On a nonparametric family of life distributions and its dual

Journal of Statistical North-Holland

Planning

and Inference

On a nonparametric and its dual Murari

385

39 (1994) 385-397

family of life distributions

Mitra

Indian Statistical

Institute,

Calcutta,

India

Sujit K. Basu Indian Institute Received

of Management

Calcutta,

India

22 June 1992/ Revised manuscript

received

25 January

1993

Abstract In this paper, we discuss a nonparametric family of life distributions (called the NWBUE family) which includes the IDMRL class ofdistributions introduced by Guess, Hollander and Proschan as well as all BFR distributions. We prove two inequalities which are then used to obtain bounds for moments of a NWBUE life distribution. The bounds thus obtained are shown to be related to the moments of an appropriate negative exponential distribution, and a characterization of the exponential distribution is derived as a consequence. Closure under weak convergence and the equivalence of weak convergence and moment convergence in the NWBUE family are established under mild conditions. Similar properties in respect of the dual family comprising of NBWUE distributions have also been explored. AMS Subject Classification:

Primary

62N05, 90B25; secondary

Key words: Different nonparametric classes convergence; convex and concave functions.

60F05.

of life distributions

based

on notions

of ageing;

weak

1. Introduction The concept of ageing is of vital importance in reliability theory. ‘No ageing’ means that the age of a component has no effect on the distribution of its residual lifetime. ‘Positive ageing’ describes the situation where the residual lifetime tends to decrease, in some probabilistic sense, with increasing age of the component. This phenomenon has been described in the literature in terms of differnt ageing notions which have given rise to a whole variety of nonparametric classes of life distributions, e.g. IFR, IFRA, DMRL, NBU, NBUE, etc. (see e.g. Barlow and Proschan, 1981). We have

Correspondence to: Dr. SK. Basu, Indian Institute P.O. Box No. 16757, Calcutta 700027, India.

037%3758/94/$07.00 0 1994-Elsevier SSDI 0378-3758(93)E0048-L

of Management,

Science B.V. All rights reserved.

Joka,

Diamond

Harbour

Road,

386

M.

a corresponding concept of negative (beneficial) like DFR, DFRA, IMRL, NWU and NWUE. In many

practical

situations,

however,

ageing and the resulting

the effect of age is initially

dual classes beneficial

(a

‘burn-in’ phase where negative ageing takes place), but after a certain period, the effect of age is adverse indicating a ‘wear-out’ phase where the ageing is positive. Situations displaying infant mortality or work hardening such ageing. This kind of nonmonotonic

of certain tools are common examples of ageing is typically modelled using life

distributions displaying bathtub failure rates (BFR). Guess et al. (1986) have introduced the class of IDMRL distributions to study the above phenomenon through the mean residual life (MRL) function. Let us introduce some notations and a few definitions to make these notions more explicit. Let F be a life distribution function (d.f.) and let F( .)= 1 -F( .) denote the corresponding survival function. Let the cumulative hazard function be denoted by R(t)+ -log,F(t). The mean residual life (MRL) function at age t20 is defined as

e&):=8(X-t,x>+ whenever

m

F(x) dx.

F(t) > 0.

Definition 1.1. A life distribution F is said to be a BFR (‘upturned bathtub failure rate’ (UBFR)) distribution if there exists a to 20 such that R(t) is concave (convex) on [0, to) and convex (concave) on [to, co). Definition 1.2. A life distribution F is said to be an ‘increasing initially, then decreasing mean residual life’ (IDMRL) (‘decreasing initially, then increasing mean residual life’ (DIMRL)) distribution if there exists a co 20 such that eF(t) is nondecreasing (nonincreasing) on [0, to) and nonincreasing (nondecreasing) on [to, co); in such a case, we shall say that F is IDMRL (to) (DIMRL(t,)). It is to be noted that the above definition of a BFR distribution is extremely general and extends the idea of distributions possessing a bathtub-shaped failure rate to situations

where the failure rate itself does not exist.

