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Tailweight and life distributions
Statistics & Probability North-Holland
Letters 8 (1989) 381-387
September
1989
Abstract: We use the concept of IL,-tailweight to classify the usual life distribution classes. This methodology allows define new classes and to characterize the NBUE class by comparison between the initial distribution and the exponential
us to one.
TAILWEIGHT
J. AVEROUS Lnboratoire
AND LIFE DISTRIBUTIONS
and M. MESTE
de Siaiistique
et Probabilittk,
lJ.A.-C. N. R.S. 745, Universith Paul Sabatier,
31062 Toulouse, France
Received April 1988 Revised October 1988
AMS
1980 Subject Classifications:
Keywords:
life distributions,
Primary
62N05;
Secondary
60ElO.
failure rate, total time on test function,
tailweight,
IL,-norm
1. Introduction We consider in this note the problem of characterizing some non-parametric classes of life distributions. To identify the type of a life distribution F (e.g. IFR, IFRA, . .), the most commonly considered tools are the hazard function R, = -log 3 (where F = 1 - F), the failure rate function r, = R; (for differentiable F) and the scaled total time on test transform ~~(r.4) = /~~‘(‘)- F(s) ds/pF (for F with finite expectation pLF) (see, e.g., Hollander and Proschan, 1984), or the scaled total time on test function T,(x) = GF 0 F(x) (also called equilibrium distribution by the economists). The characterization of a class by a property of these functions can often be equivalently expressed by a comparison either between F and the exponential distribution or between distributions associated with F. For instance, the IFR class is characterized by: (a) rF non-decreasing, (b) c$~ concave, (c) F cc d (cc being the Van Zwet skewness ordering and d an exponential distribution), (d) F,, cbt F,, (0 =Ct, < tz) (where E is the survival function at age t and cs, is the usual stochastic ordering). _ For such characterizations, Loh (1984) uses a tailweight ordering to compare the symmetrized distribution associated with F: F*(x) = i(l + sgn(x)F( 1X I)) with the double exponential. This is equivalent to use the corresponding skewness ordering to compare F with an exponential distribution. Nevertheless, Loh notes that the characterization of the NBUE class with such orderings remains an open question. Deshpande et al. (1986) base a classification of the usual families on the concept of stochastic dominance. Introducing different tailweight or stochastic orderings, these authors define new classes of life distributions: NBAFR (Loh, 1984) IFR(2), NBU(2), . . . (Deshpande et al., 1986). Another approach is used by Abouammoh (1988) or Basu and Ebrahimi (1984) who introduce families by new properties of the mean residual life distribution. In a paper to appear, we introduce the concept of I_,Y-description of any distribution F, based on IL,-tailweight (s > 1). The case s = 1 corresponds to the usual tailweight (F(x) is the weight of the left tail (- cc, x) and 1 - F(y) those of (y, + cc)). For the case s = 2 (when pLF exists), the weight /? m F(t) dt is assigned to the left tail (- cc, x) and /,? w (1 - F(t)) dt to the right tail (y, + cc). These concepts allow one to clarify the classification of the various life distribution classes and also to construct a methodology 0167-7152/89/$3.50
0 1989, Elsevier Science Publishers
B.V. (North-Holland)
381
Volume
8, Number
STATISTICS&PROBABILITY
4
LETTERS
for making new classes. Moreover to complete the Loh characterization, which answers to the Loh remark about the NBUE class.
September
we propose
a tailweight
1989
ordering
2. Preliminaries Let 9 be the family of absolutely continuous distributions F, such that F(0 - ) = 0 and with finite mean Pi. We note TF the d.f. obtained by normalizing with pF the total time on test function,
T,(x) = &ix[l-F(u)] du, and F * the symmetrized
version
(1)
of F,
(2)
F*(x)=:[l+sgn(x)F(Jx1)]. orderings
We recall the usual tailweight
on S*
= {F*;
FEF}.
Definition 1. F*<,,G*
Go ’ 0 F convex
0
F*<,G*
G-’ 0 F * -shaped
w
F * -cSuG * = F*
(Van Zwet ordering),
G-’
e
G-’
(Lawrence
ordering),
0 F super additive, 0 F( x)/x
af(O)/g(O)
Vx f 0
We give the following current classes of life distributions > t} (t E [0, l[), F-‘(l) = sup{x; F(x) < l}.
