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Department of Mathematical Statistics University of Umeå S-901 87 Umeå Sweden
No. 1980-4 March 1980
IFRA OR DMRL SURVIVAL UNDER THE
PURE BIRTH SHOCK PROCESS
by
Bengt Klefsjö
American Mathematical Society 1970 subject classification: 60K10, 62N05.
Key words and phras es: Shock model, birth process, survival function,
variation diminishing property, total positivity, IFRA, DFRA, DMRL, IMRL.
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ABSTRACT
Suppose that a device is subjected to shocks and that P^, k • 0, 1, 2, 00
denotes the probability of surviving k shocks. Then H(t) = E P(N(t) = k)P, k=0
is the probability that the device will survive beyond t, where N =
(N(t): t > 0} is the counting process which gover ns the arrival of shocks.
A-Hameed and Proschan (1975) considered the survival function H(t) under
what they called the Pure Birth Shock Model. In this paper we shall prove
that H(t) is IFRA and DMRL under conditions which differ from those used
by A-Hameed and Proschan (1975).
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1.
1. Introduction
Suppose that a device is subjected to shocks and that the probability of
s u r v i v i n g k s h o c k s i s P ^ , k = 0 , 1 , 2 , w h e r e 1 = P q > > • • .
If N » {N(t): t >0} is the counting process which governs the arrival
of shocks, the probability H(t) that the device will survive beyond t
is given by
00
(1.1) H(t) = I P(N(t) - k)P . k=0
Shock models of this kind have been studied by se veral authors; e.g. Esary
Marshall and Proschan (1973), A-Hameed and Proschan (1973, 1975), Block
and Savits (1978) and Klefsjö (1980).
In this paper we shall study the following Pure Birth Shock Model con
sidered by À-Ha&eed and Proschan (1975):
Shocks occur according to a Markov process; given that k shocks have
occurred in (0,t], the probability of a shock in (t, t + At] is
A^A(t)At + o(At), while the probability of more than one shock in
(t, t + At] is o(At).
For this shock model the survival function H(t) in (1.1) can be written
as
(1.2) H(t) = S(A(t))
with
00
(1.3) S(t) = I Mt)P, » k=0
where
(1.4) zk^c^ = P(e*âctly k shocks have occurred in (0,t] when A(t)
and
t A(t) * I X(x)dx
0
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2.
This means that S is the survival function when the shocks occur according
to a stationary birth process for which the interarrivai times between
the shocks number k and k + 1, k = 0, 1, 2, are independent and
exponentially distributed with mean . 00
We are only interested in sequences (^^k-0 ^°r t*ie Pr°bability CO
of infinitely many shocks in (0,t] is zero, i.e. for which E = !• k=0
By the Feller-Lundberg Theorem (see Feller (1968), p. 452) this is equiva-OO
lent to the con dition that E X, = This condition is not explicitly k=0
mentioned by A-Hameed and Proschan (1975).
A—Hameed and Proschan (1975) proved that H(t) in (1.2) is IFR, IFRA,
NBU, NBUE and DMRL (for definitions see e.g. Barlow and Proschan (1975))
00 — °o under suitable conditions on anc* ^k^k=0* ^or instance
they proved that
(a) H(t) is IFRA if (^.>£=*0 is increasing> is starshape-d (i.e.
_ OO
A(t)/t is increasing for t > 0 and A(0) = 0) and (^^=0 ^as
the discrete IFRA property. (Theorem 2.6 in A-Hameed and Proschan
(1975).)
/ 00 \
(b) H(t) is DMRL if A(t) is increasing and ( I P.XT )/P, is decreasing \js=k
in k = 0, 1, 2, .i. .(Theorem 2.10 in A-Hameed and Proschan (1975).)
In this paper we shall prove that H(t) is IFRA under weaker condi
tions on (A^)^_q and (*^k=0 than those used by A-Hameed and Proschan 00
(1975). We shall also prove that H(t) is DMRL when A(t) and ^^^k=0
are increasing and has the dis<irete KIRL property. Finally, we
present dual theorems in the DFRA and IMRL cases.
2. IFRA survival
The main theorem of this section is the following one.
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3.
