then follows that the survival function (2.5) is such that

(@k('+) = ( 1)k

k

j ! [ l n ( l + r ; , ,3) ] A,[ b + l n ( l +

/)yj~,(~) "

(3.3)

When s = l ( j = k = 0), the survival function (2..5) then becomes, using (3.3), of the form

P[R~>1+[r,,,]

Qoo(l+)/(2oo(r,,,)

[ ~ + i n ( l + r,~,.)] ,,,+o+1

+ h~(1+ ,,.,)]

(3.4)

Substituting Expression (3.4) in Equation (2.10) and solving, explicit lower and upper bounds of predictive interval with cover v for the first future record R~ can lye shown to lye of the form:

+ in(l + r;,,)] b

1

+ ln(1 + r'~,,)]

1

b

On Bayesian Interval Prediction of Future Records

87

In a sequence of generated Burr type XII(5,6) variates, the following 10 records have been reported (m = 9): 0.56079, 0.58781, 0.65926, 0.69784, 0.78322, 0.78479, 0.89069, 0.89907, 0.99047, 1.00271. It can be seen th~l,, M t h prior parameters a - i0 and b - 2, the 95% predictive limil, s of the first future record (11 th in the sequence) are given by L 1.00384 and U 1.16382.

3.1.3

Pareto type I(0,3 ) model

In this model A(x) = 01n(.d/x), X(x) = 0/x, x > 3, 0 > 0, 0 < .3 < A . It follows, from Equation (1.7), t h a t A(O,r) 5,,0'"+1, 5,,, H'"i . = O ~1/,-~ Set C(0;~) 0 o ~, D(0;~') bO, ~" (~,b), ~ and b are \ I W 9 positive. It follows, from Equation (1.10), t h a t 9(0 it_) = 6..,0 '''.+~ and

4(0 I r,,,) = O[b

i n . d + lnr,,], where 0 > 0, 0 < .d < M, M = rain{r0, &}.

Application to (2.7) t h e n yields

Jj(t/)

(- 1)k-JG,.(,3)[h~(,3/~/)];F[ej~,(s)]/[h~(,,J/r,.)]k[b

- in,,3+ ln~/] ~J~(~>,

where

G,,(.3)=(1)

~

l[ln(.3/r,,)]~ 15,,,

and

cj~.(s)=m+a+s+j

k.

It t h e n follows t h a t the survival function (2.5) is such t h a t

s (@k (~)

(-1) 9

1) k

l,:![ln(,3/T/)]JF[cj~.(s)] j![ln(.3/r,,,.)]k[b 2 ln..3 + in t/]~J~ (') '

(3.5)

W h e n s = l ( j = k = 0), the survival function (2..5) then becomes, using (3.5), of the form

[b _!n,~+_!n>] ' '+~ e[R2 > ,J I ~,,,] = (?oo(,4/(?oo(~,,,) = L b - l n . ~ +

ln,~

(3.6)

Substituting Equation (3.6) in Equation (2.10) and solving, explicit lower and upper bounds of predictive interval ~ t h cover r for the first

E. K. AL-Hussainiand A. E. A. Abroad

88

future record R{ can be shown to be of the forms: L

=

exp{ln.3

b+[b

ln.~+lnr,~](~-)

U

=

exp{ln.3

b+[b

ln.~+lnr,~](~@

~} f-) ~ } .

For this model, the computations were carried out in two cases: when . 3 < l a n d . 3 > 1. 1..3 0.5 < 1: In a sequence of generated Pareto type I(2,0.5) variates, the following 30 records have been reported (m ---- 29): 0.57890, 0.68168, 0.99801, 1.65748,

0.58641, 0.73403, 1.01222, 1.71499,

0.58803, 0.59549, 0.59761, 0.64604,0.65597, 0.67666, 0.73482, 0.75464, 0.83683, 0.86002, 0.89371, 0.91593, 1.10432, 1.13228, 1.16246, 1.33156, 1.46805, 1.62221, 1.81369, 2.41121, 2.63523, 3.37100.

It can be seen, with prior parameters a 10 and b 1, t h a t 95% predictive limits of the first future record (31 "1 in the sequence) are given by L 3.37721 and U 4.46460. 2 . . d = 2 > 1: In a sequence of generated Pareto type I(2,2) variates, the following 30 records have been reported (m 29): 2.31560, 2.72672, 3.99204, 6.62992,

2.34564, 2.93612, 4.04888, 6.85996,

2.35212, 2.38196, 2.39044, 2.58416, 2.62388, 2.70664, 2.93928, 3.01856, 3.34732, 3.44008, 3.57484, 3.66372, 4.41726, 4.52911, 4.64992, 5.32624, 5.87220, 6.48884, 7.25476, 9.64484, 10.54092, 13.48400.