(Loh ordering)
in 9,
where 2 = 1 - F, Fp ‘( t) = inf{ x; F(x)
Definition 2. FEIFR
S+ F is log-concave,
FEIFRA
*
FENBU
e
- log( F( t ))/t
+t2)d(tl)F(t>)
F(t,
FENBUE
CJ
FEDMRL
a
is increasing
/0
vt,,
+mF(t+x)dx
-!-
F(t)
/f
+oO_ F(x)
in t > 0, t,ao, VtaOsuchthat
dx is decreasing
+30_
FE HNBUE
a
/f
F(x)dx,(~~exp{-1,‘~~)
(Rolski, FE NBUFR FE NBAFR 382
=
(or NBUFRA)
in t 2 0,
VtaOsuchthat
1975),
T-F(t) a+(O)
vt>o, e
(l/t)
log F(t)
F(t)>O,
2 rF(O).
F(t)>0
Volume
8, Number
STATISTICS
4
It is known (Loh, 1984; Hollander implications hold:
& PROBABILITY
and Proschan,
September
LETTERS
1984; Basu and Ebrahimi,
FENBUFR FEIFR
3
Using comparisons Loh, 1984):
FEIFRA
with the double
FEIFR
F* <_
=
FEIFRA
=
exponential
F’
=
FENBUE
(DE), the following
1989
1984) that the following FE NBAFR
=
characterizations
FEHNBUE
have been given (e.g.
DE,
CJ F*<,DE,
FENBU
=
(3)
F*<,,DE, c=, F* ct DE.
FENBAFR
As far as we know, no such characterization exists for the NBUE class. In Averous and Meste (1989) we pointed out that the usual tailweight orderings (given in Definition 1) rather correspond to orderings in IL, sense. In this case, F(x) is the weight of the left tail (- cc, x) of F, 1 - F(y) being the weight of the right tail (y, + cc). We introduced a concept of tailweight in ILs sense (S > 1) which is closely related to the total time on test function when s = 2. Let us recall the main definitions and properties. 3. For any s >, 1, and any F with finite CO, x) (resp. the right tail (y, + 00)) is
Definition
(-
w,“((-, rev
4)
determines two neighbourhood To facilitate the distribution
w,“((Y>
+M))=~y,+p)lf-yl~~-’
arg min jt-mlJ m /R
s - 1, the L ,-weight
of the left tail
dF(t)
i
that the s-mean dF(t)
equally IL,-weighted tails (with usual restrictions when s = 1 and F non-continuous in a of the median). the description of F in IL,Tsense, for s > 1, we have introduced (Averous and Meste, 1989) function 6. defined by: W,“((-a,
F,(x)
of order
=~_m_~,l-I~y-’dF(f)
It follows from this definition ms=
moment
=
1 1 - y,‘((x,
x))/~@((-M, + m))/2WsF((
m.,)), - 00, m,<)),
if x < m,, if x > m,?.
When s = 1 and F is continuous, 2W,(( - co, med( F))) = 1). So the d.f. Ts describes normalized s-tailweights of F just as F does when s = 1. Then a description of c’ in [L, sense is equivalent to a description of F in O_r sense. For instance the median of FY (i.e. the IL, location parameter of F.) equals the s-mean of F. 383
Volume
8, Number
STATISTICS&PROBABILITY
4
LETTERS
W,’ and F, are highly connected with functions If SEN*, economists. For instance, when s = 2 and for FE 9, we have w;((-moo,
x))=x-p&(x)
(where TF is the scaled TTI Propositionl.
ForanyFin
V’xaO,
function 9,
w:((x,
already
used in reliability
+ O”)) =
given in (l)), and so we obtain
T,*=(F*),.
September
pF(
1 - T,(x))
the following
theory
1989
or by
vx >, 0
result:
0
Furthermore these functions arise in the concept of stochastic dominance of order k. Deshpande et al. (1986) use the orders 1, 2 and 3 to characterize usual or new classes of life distributions. For instance, a second order stochastic dominance relation is given by FSSD,
G a
+Qi_ F(r) /X
CJ wI((x, which allows one for instance FE HNBUE where &(pF)
dt > t-00))
+m_ G(t) /X 2 w:((x,
to characterize
=
F(pF)
is the exponential
dt
Vx>O
+co))
the HNBUE
Vx > 0 class by:
SSD, F
with mean
pF.