In this paper, we discuss a nonparametric class of life distributions which not only provides models for the kind of nonmonotonic ageing we are talking about, but also includes the IDMRL class of Guess et al. (1986) as well as the family of BFR distributions. To this end, we have the following definition. Definition 1.3. A life distribution F having support on [0, co) (and finite mean P) is said to be ‘new worse then better than used in expectation’ (NWBUE) (‘new better then worse than used in expectation’ (NBWUE)) if there exists a point x0 BO such that eF (-4

>(<)eF(0)

for

<(>)ep(0)

for x2x0.

x
(1.1)

M. Mitra. S.K. Basu 1 Nonparametric

387

.family of l(f>distributions

We shall refer to such an x,, (which need not be unique)

as a change point of the d.f. F;

we shall write F NWBUE(xO) (NBWUE(x,)) to indicate that the life distribution F is NWBUE (NBWUE) in the above sense and x0 is a change point of F. Let %‘Fbe the collection of all change points of an NWBUE (or NBWUE) life d.f. F; it is plain to see that for a continuous NWBUE (NBWUE)

d.f. F, %YFis either a singleton or a closed interval. Note that an F is NBUE (NWUE) if 0~57~ while it is NWUE (NBUE) if

COEWF. Example

1.4. Consider F(x)=

The corresponding

eF (x)

the life distribution

(2/V + 4)’ >

Odx
{ (4x/9)exp(l-x2)/6,

lbx
MRL function

=

whose survival

function

is given by

is given by

2+x,

Odx
i 31%

x> 1.

It is clear that F is NWBUE(xO)

with x0=3/2.

In the next section, we discuss the interrelationships amongst NWBUE, BFR and IDMRL classes of life distributions. In Section 3, we obtain some useful bounds for the moments of NWBUE distributions; we also provide a characterization of the exponential distribution as a consequence of some of the results discussed in this section. In Section 4, we prove closure under weak convergence and the equivalence of weak convergence and moment convergence in the NWBUE family of life distributions and also furnish a corollary which is an interesting by-product of our results. Our observations relating to the dual class comprising of NBWUE distributions are given in Section 5.

2. Relationships In this IDMRL

amongst NWBUE,

section,

we explore

BFR and IDMRL

the interrelationships

classes among

NWBUE,BFR

and

classes of life distributions.

Proposition 2.1. If a continuous then F is NWBUE.

and strictly increasing

Proof. Since F is continuous and strictly increasing, (TTT)-transform Ic/F(.) of F as

life d.jI F is BFR with mean p,

we can write the total time on test

388

M. Mitra, S.K. Basu 1 Nonparametric

where F(x)=t. O
family

that F is NWBUE

It then follows

of life disrributions

if and only if there exists a tb,

1, such that for t-c&,

dt *F(t) =

(2.1)

for tat;.

i 3t

Theorem 3.1 of Deshpande and Suresh (1990), to be referred to as DST below, informs us that F is BFR if and only if there exists a t0, 0 < t,, < 1 such that $F(t) is convex in [0, to) and concave in [to, 11. We now discuss the proof in 4 different cases as follows: Case I: qF(t) does not intersect the line g(t) = t, 0 6 t < 1. In view of DST, then F is either NBUE or NWUE according as $F(t) 3or d t, tE[O, l] and the conclusion of the theorem follows. Case II: First crossing ofg(t)=t by the curve GF(t) isfrom above. Let this point of intersection be t*; then, $F(t)= t, t~[t*, I], for, otherwise, either the DST or the fact that (cIF(l)= 1 will be violated. As such, F is NBUE and the theorem follows. Case III: First crossing ofg(t) = t by the curve GF(t) is from below and there are no further crossings. Here also tiF(t) = t, V tE[t *, 11, and the theorem follows. Case IV: First crossing of g(t) = t by the curve $F(t) is from below and there is a second crossing. In this case, in view of the DST the second crossing must be from 0 above, and the proof will follow as in Case II. The converse example.

of the above proposition

Example 2.2. Consider

is false as can be seen from the following

the life distribution

whose survival

I

e- ‘/x2,

x/*exp(

The corresponding

l
MRL function

!

+(x)=

r(x)= 1

Odx
X,

16x<2,

4/x,

2
co,

with x0 = 4. The failure rate function

1,

Odx
2/x,

16x<2,

(x/4-

reveals that F is not BFR.

26x
is given by

1,

SO that F is NWBUE(xO),

is given by

O
[ ePx,

F(x) =

function

l/x),

2
00,

which is given by

389

M. Mitra. S.K. Basu/ Nonparametric family of lifr distributions

Proposition

is IDMRL(t,),

2.3. IfF

then F is NWBUE(tb)

We omit the proof since it is trivial. Below we provide an example to convey without

with tb> to

that a life distribution

can be NWBUE

being IDMRL.