THEOREM 2.1 The survival function H(t) in (1.2) is IFRA if
(i) A(t) is starshaped
and
- k _ 1 -1 (ii) for every 0 with 0 < 0 < Xn the sequence P. - II (1-0X. ),
v K • /\ 1 j=o k= 1, 2, 3, ..., changes sign at most once, and if onc e, from + to -
and
(iii) there is a jn such that X. > \n for j > jrt. J0 j - 0 J - J0
To prove that theorem we use the variation diminishing property of the to
tally positive kernel k = 1, 2, 3, ..., and t € [O, 00) (see Karlin
(1968), Ch. 1). The fact that is totally positive follows from Theorem
3 in Karlin and Proschan (1960). We also need the following lemma.
LEMMA 2.2 Suppose that there is a Jq such that 0 < 0 < X. for j > jg. k-1 J oo
Let Eq = 1 and E^= II (1-0X. ), k=l,2,3, ..., and let L(t) = E z^(t) E^, j=0 -1 » k=0
where z, (t) is given by (1.4). Then L(t) = exp(-0t) if Z X, = °°. k=0 k
PROOF Let denote the Laplace transform of z^(t), i.e.
00
\(s) • / e stzk(t)dt.
Then
V'> - 1^7
\'s' = X. + s) X + s £or k " l' 2' 3-J=0 j ' k
(see Feller (1971), p. 489). From this follows that the Laplace transform
g(s) of L(t) is given by
00 00 i °° /k_1 " 6\
(2.1) S,(s) « / e s L(t)dt = Z E,it (s) » *—— + Z (II Y J ) r—— n . n k k A n + s 1 - \ . n A . + s / A 1 + s 0 k=0 0 k=l Nj=0 j ' k
is this series converges.
With
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4.
and
/k-1 A .. - 6 \ ßk = 7^3" ( n rT7 ) for k = 1, 2, 3, ...
we can write the terms of the series in (2.1) as
1 - ßn - ß, XQ+s ^0 "1
^k-1 X. - 0X for k = 1, 2, 3, ....
/K-l A . - D\ ,
(a v7) = ßk 'ßk+i
Accordingly, the nth partial sum is equal to
n x / n X.-0V
EAV S ) • eo - * TTê v 1 -
Using the uniqueness of the Laplace transform (see Felle r (1971), p. 430)
it is now sufficient to prove that, for s > 0,
n X. - 0 00 , (2.2) lim n -r^-T— = 0 if E X.
5-0 Vs j»o J
But from the Mean Value Theorem we get that
Än(Xj + s) - An(Xj - 0) > £—• for j > jQ
and hence that
n X. - 0 n 1
-£n n ,J ^ > (s + 0) I . . . . X. + s - •' X. + s J=J 0 J J
From this inequality follows (2.2) and the proof is complete. •
PROOF OF THEOREM 2.1 From Lenma 2.1 in A-Hameed and Proschan (1975) it
follows that the survival function H(t) in (1.2) is IFRA if S(t) in
(1.3) is IFRA and A(t) is starshaped. Accordingly, it is sufficient to
-m 00 — OO prove that S(t) is IFRA if ^^^=0 a™* ^k^k«0 sat^sfy condi
tions (ii) and (iii). This will be done by showing that, for every 0 > 0,
S(t) - exp(-0t) changes sign at most once, and if once, from + to - .
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That this is sufficient for S(t) to be IFRA follows from Th eorefi 2.18
in Barlow and Proschan (1975), p. 89.
Since
00
SCO = + E Zjc(t)Pk > zQ(t) = exp(-Xpt) k=l
we get that S(t) > exp(-0t) if 0 > Xq. NOW suppose 0 < 0 < Xq. Then
Lemma 2.2 gives that
00
S(t) - exp(-0t) • S(t) - L(t) = Z z^(t){Pk - Efc}, k* X
where
Eo = 1
k _ 1 -1 E = ÏÏ (1 - ex. ) for k = 1, 2, 3 k j=0 3
Since is totally positive in k = 1, 2, 3, ... and t > 0 it
follows from the variation diminishing property that S(t) - exp(-Qt) has
the sane sign changes as - Efc, k «= 1, 2, 3, ... . This completes the
proof. o
We shall now-prove that our conditions (ii) aad (iii) in Theorem 2.1 hold
00 — 00
when an(* ^k^k-Q satisfy the conditions used by A-Hameed and Proschan
(1975) in their Theorem 2.6 (cf. (a) on p. 2).