3. A classification of non-parametric life distribution classes The characterizations by tailweight orderings given in (3) depend on properties of G-’ 0 F. This function associates with any F left tail (- co, x) the G left tail equally weighted (with the usual weight definition, i.e. in IL, sense). Then the classes IFR, IFRA, NBU and NBAFR may be considered as “L,-classes”. Therefore we can define in a natural way the corresponding IL,-classes for any s > 1 using (F *), instead of F * in Definition 1, or equivalently replacing F ~g by T,(F) where T,(F)(x)=l-~+W(r-~)“dF(t)/~Wt.FdF(f) (for s = 0, T,(F) properties:
= F and
for s = 1, T,(F)
Vx>O,Vs>,O = TF). Such
Proposition 2. For any (s, s’) E [l, + 00)~ and any FE% T,+,,(F)
a generalization
with finite
= T,&(F)).
Proof. vx 2 0, 1and
384
T,(I;.(F))(x)=~+~(u-x)~
dT,.(F)(u),Jo+PU’dt,,(F)(u)
is based
(s + s’)-mean,
on the following
Volume
8, Number
STATISTICS&PROBABILITY
4
So, using Fubini’s
theorem
and integration
September
LETTERS
1989
by parts,
I-~(T,,(F))(x)=[+~[[‘(u, x)‘-‘(t-u)“du]
dF(t),‘c
where ~=j,~/uI,‘-~(t-u)” Moreover
du dF(t).
for t > x, X’(,-x)t~L(t-U)I’du=(t-x)S’+‘j3(s, $
s’+l)
and so 1 - T,(T,@))(x) It is straightforward
=LtW(t-x)“+’
dF(t)/&?‘+.‘dF(t)
= 1 - c,+S(F)(x).
0
to verify the following:
Proposition 3. For any s >, 0 and any F E 9 (T,(F))*
with finite
(s + 1)-mean,
q
= (F*)s+l.
So, we have: Corollary. For any (s, s’) E [l, + 00)~ and any H symmetric H s+s,= (H&+1
= (H,,+,).s.
w.r. to zero with finite
(s + s’)-mean,
we have
0
Now, as the double exponential is invariant by the transformation H + H, (s > l), we can define following IL,-classes by any one of the four orders introduced in Definition 1.
the
Definition 4. For any s 2 1, FE s-IFR
w
(F*).%
FE s-IFRA
e
FE s-NBU
*
(F*).<, (F*)s
FE s-NBAFR
DE,
=
DE,
(F*)s
ct DE.
For any integer k, the above propositions show that Tk( F) and (F *)k can be obtained by iterations of the total time on test procedure. The stochastic dominances of order k used by Deshpande et al. (1986) are based on such iterations. When k = 2, Proposition 1 allows to see that the classes 2-IFR, 2-IFRA and 2-NBAFR are no more than the DMRL, DMRLHA and HNBUE respectively (see Deshpande et al., 1986). When k = 3, the class 3-NBAFR is the class HNBUE(3) defined in Deshpande et al. (1986) with a concept of third order stochastic dominance by F E HNBUE(
3)
F, TSD,
a
*
a(pF)
TSD,
F
where +M
forO
F,
jx
+oO_
1t
F,(u)
du dt 2 jtm X
j+“F2(u) f
du dt
-too. 385
Volume
8, Number
4
STATISTICS
& PROBABILITY
September
LETTERS
To complete this classification by using tailweight orderings, it remains to characterize So it is sufficient to define NBUFR classes. We know that FE NBUE e TF~ NBUFR. such that FENBUFR
w
the condition f(0) = 0 implies FE NBUFR. Then it F and G we define the following tailweight ordering s(x) = F(x/f(O)),
cJ rp>rc_
F* c7G*
= 1, and so the NBUFR
FE NBUFR
*
f(O)=0
class can be characterized
*
by
or F*<,DE.