It can be seen, with prior parameters a = 10 and b = 1, t h a t 95% predictive limits of the i st future record (31 s~ in the sequence) are given by L 13.50885 and U 17.85840. R e m a r k 3.3. It may be noticed t h a t better intervals (~ith shorter widths) can be obtained if at least one of a or m is large. It can be seen t h a t when s 1, b o t h of the lower and upper limits, in each of the previous examples, t e n d to r.~. as m + a + 1 --+ on.

89

On Bayesian Interval Prediction of Future Records

3.2

Bayesian

Conditional

Survival

Function

of the

s th F u t u r e

Record

It has been shown, in the previous examples, that the conditional Bayesian survival function of the S th future record given the previous (m + 1) records is given in all cases by Equation (2.5), with (@k(1/) as in (3.1), (3.3) and (3.5) for the V~.~ibull, Burr type XII and Pareto models, respectively. It can be sho~n~, by induction, that the survival function (2.5) may be rewritten in the following simpler form:

P[R* > ,., I",,,] = L A(,.,) ]

k

A(,-,,,)

j ,

(3.7)

9i = 0

"where

c(s) = m + a + s , and A(.) differs according to the model used. Explicitly', (1) for the a.~ibull (0,.d) model, A(t) = b + t~, (2) for the Burr type XII(O,.J) model, A(t) = b + ln(1 + t3), (3) for the Pareto (0,.3) model, A (t ) = b

ln.d+lnt.

Indeed, expression (3.7) %r the conditional Bayesian survival function is easier to use than (2.5) in the computations involving the lower and upper limits of the predictive interval of the S th future record. It should be noticed here that expression (3.7) is obtained under the assumption that only one of the parameters is unknown in each of the three studied models. 3.3

Two

Unknown

Parameters

We shall apply the results of Section 2 to the V~.%ibull(a.,3), Burr type XII(a., .3) and Pareto type I(a.-, .3) models. In all cases both of the parameters c~ and .3 are assumed to be unknown.

E. K. AL-Hussaini and A. E. A. Ahraad

90 3.3.1

Weibull

(ct,3) m o d e l

In this model A(x) = a x 3, A'(x) = a.dx ,3-~, x > 0, (a, .3 > 0). V~.k'ite 0 = (a-, 3) so that Equation (1.7) yields A(O, r_) = 6/,, la'"+~.d ''+~, a,,, = I1Lo ri. Set C(0;7 ) : a~+r ~-~ and D(0;7 ) = a ( d +b.3), 7 : (a,b,c,d), where all of the prior parameters a, b, c, d are positive. The prior density function 7r(0; 7), given by (1.8), then takes the form of the prior suggested by ALHussaini and Jaheen (1992). LIn this prior ct has a gamma (c,d) distribution and the conditional distribution of .3 given c~ is gamma (a, ba)]. It follows, from (1.10), that ,1(0 I r ) 53,,, aa"+a+e3''+a., and ~(Olr,,,) c~[rr~, q- (d 4b.3)]. Application to (2.7) then yields

J5

P[~;k (~)] :~, (,.)/a,,,, where

cc}~..3"'.+~d.3

[d + b,3 + u,J]~J~(~) ' cJj~. = vJ6,,r~,7 k a

and

ejk(s)=m+a+c+s+j

k.

It then follows that the survival function (2.5) is such that

%~(,.)

(-1) k ( s - l~.)

k'F leA,(s)]IA, (,/) j!

(as)

When s = l(k = j = 0), the survival function (2.5) 1,hen becomes, using (3.8), of the form

P[R~ > ,~ I ,-,,,] = i00(.)/i00(,-,,,),

,. > ,-,,,,

where

3,,,+o~, d3 [d + b3 + v,X]~ooO)'

e00(1)

m + a + c + 1.

In a sequence of generated V~:eibull (3,4) variates, the following 30 records have been reported (m = 29) : 0.35050, 0.42047, 0.47544, 0.49870, 0.50225, 0.51188, 0.51412, 0.54427, 0.56012, 0.56168, 0.59977, 0.60232, 0.62651, 0.66038, 0.67098, 0.67872,

On Bayesian Interval Prediction of Future Records

91

0.68987, 0.69767, 0.70391, 0.73458, 0.76075, 0.76745, 0.76907, 0.77553, 0.80428, 0.82125, 0.84863~ 0.88827~ 0.90865~ 0.96815. It can be seen, with prior parameters a = 10, b = 2, c = 8 and d = 4, that 95% predictive limits of the first future record (31 "t in the sequence) are given by L = 0.97085 and U = 1.16308. 3.3.2

B u r r t y p e X I I ( a , ~) m o d e l

~n tt~s model

;~(~:) ~ln(1 + ~:,~), ~'(~:)