THEOREM 2.3 The conditions (ii) and (iii) in Theorem 2.1 hold if
• — 00
is increasing and ^as t^ie discrete IFRA property.
PROOF That condition (iii) is satisfied is obvious. To prove that (ii) holds,
00 take a 0 with 0 < 0 < Aq, Since ^-ncreâS^në it can proved
by induction that the geometric mean
fk-1 . il/k f -1 M. (9) = <! ii (i-ex/) =o J
"*"l/k ® ^ oo is increasing in k. Since further (P^ ^-1 decreasing if
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6.
-1/k is IFRA we get that - (̂9) is decreasing in k. Accordingly,
- Mj^(Q), k = 1, 2, 3, changes sign at most once, and if one
- -i change occurs, it occurs from + to - . Since P, - Il (1-9X. ) has
1=0 J
-1/k . the same sign as P^ - ̂ (9) Pro°f is complete. •
We shall now give an example where conditions (ii) and (iii) in Theorem 2.1
» CO f
hold although the sequence of survival probabilities is not IFFA and
00
is not increasing. Together with Theorem 2.3 this shows that the
OO _ 00 conditions on and ^k^k=0 used Theorem 2.1 are weaker than
those used by A-Haneed and Proschan (1975) in their Theorem 2.6.
Choose PQ = 1, = 3/4, ?2 = 5/8, ?3 = 2/5, Pfe = 0 for k > 4, XQ = 1,
X^ » 2, X2 = 1 and \ = k for k - 3* Then ^k^k=0 is not IFRA since
P < Furt^er ^k^k»0 DOt * But holds and
- k'X -1 Y, » P, - II (1-0X. ) for k = 1, 2, 3, ... k k j-0 J
satisfies the condition (ii) in Theorem 2.1 since straight-forward calcula
tions show that y1 > y2 > Y 3 ^ \ < 0 for k -
3. DMRL survival
In this section we shall prove the following the orem.
THEOREM 3.1 The survival function H(t) in (1.2) is DMRL if X(t) and
00 " 00
^\^k=0 are increas"*S and ^Pk\=0 haS the discrete DMRL property.
PROOF Fran Lenma 2.2 in A-Haneed and Proschan (1975) it follows that H(t)
is DMRL if X(t) is increasing and S(t) in (1.3) is DMRL. Furthermore,
Theorem 2.10 is A-Hameed and Proschan (1975) gives that S(t) is DMRL if / °° \
( E P.XT1j/P is decreasing in k=0, 1, 2, ... . Accordingly, it is suff i-\j=k J J / • * cient to prove that
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V* Vj 1 " - 0 f 0 r k = °' l f 2' " j=k J J j=k+l
if ^k^k=0 an<* ^k^k=0 *"S increasin§* °° —1 ""1
Since (\)k_o is increasing lim X fc = X >0 exists. From this k-*»
follows that
00 1 oo p" z p.xT - p~* z p.xT1 = k • i J J k+1 . i , 1 1 jsk J J j=k+l J J
= [a"1/?-1 L . . . L V k ^_T, J k+l _•
P, " Z P. - p"1
j-k z P-.y
j-k+l J/
• [f"1 i p «"'-A"1) - PT' i P UT'-X"1) L j=k J J k 1 j=k+l J J ' J
= I + II,
— OO
Here the expression I is non-negative since ^^^=0 *"s ^ we wr:*-te
00
xT1 - x"1 = i 1 .V v+1
V=J
and then change the sunmation order we can write the expression II as
oo V 00 _ V -z (X"1-x"î;1) p"1 z p. - z (x'1-x"J;1) p.-* z p. = „ , V V+1 k . , j v V+l k+1 . . i V=k j=k J V=k+1 J^k+l J
-1 -i i v _ 00 -i f—i v _ —i v _ 1 • (xi - K ̂ ) Pi S P • + 2 ()I -H ;,) P. ÏP.-P.:. z p. > k k+1 k . , i , v v+1 I k , . i k+1 . , i j J j=k J v=k+l "• j=k J j=k+l JJ
If
v . v _ $, (v) = P, Z P. - P. _7- Z P. > 0 k k . j k+1 . , ., j -j=k J j=k+l J
for v = k + 1, k + 2, k + 3, ... the proof is complet e. But this follows
from the facts that
* kcv) - ̂ (VD - P v + 1(P;Ì! - p" 1) > o
— oo since g "^s decreasing, and that
00 00
lia» * (v) = p"1 z p - p~* Z P > 0 V-*» j=k J j=k+l J
since (P. ), . is DMRL. k k=0
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8.