The scaled d.f. F” has been used by Loh (1984) to define the tailweight that c7 is stronger than ct. Then the NBUE class is characterized by FENBUE
and c7
F* <,DE.
Let us remark that for d.f. belonging to 9, are such that f(0) # 0 and g(0) f 0, noting (for d.f. in F*):
We note that rz
the NBUE an ordering
1989
ordering
ct
and it is easy to prove
T;i,DE
(the density of T, at the origin is 1,‘~~). If we want to characterize the NBUE class by comparison of F * and DE with a tailweight ordering, it is clear by the previous results that this order must be based on tailweight in IL, sense. One way to do that is to use the scaled total time on test transform GF= TF 0 F-’ and to define F* cT G* so
@
+F
that, if we take G* = DE, the characterization FENBUE
e
F*<,DE
is no more than those given by comparison FENBUE
0
of TF and F with the usual stochastic
ordering:
TFcs, F.
The concept of [L,-tailweight allows one to rearrange the classes of life distributions, grouping in the same family those involving in their definitions the same tailweight. We summarize this classification in the following diagram:
IFR u IFRA II NBU II NBUFR u NBUFRA
Remarks.
distributions 386
3-classes. . _
2-classes
1 -classes 3
2-IFR
= DMRL
II 2-IFRA u 2-NBU a = NBAFR
t-NBUFR II 2-NBAFR
= DMRLHA
II 3-IFRA II 3-NBU
= NBUE
3-NBUFR
= HNBUE
II 3-NBAFR
= HNBUE(3)
(i) The introduction of s-tailweight allows unification of the various definitions of the usual life and simplifies their classification. Indeed the only reference distribution remains the double
Volume
8, Number
4
STATISTICS
& PROBABILITY
LETTERS
September
1989
exponential so that the classification only depends on the transformed distribution T,(F) and on IL,-tailweight orderings. (ii) The properties of the new classes corresponding to any s > 0 will be studied later. (iii) We have chosen to work with absolutely continuous distributions for sake of simplicity in the presentation, but it is easy to generalize to any distribution. (iv) The methodology here adopted, based on the IL, loss function, may be extended in a similar way to other loss functions.
References Abouammoh, A.M. (1988) On the criteria of the mean residual life, Statist.Probab. Lett. 6, 205-211. Averous, J. and M. Meste (1984), Parties centrales optimales dune loi de probabilite rtelle: M-fonction de centrage, C.R. Acad. Sci. Paris S&r. I Math. 298 (15) 365-368. Averous, J. and M. Meste (1989), Location, skewness and tailweight in O_, sense: a coherent approach (to appear). Barlow, R.E. and F. Proschan (1981) Statistical Theory of Reliability and Life Testing (To Begin With, Silver Spring, MD) Basu, A.P. and N. Ebrahimi (1984), On k-order harmonic new better than used in expectation distributions, Ann. Inst. Statist. Math. A 36, 87-100. Cao, J. and Y. Wu (1986) HDMRL life distribution class, Chinese J. Appl. Probab. Statist. 2, 314-321. Deshpande, J.V., SC. Kochar and K. Singh (1986), Aspects of positive ageing, J. App[. Probab. 23, 748-758. Hollander, M. and F. Proschan (1984) Non parametric concepts and methods in reliability, in : P.R. Krishnaiah and
P.K. Sen, eds., Handbook of Statistics, Vol. 4 (Elsevier Science Publishers, Amsterdam) pp. 613-655. Klefsjo, B. (1982), The HNBUE and HNWUE classes of life distributions, Naval Res. Logist. Quart. 29, 331-344. Langberg, N.A., R.V. Leon and F. Proschan (1980) Characterization of non parametric classes of life distributions, Ann. Probab. 8 (6), 1163-1170. Loh, W.Y. (1984). Bounds on ARE’s for restricted classes of distributions defined via tail orderings, Ann. Statist. 12, 685-701. Marshall, A.W. and F. Proschan (1972) Classes of distributions applicable in replacement, with renewal theory implications, in: L. Le Cam, J. Neyman and E.L. Scott, eds., Proc. Sixth Berkeley Symp. Math. Statist. Probab. Z (Univ. California Press, Berkeley, CA) pp. 395-415. Rolski, T. (1975), mean residual life, Bull. Znt. Statist. Inst. 46, 266-270.