Example 2.4. Consider

F(x) =

the life distribution

I

{e/(e+cc)}*

{ej(e+cc))2 exp

i

--

having

the following

( e+2c(-p)z, ’f ( e+2c(+)O

function:

BGX
e+*u-*p+x

e+*a--2p+y

1

(x2-y2)

8+x,

o
e+2cr-x,

cr
e+2a-2p+x,

BbX
Y(e+2a-2p+y)jx,

x3y.

I

X3Y,

I ’

2 y(e+2c(-2p+y)

where the parameters CI,fi, y and f3 are such that 0
44 =

survival

B d y, 2cr > B and 8 > 0. The

It is easy to observe that the life distribution F is NWBUE with xo=y(8+2c(-2/3+y)/8; however, F is not IDMRL whenever cc
for a life distribution

to be NWBUE,

its MRL function

has to lie above

the mean life ,u till x0 and below it then onwards. This property seems to imply a natural nonmonotonic ageing phenomenon which cannot be captured, as illustrated in Examples 2.2 and 2.4, through either BFR or IDMRL notions.

3. Inequalities

and moment bounds

We would need the following Lemma 3.1. If F is NWBUE(x,)

basic inequality. with (jnite)

mean p, then (3.1)

390

M. Mitra, S.K. Basu / Nonparametric family of life distributions

Further, if ep(x) is bounded above by M, then m

F(u)du d pe -xlP,-xo(M-‘-K’),

Proof. The probability fci,(x)=F(x)/p,

x30

vx>xO.

(3.2)

density function of the first derived distribution is given by and let r C1J(.) denote the corresponding failure rate function.

Using the fact that F is NWBUE(xe), x < or 3 x0. Writing

it follows that r(,,(x)
> l/p according

and using the above inequalities, (3.1) follows easily. If eF(x)d M, then (3.2) follows simply by noting that rC1,(x)= l/eF(x).

as

0

Proposition 3.2. Let F be NWBUE(x& x0 < co with (finite) mean ,u and 4( .) be a nondecreasing (nonincreasing) function on [0, co). Then

som

F(x)dx

4(x)

d jXy $(x)exp (~)

(F)dx.

(3.3)

Proof. We shall prove the result for nondecreasing 4; the other case can be treated similarly. Without loss of generality, we take 4 to be nonnegative. Let Z be a random variable whose d.f. is FC1,, defined in the previous proof and U be another random variable such that U-x0 is exponentially distributed with mean p. Then, as 4 is nondecreasing, in view of (3.1), we have

s,”

4 (4 6) dx

=P

j

0

(h9(4(Z)

>x)dx

P(Z>d-‘(x))dx

s

s m

b(m)

=P

9(Z>4-‘(x))dx+p

0

0(-w)

cc d

&(x0)

+

P(U

P s

@(+I)

>

4-‘(x))dx

P(Z>d-‘(x))dx

391

M. Mitra. S.K. Basu / Nonparametric family of life distributions

Now,

s

s

Q(.w)

m

9($(U)> Q(xo)

x)dx=b&U)-

9(4(u)>

x)dx

0

4(x0) =ac$(u)-

WW)>x)dx s0 Q(m)

ccP(U> 4-‘(x))dx,

=&c#I(U)s0

the integrand

being unity,

s

since 4 is nondecreasing.

Thus,

om 6(x)~(x)dxQ&(xo)+~C~@(U)--(xo)l=@’+(U),

which completes Corollary 3.3.

the proof.

0

IfF is NWBUE(xe),

x0 < co, with (finite) mean ,u, then x*-l exp(-(x-xo)/p)dx

for r>

1.

(3.4)

(')

Proof. Take &J(X)= x r-1 in Proposition and nondecreasing for r > 1. 0

3.1 and note that 4 is nonincreasing

The above result leads to the following distribution where x0 is finite. Corollary 3.4. Zf F is NWBUE(xo),

bounds

for moments

for r < 1

of a NWBUE(xo)

x0 < co with (finite) mean p, then

a, xI-1 dFX* < reXoip e-“‘“dx, (3) s x0

for r > 1. cc)

(3.5)

Given a NWBUE(xo) distribution, x0 < co, with finite mean p, we shall now obtain bounds for the moments of the above distribution in terms of the moments of an appropriate negative exponential distribution. Consider the following negative exponential distribution defined by the survival function

392

M. Mirra. S.K. Basu 1 Nonparametric family of life disrributions

Simple calculations

show that for r> 1, 30 x’-l

BGX’=xh+rexoiP

eexiP dx.

