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.

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).

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

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.

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

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 - .

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

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

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

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.

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

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