387
Letters 8 (1989) 381-387
September
1989
Abstract: We use the concept of IL,-tailweight to classify the usual life distribution classes. This methodology allows define new classes and to characterize the NBUE class by comparison between the initial distribution and the exponential
us to one.
TAILWEIGHT
J. AVEROUS Lnboratoire
AND LIFE DISTRIBUTIONS
and M. MESTE
de Siaiistique
et Probabilittk,
lJ.A.-C. N. R.S. 745, Universith Paul Sabatier,
31062 Toulouse, France
Received April 1988 Revised October 1988
AMS
1980 Subject Classifications:
Keywords:
life distributions,
Primary
62N05;
Secondary
60ElO.
failure rate, total time on test function,
tailweight,
IL,-norm
1. Introduction We consider in this note the problem of characterizing some non-parametric classes of life distributions. To identify the type of a life distribution F (e.g. IFR, IFRA, . .), the most commonly considered tools are the hazard function R, = -log 3 (where F = 1 - F), the failure rate function r, = R; (for differentiable F) and the scaled total time on test transform ~~(r.4) = /~~‘(‘)- F(s) ds/pF (for F with finite expectation pLF) (see, e.g., Hollander and Proschan, 1984), or the scaled total time on test function T,(x) = GF 0 F(x) (also called equilibrium distribution by the economists). The characterization of a class by a property of these functions can often be equivalently expressed by a comparison either between F and the exponential distribution or between distributions associated with F. For instance, the IFR class is characterized by: (a) rF non-decreasing, (b) c$~ concave, (c) F cc d (cc being the Van Zwet skewness ordering and d an exponential distribution), (d) F,, cbt F,, (0 =Ct, < tz) (where E is the survival function at age t and cs, is the usual stochastic ordering). _ For such characterizations, Loh (1984) uses a tailweight ordering to compare the symmetrized distribution associated with F: F*(x) = i(l + sgn(x)F( 1X I)) with the double exponential. This is equivalent to use the corresponding skewness ordering to compare F with an exponential distribution. Nevertheless, Loh notes that the characterization of the NBUE class with such orderings remains an open question. Deshpande et al. (1986) base a classification of the usual families on the concept of stochastic dominance. Introducing different tailweight or stochastic orderings, these authors define new classes of life distributions: NBAFR (Loh, 1984) IFR(2), NBU(2), . . . (Deshpande et al., 1986). Another approach is used by Abouammoh (1988) or Basu and Ebrahimi (1984) who introduce families by new properties of the mean residual life distribution. In a paper to appear, we introduce the concept of I_,Y-description of any distribution F, based on IL,-tailweight (s > 1). The case s = 1 corresponds to the usual tailweight (F(x) is the weight of the left tail (- cc, x) and 1 - F(y) those of (y, + cc)). For the case s = 2 (when pLF exists), the weight /? m F(t) dt is assigned to the left tail (- cc, x) and /,? w (1 - F(t)) dt to the right tail (y, + cc). These concepts allow one to clarify the classification of the various life distribution classes and also to construct a methodology 0167-7152/89/$3.50
0 1989, Elsevier Science Publishers
B.V. (North-Holland)
381
Volume
8, Number
STATISTICS&PROBABILITY
4
LETTERS
for making new classes. Moreover to complete the Loh characterization, which answers to the Loh remark about the NBUE class.
September
we propose
a tailweight
1989
ordering
2. Preliminaries Let 9 be the family of absolutely continuous distributions F, such that F(0 - ) = 0 and with finite mean Pi. We note TF the d.f. obtained by normalizing with pF the total time on test function,
T,(x) = &ix[l-F(u)] du, and F * the symmetrized
version
(1)
of F,
(2)
F*(x)=:[l+sgn(x)F(Jx1)]. orderings
We recall the usual tailweight
on S*
= {F*;
FEF}.