GO * OO
Note that the conditions on anc* ^k^k=0 âre eclu^va^ent: anc*
analogous, respectively, to those used by A-Hameed and Proschan (197 5) in
the IFR, IFRA and NBU cases and by Block and Savits (1978) in the NBUE case.
/ 00 \ We shall now illustrate that ^ E k = 1, 2, 3, may be
oo ^ — OO decreasing although neither (X^^-q is increasing nor
00 / — .00
This means that the conditions on (X, ). n and (P. ). n used in Theorem kk=0 k k=0
3.1 is stronger than those used by A-Hameed and Proschan (1975) in their
Theorem 2.10 (cf. (b) on p. 2).
Choose P0 = 1, P^ = 3/4, P^ = 1/2, P^ = 2 ^ for k > 3, Aq = = 1,
00 X'2 Ä 1/2 and X^ = 1 for k > 3, Then (^k^k=0 not increasing and
(P^k^o not ^MRL since ^ £ V^j/V^ = 3/2 and ^ E P^/P^ = 2. But
00
E P L 1 / P = \ j=k k
( 1 \ and therefore ^ y^k* k = 0, 1, 2, 3, is decreasing<
3 for k = 0
8/3 for k = 1
5/2 for k = 2
2 for k > 3
^j=k
4, The dual cases
By reversing all inequalities and the directions of monotonicity we get
dual results to Theorem 2.1 and Theorem 3.1,
THEÜREH *+.1 ìtie survival function H(t) in (1.2) is DFRA if
(i) A(t) is antistarshaped (i.e. A(t)/t is decreasing for t > 0
and A(0) ® 0)
and
- k"1 -1 (ii) for every 0 with 0 < 6 < X the sequence P, - H (1 - 0X. ),
j*o J
k*l, 2, 3, ... , changes sign at most once, and if once, from - to +
and
(iii) there is a Jq such that Xj > Aq for j > jg.
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9.
THEOREM 4.2 The survival function H(t) in (1.2) is IMRL if X(t) and
00 — 00
are decreasing and k-0 ^as t'ie ^lscrete 1*®^ property.
Acknowledgement
The author is very grateful to Dr Bo Bergman, Professor Gunnar Kulldorff
and Dr Kerstin Vännman for valuable comments.
References
A-Hameed, M.S. and Proschan, F. (1973). Nonstationary shock models. Stochastic
Process. Appi., 1, 383-404.
A-Hameed, M.S. and Proschan, F. (1975). Shock models with underlying birth
process. J. Appi. Probab., 12, 18-28.
Barlow, R.E. and Proschan, F. (1975). Statistical Theory of Reliability and
Life Testing. Probability Models. Holt, Rinehart and Winst on,
New York.
Block, H.W. and Savits, T.H. (1978). Shock models with NBUE survival. J.
Appi. Probab., 15, 621-628.
Esary, J.D., Marshall, A.W. and Proschan, F. (1973). Shock models and wear
processes. Ann. Probab., 1, 627-649.
Feller, W. (1968). An Introduction to Probability Theory and its Applications.
Vol. 1, 3rd ed. John Wiley & Sons, New York.
Feller, W. (1971). An Introduction to Probability Theory and its Applications.
Vol. II, 2nd ed. John Wiley & Sons, New York.
Karlim, S. and Proschan, F. (1960). Polya type distributions of convolutions.
Ann. Math. Statist., 31, 721-736.
Karlin, S. (1968). Total Positivity. Vol. 1. Stanford University Press, Stanford.
Kief sjö, B. (1980). HNBUE survival under seme shock models. Research Report
1980-3, Department of Mathemat ical Statistics, University of Umeå.
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