(3.6)

s x0 If r> 1 is an integer, ‘-I (xolP)j 1) 1 jl.

c?~Xr=X;+/L’r(r+

j=O

The following

corollary

(3.7)

.

now gives the various

Corollary 3.5. Let F be NWBUE(xe), dejined above. Then

moment

bounds.

x0 < cc with &ire)

mean p, and let G be as

(i) bcX*< cofor all r>O. (ii) &FX’<&GX’f~r all r> 1. (iii) f~Y~X’Qxb+p’r(r+ 1) CJ’&r (xo/p)j/j!for (iv) dFX’
all integers r3 1.

Proof. (i) This follows from (3.6). (ii) For r> 1, by (3.5), (3.8) co
x’-1 e-“‘“dx=d,X’.

s x0 (iii) This follows from the previous (iv) For rB 1, by (3.5),

part and (3.7).

30 X’-l eexip dx s x0 00 Gre XOIP x'-l e-X’pdx=p’r(r+ s0

8FX’
l)exO’P.

Remarks. (A) Taking x0 = 0 (i.e. if F is NBUE) in Corollary 3.3, we get the usual well-known bounds for the NBUE situation. (B) Although Corollary 3.5(ii) and (iv) give the same bounds for the NBUE case, it is clear that the former, which has an interesting probabilistic significance, is sharper than the latter, which, on the other hand, has useful applications as illustrated in the next section. The next distribution.

result

gives

an

interesting

characterization

of

the

exponential

393

M. Mitra, S.K. Basu / Nonparametric family of life distributions

Theorem

3.6. Let F be NWBUE(xe),

x0 < cc with finite mean p and let G be as in

Corollary 3.4. Then F is the exponential distribution if&FX’ = bcX’ Proof.

As &FX’= gGXr for some r > 1, then equality

(3.8) so that x,, = 0 and hence F is NBUE some r> 1. Consequently, F is HNBUE Basu and Bhattacharjee

(1984, Lemma

4. Weak convergence of NWBUE We start this section

for some r > 1.

holds in the string of inequalities

the relation 6, X’= F (r + 1) $ for the above relation and hence, by 0 2.4) F is exponential. satisfying satisfying

distributions

with the following

theorem.

Theorem 4.1. Let F,, n = 1,2, . . . be a sequence ofNWBUE(xO,) means p,,. Suppose that

life distributions with

(i) F,,-+F in law, where F is a continuous d$ and (ii) the sequences {p,,} and {x0”} are bounded. Then F is NWBUE. Further

s m

lim x*dF,(x)= rl+CC 0

s m

xrdF(x)

for every r>O.

(4.1)

0

Proof. Let ,u,,d B for all n 3 1 and let p be the mean of F. An application of Fatou’s lemma together with (i) above shows that p< co. We first prove that p,,-+p as a-03; for A >O, to be chosen suitably,

sa

{F,(x)-F(x)}dx=

0

Suppose

A (F,--)(x)dx+ s 0

cc F,(x)dxs A

xoll GM, for all n 2 1. For A > M > xOn, E> 0, by Lemma

“F(x)dx s A

3.1,

s m

12n=

F,(x)dx~~L,exp(-(A-xo,)/~~)~Be-‘A-M)’B

A

which can be made smaller than ~13 by choosing a sufficiently large A. Since p< 03, 1Z31 M, sufficiently large, we have 1Z2,,( -c&/3 for all nk 1 and II,1
(4.2)

394

M. Mitra, S.K. Basu / Nonparametric family of life distribulions

Since {x0,,} is bounded, there exists a subsequence {x0,,,} such that x0.,+/I< co as k-co. For x x, Vk B kO which implies 1

i”F,,(u)duBpL,,.

F,,(x). s Taking

as k-co,

limits

convergence

theorem 1

via condition

(i) of the theorem,

(4.2) and the dominated

we get

mF(u) du b p.