Definition 1. F*<,,G*
Go ’ 0 F convex
0
F*<,G*
G-’ 0 F * -shaped
w
F * -cSuG * = F*
(Van Zwet ordering),
G-’
e
G-’
(Lawrence
ordering),
0 F super additive, 0 F( x)/x
af(O)/g(O)
Vx f 0
We give the following current classes of life distributions > t} (t E [0, l[), F-‘(l) = sup{x; F(x) < l}.
(Loh ordering)
in 9,
where 2 = 1 - F, Fp ‘( t) = inf{ x; F(x)
Definition 2. FEIFR
S+ F is log-concave,
FEIFRA
*
FENBU
e
- log( F( t ))/t
+t2)d(tl)F(t>)
F(t,
FENBUE
CJ
FEDMRL
a
is increasing
/0
vt,,
+mF(t+x)dx
-!-
F(t)
/f
+oO_ F(x)
in t > 0, t,ao, VtaOsuchthat
dx is decreasing
+30_
FE HNBUE
a
/f
F(x)dx,(~~exp{-1,‘~~)
(Rolski, FE NBUFR FE NBAFR 382
=
(or NBUFRA)
in t 2 0,
VtaOsuchthat
1975),
T-F(t) a+(O)
vt>o, e
(l/t)
log F(t)
F(t)>O,
2 rF(O).
F(t)>0
Volume
8, Number
STATISTICS
4
It is known (Loh, 1984; Hollander implications hold:
& PROBABILITY
and Proschan,
September
LETTERS
1984; Basu and Ebrahimi,
FENBUFR FEIFR
3
Using comparisons Loh, 1984):
FEIFRA
with the double
FEIFR
F* <_
=
FEIFRA
=
exponential
F’
=
FENBUE
(DE), the following
1989
1984) that the following FE NBAFR
=
characterizations
FEHNBUE
have been given (e.g.
DE,
CJ F*<,DE,
FENBU
=
(3)
F*<,,DE, c=, F* ct DE.
FENBAFR
As far as we know, no such characterization exists for the NBUE class. In Averous and Meste (1989) we pointed out that the usual tailweight orderings (given in Definition 1) rather correspond to orderings in IL, sense. In this case, F(x) is the weight of the left tail (- cc, x) of F, 1 - F(y) being the weight of the right tail (y, + cc). We introduced a concept of tailweight in ILs sense (S > 1) which is closely related to the total time on test function when s = 2. Let us recall the main definitions and properties. 3. For any s >, 1, and any F with finite CO, x) (resp. the right tail (y, + 00)) is
Definition
(-
w,“((-, rev
4)
determines two neighbourhood To facilitate the distribution
w,“((Y>
+M))=~y,+p)lf-yl~~-’
arg min jt-mlJ m /R
s - 1, the L ,-weight
of the left tail
dF(t)
i
that the s-mean dF(t)
equally IL,-weighted tails (with usual restrictions when s = 1 and F non-continuous in a of the median). the description of F in IL,Tsense, for s > 1, we have introduced (Averous and Meste, 1989) function 6. defined by: W,“((-a,
F,(x)
of order
=~_m_~,l-I~y-’dF(f)
It follows from this definition ms=
moment
=
1 1 - y,‘((x,
x))/~@((-M, + m))/2WsF((
m.,)), - 00, m,<)),
if x < m,, if x > m,?.
When s = 1 and F is continuous, 2W,(( - co, med( F))) = 1). So the d.f. Ts describes normalized s-tailweights of F just as F does when s = 1. Then a description of c’ in [L, sense is equivalent to a description of F in O_r sense. For instance the median of FY (i.e. the IL, location parameter of F.) equals the s-mean of F. 383
Volume
8, Number
STATISTICS&PROBABILITY
4
LETTERS
W,’ and F, are highly connected with functions If SEN*, economists. For instance, when s = 2 and for FE 9, we have w;((-moo,
x))=x-p&(x)
(where TF is the scaled TTI Propositionl.
ForanyFin
V’xaO,
function 9,
w:((x,
already
used in reliability
+ O”)) =
given in (l)), and so we obtain
T,*=(F*),.