~,,~:,~-~/(1 + ~:,~), ~: > 0,

(a,..J > 0). V~.%ite 0 = (a,..J) so that Equation (1.7) yields

a(O,,-) O, 0 < '.3 < d). 5~ Let 0 = (a-,..3). From Equation (1.7) A(O,r_) = 5~,a"'+~,8,, = [I;~ 0(1/ri). Arnold and Press (1983) criticised the prior suggested by Lwin (1972) as being unnaturallg restrictive. T h e y proposed the use of a generalized prior, in which a has a g a m m a distribution while the conditional distribution of .3 given a. is of the power function form. Arnold (1983) pointed out some problems associated ~qth this generalized L~qn's prior and suggested the use of independent prior densties for a and .3. However, Arnold and Press (1989) proposed the use of a g a m m a (a, b) prior for a and an inverse Pareto (ac, d) for the conditional distribution of .3 given a. Using this prior, we can ~7"ite 7r(a, .3)

7r1 (a)Tr2(.3]a) oc a~.3 *~-~ e x p [ - b a ] / d *~, a > O, 0 < .3 < d.

So t h a t C(0; ?,)

a~.3 ~ r

~r and D(#; ?')

ha. It follows, from (1.11),

Application to (2.8) yields

4(,4 where

( - 1 7 +*' ~ %,r [ ~; , (4]Ij k(d; -),

/~ ,,3[b+clnd+ln,,-(c+ d

Ijk(d; ,/)

[ln d _ l n m ] ~

k 1[ln(3/,4]~d 3 1) ln,,3]~J~(~)'

93

On Bayesian Interval Prediction of Future Records

eak(s)

m+a+s+j-k+l.

It t h e n follows t h a t the survival function (2.5) is such t h a t

= (1/( When s= 1@=~=0),

P[R

>

I,.,.]

Ioo(d;,4/Ioo(d;,-,,,.),

12 > Fro.,

where e

jo

Ioo(d;,/) . .3

d,,3

,,3[b + c l n d + h~ v , - ( c + 1) ln,,3]~oo(a) '

Coo(l)

re+a+2.

0.5 < 1:

In a sequence of generated Pareto type I(2,0.5) variates, the following 30 records have been reported (m = 29) : 0.51145, 0.60169, 0.75066, 1.09991,

0.52400, 0.60560, 0.77542, 1.18080,

0.53983, 0.54861, 0.55008, 0.56555, .58372, 0.59741, 0.63002, 0.64938, 0.65694, 0.65868, 0.69027, 0.71932, 0.82625, 0.84011, 0.87921, 0.93044, 0.94799, 1.05297, 1.19385, 1.22771, 1.46399, 1.85173.

It can be seen, with prior parameters a = 10, b = 2, c = 2 and d 2, t h a t 95% prectictive limits of the first future record (31 st in the sequence) are given by L 1.854319 and U 2.018324. . 3=2>

1:

In a sequence of generated Pareto type I(2,2) v~riates, the f o l l o ~ n g 30 records have been reported (ra 29) : 2.04580~ 2.40676, 3.00263, 4.39963,

2.09600, 2.42238, 8.10170, 4.72320,

2.15932, 2.52010, 3.30500, 4.77541,

2.19445~ 2.20031, 2.26220~ 2.33486, 2.38965~ 2.59754, 2.62778, 2.63473, 2.76107, 2.87728, 3.36046, 3.51684, 3.72177, 3.79198, 4.21191, 4.91086, 5.85596, 7.40692.

It can be seen, with prior parameters a = 10, b = 2, c = 1 and d = 5, t h a t 95% predictive limits of the first future record (31 st in the sequence) are given by L = 7.41999 and U = 9.63299.

94

E. K. AL-Hussaini and A. E. A. Ahrnad

R e m a r k 3.4. It might be anticipated that the form given by Equation (3.7) for the conditional Bayesian survival function of the stl~ future record holds true for any of the members of class (1.1) when one of the parameters is the only unknown1 parameter involved. R e m a r k 3.5. Bayesian prediction bounds of future records based on other distributions t h a t are members of class (1.1) can be similarly obtained. R e m a r k 3.6. In Bayes theory, the vector 7 of prior parameters is assumed to be known1. If such prior parameters are unknown, we may use the empirical B~yes approach to estimate t h e m using past samples, [see, for example, Maritz and bwin (1989). Alternatively, one could use the hierarchical B~yes approach in which a suitable prior for 7 is used (see Bernardo and Smith, 1994). R e m a r k 3 . r . As we have remarked earlier (Remark a.3), in the one parameter case, if either a or m gets large, b o t h of the lower or upper limits of the predictive intervals tend to r,, as m + a + 1 --+ oo. In other words, the interval leng-~hs get shorter as ra (or a) gets larger. R e m a r k 3.8. It might be expected, as in the one parameter case, that the prediction intervals for the first future records become better (shorter), in the two parameter case, for large values of r R e m a r k 3.9. The general class of distributions (1.1) and general conjugate priors (1.9) were also used in the multisample prediction problem that was presented by AL-Hussaini (2001).

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