September
pF(
1 - T,(x))
the following
theory
1989
or by
vx >, 0
result:
0
Furthermore these functions arise in the concept of stochastic dominance of order k. Deshpande et al. (1986) use the orders 1, 2 and 3 to characterize usual or new classes of life distributions. For instance, a second order stochastic dominance relation is given by FSSD,
G a
+Qi_ F(r) /X
CJ wI((x, which allows one for instance FE HNBUE where &(pF)
dt > t-00))
+m_ G(t) /X 2 w:((x,
to characterize
=
F(pF)
is the exponential
dt
Vx>O
+co))
the HNBUE
Vx > 0 class by:
SSD, F
with mean
pF.
3. A classification of non-parametric life distribution classes The characterizations by tailweight orderings given in (3) depend on properties of G-’ 0 F. This function associates with any F left tail (- co, x) the G left tail equally weighted (with the usual weight definition, i.e. in IL, sense). Then the classes IFR, IFRA, NBU and NBAFR may be considered as “L,-classes”. Therefore we can define in a natural way the corresponding IL,-classes for any s > 1 using (F *), instead of F * in Definition 1, or equivalently replacing F ~g by T,(F) where T,(F)(x)=l-~+W(r-~)“dF(t)/~Wt.FdF(f) (for s = 0, T,(F) properties:
= F and
for s = 1, T,(F)
Vx>O,Vs>,O = TF). Such
Proposition 2. For any (s, s’) E [l, + 00)~ and any FE% T,+,,(F)
a generalization
with finite
= T,&(F)).
Proof. vx 2 0, 1and
384
T,(I;.(F))(x)=~+~(u-x)~
dT,.(F)(u),Jo+PU’dt,,(F)(u)
is based
(s + s’)-mean,
on the following
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So, using Fubini’s
theorem
and integration
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LETTERS
1989
by parts,
I-~(T,,(F))(x)=[+~[[‘(u, x)‘-‘(t-u)“du]
dF(t),‘c
where ~=j,~/uI,‘-~(t-u)” Moreover
du dF(t).
for t > x, X’(,-x)t~L(t-U)I’du=(t-x)S’+‘j3(s, $
s’+l)
and so 1 - T,(T,@))(x) It is straightforward
=LtW(t-x)“+’
dF(t)/&?‘+.‘dF(t)
= 1 - c,+S(F)(x).
0
to verify the following:
Proposition 3. For any s >, 0 and any F E 9 (T,(F))*
with finite
(s + 1)-mean,
q
= (F*)s+l.
So, we have: Corollary. For any (s, s’) E [l, + 00)~ and any H symmetric H s+s,= (H&+1
= (H,,+,).s.
w.r. to zero with finite
(s + s’)-mean,
we have
0
Now, as the double exponential is invariant by the transformation H + H, (s > l), we can define following IL,-classes by any one of the four orders introduced in Definition 1.
the
Definition 4. For any s 2 1, FE s-IFR
w
(F*).%
FE s-IFRA
e
FE s-NBU
*
(F*).<, (F*)s
FE s-NBAFR
DE,
=
DE,
(F*)s
ct DE.
For any integer k, the above propositions show that Tk( F) and (F *)k can be obtained by iterations of the total time on test procedure. The stochastic dominances of order k used by Deshpande et al. (1986) are based on such iterations. When k = 2, Proposition 1 allows to see that the classes 2-IFR, 2-IFRA and 2-NBAFR are no more than the DMRL, DMRLHA and HNBUE respectively (see Deshpande et al., 1986). When k = 3, the class 3-NBAFR is the class HNBUE(3) defined in Deshpande et al. (1986) with a concept of third order stochastic dominance by F E HNBUE(
3)
F, TSD,
a
*
a(pF)
TSD,
F
where +M
forO
F,
jx
+oO_
1t
F,(u)
du dt 2 jtm X
j+“F2(u) f
du dt
-too. 385
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To complete this classification by using tailweight orderings, it remains to characterize So it is sufficient to define NBUFR classes. We know that FE NBUE e TF~ NBUFR. such that FENBUFR
w
the condition f(0) = 0 implies FE NBUFR. Then it F and G we define the following tailweight ordering s(x) = F(x/f(O)),
cJ rp>rc_
F* c7G*
= 1, and so the NBUFR
FE NBUFR
*
f(O)=0
class can be characterized
*
by
or F*<,DE.
The scaled d.f. F” has been used by Loh (1984) to define the tailweight that c7 is stronger than ct. Then the NBUE class is characterized by FENBUE
and c7
F* <,DE.
Let us remark that for d.f. belonging to 9, are such that f(0) # 0 and g(0) f 0, noting (for d.f. in F*):
We note that rz
the NBUE an ordering
1989
ordering
ct
and it is easy to prove
T;i,DE
(the density of T, at the origin is 1,‘~~). If we want to characterize the NBUE class by comparison of F * and DE with a tailweight ordering, it is clear by the previous results that this order must be based on tailweight in IL, sense. One way to do that is to use the scaled total time on test transform GF= TF 0 F-’ and to define F* cT G* so
@
+F
that, if we take G* = DE, the characterization FENBUE
e
F*<,DE
is no more than those given by comparison FENBUE
0
of TF and F with the usual stochastic
ordering:
TFcs, F.
The concept of [L,-tailweight allows one to rearrange the classes of life distributions, grouping in the same family those involving in their definitions the same tailweight. We summarize this classification in the following diagram:
IFR u IFRA II NBU II NBUFR u NBUFRA
Remarks.
distributions 386
3-classes. . _
2-classes
1 -classes 3
2-IFR
= DMRL
II 2-IFRA u 2-NBU a = NBAFR
t-NBUFR II 2-NBAFR
= DMRLHA
II 3-IFRA II 3-NBU
= NBUE
3-NBUFR
= HNBUE
II 3-NBAFR
= HNBUE(3)
(i) The introduction of s-tailweight allows unification of the various definitions of the usual life and simplifies their classification. Indeed the only reference distribution remains the double
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1989
exponential so that the classification only depends on the transformed distribution T,(F) and on IL,-tailweight orderings. (ii) The properties of the new classes corresponding to any s > 0 will be studied later. (iii) We have chosen to work with absolutely continuous distributions for sake of simplicity in the presentation, but it is easy to generalize to any distribution. (iv) The methodology here adopted, based on the IL, loss function, may be extended in a similar way to other loss functions.
References Abouammoh, A.M. (1988) On the criteria of the mean residual life, Statist.Probab. Lett. 6, 205-211. Averous, J. and M. Meste (1984), Parties centrales optimales dune loi de probabilite rtelle: M-fonction de centrage, C.R. Acad. Sci. Paris S&r. I Math. 298 (15) 365-368. Averous, J. and M. Meste (1989), Location, skewness and tailweight in O_, sense: a coherent approach (to appear). Barlow, R.E. and F. Proschan (1981) Statistical Theory of Reliability and Life Testing (To Begin With, Silver Spring, MD) Basu, A.P. and N. Ebrahimi (1984), On k-order harmonic new better than used in expectation distributions, Ann. Inst. Statist. Math. A 36, 87-100. Cao, J. and Y. Wu (1986) HDMRL life distribution class, Chinese J. Appl. Probab. Statist. 2, 314-321. Deshpande, J.V., SC. Kochar and K. Singh (1986), Aspects of positive ageing, J. App[. Probab. 23, 748-758. Hollander, M. and F. Proschan (1984) Non parametric concepts and methods in reliability, in : P.R. Krishnaiah and
P.K. Sen, eds., Handbook of Statistics, Vol. 4 (Elsevier Science Publishers, Amsterdam) pp. 613-655. Klefsjo, B. (1982), The HNBUE and HNWUE classes of life distributions, Naval Res. Logist. Quart. 29, 331-344. Langberg, N.A., R.V. Leon and F. Proschan (1980) Characterization of non parametric classes of life distributions, Ann. Probab. 8 (6), 1163-1170. Loh, W.Y. (1984). Bounds on ARE’s for restricted classes of distributions defined via tail orderings, Ann. Statist. 12, 685-701. Marshall, A.W. and F. Proschan (1972) Classes of distributions applicable in replacement, with renewal theory implications, in: L. Le Cam, J. Neyman and E.L. Scott, eds., Proc. Sixth Berkeley Symp. Math. Statist. Probab. Z (Univ. California Press, Berkeley, CA) pp. 395-415. Rolski, T. (1975), mean residual life, Bull. Znt. Statist. Inst. 46, 266-270.
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