(4.3)

F(x), s Similarly

for x>fi, 1

we can show that coF(u) du
(4.4)

F(x), s

The proof of the first part of the theorem now follows from (4.3) and (4.4). To prove (4.1) we first note via (4.2) that it holds for r = 1. Let X,, y1= 1,2, . . . , and X be random variables having d.f.s F, and F, respectively. Then, as sup,, 6X, < co, {X.‘} is uniformly integrable for Y< 1, this together with (i) establishes (4. l), for r < 1. On the other hand, for r> 1, by Corollary 3.5 (iv), dX,‘<~~~(r+l)exo”‘@~. As {x0,,} is bounded and {pn} converges, { xOn/p,,} is bounded Vr> 1, by the above relation. So, (XL} is uniformly integrable proves (4.1) for r> 1. 0 Lemma 4.2. A d.jYwhich is NWBUE(x,J, sequence.

so that sup” 8X,* < cc for each r> 1, and this

x0 < cc, is uniquely determined

by its moment

Proof. Let F be NWBUE(xo), x0 < co with mean p. By Corollary 3.5(iv), it follows easily that the power series clY!o (u’/r!) bFX’ has a nonnull radius of convergence. The lemma now follows from a result of Loeve (1963, p. 217). 0 In Theorem 4.1, we showed that under a couple of conditions, weak convergence of a sequence of NWBUE d.f.s implies convergence of moments of the sequence of d.f.s to the corresponding moments of the limiting d.f.; the following deals with the converse of this result. d.js with xon < CC Theorem 4.3. Let F,,, n= 1,2, . . . , be a sequence of NWBUE(xon) Vn 2 1 and suppose that F is NWBUE(xo), x0 < co such that for all integers r 2 1, m m xrdF(x). lim x’dF,(x)= (4.5) n+30 s 0 s0 Then F,+F

in law.

M. Mitra, SK.

Proof. By Lemma

4.2, F is uniquely

this and (4.5) the limiting (F,,} happens

determined

distribution

to be necessarily

If condition

395

Basu / Nonparametric ,famiiy of life distributions

(ii) in Theorem

F and this completes 4.1 is replaced CC xdF(x)

Cc lim xdF,,(x)= n-tm s 0

by its moment

of every weakly

sequence.

convergent

the proof.

In view of

subsequence

of

0

by (
(4.6)

s0

then we will have an interesting a proposition.

conclusion

which we present

below in the form of

Proposition 4.4. Suppose in Theorem 4.1, we replace condition (ii) by (4.6). lf {x0,,}

is

unbounded, then F is N WUE. Proof. Let p be the mean of F; consider

there exists

(xonk} such that xOnr >x, Vk3 1. So,

a subsequence

Taking

any x > 0. As {x0”} is unbounded,

limits as k+ CC and arguing 1

F(x).

as in Theorem

4.1, we get

s m-

F(u) du >, pu,

so that F is NWUE.

0

Remark. It is trivial to observe from Proposition 4.4 that under (4.6), the weak limit of a sequence of NWUE d.f.s is also NWUE, provided the limiting distribution is continuous. The counterpart by Basu and Bhattacharjee

of this result concerning NBUE (1984) with more generality.

distributions

was given

5. The dual class In this section, family comprising

we extend similar investigations of NBWUE distributions.

as in Sections

3 and 4 to the dual

Remark 5.1. Through the same approach as in the proof of Proposition 2.1, it can easily be seen that if a continuous and strictly increasing life distribution F is UBFR (with finite mean p), then F is NBWUE. Also, as in Proposition 2.3, if F is DIMRL(t,), then F is NBWUE(tb) with t&2to.

396

M. Mitra, S.K. Basu 1 Nonparametric family

oflife distributions

Remark 5.2. The following example confirms that unlike NWBUE life distributions, an NBWUE distribution having a finite change point may not possess finite moments of order higher than one. Example

5.3. Consider

the life distribution

whose

F(x) = a(x 2 + c()- ‘, x > 0, where a > 0 is a constant. (9+/2

< co; however,

for I> 1, &rX’=

is given

by

The mean of this distribution

survival

function

is

co.

On the other hand, its failure rate function

is of the form

-1

r(x)=2

( > x+”

X

which is strictly increasing in (0, &) and strictly decreasing in (&, UBFR and hence, from Remark 5.1, we conclude that F is NBWUE. its MRL function

we even conclude

that F has a unique

finite change

co). As such, F is In fact, analysing

point,

In view of the above example, it is evident that a result analogous to Corollary 3S(ii) is, in general, false for NBWUE life distributions having finite change points. As such, issues contained in subsequent results in the same corollary do not seem relevant in the context of NBWUE distributions. Remark 5.4. From version of Theorem

the following 4.1 is false.

example,

Example 5.5. For n = 1,2, . . . , consider defined by the survival functions

we further

the sequence

conclude

of NBWUE(0)

that the NBWUE

life distributions

i.(x)=r,exp(-i)+&exp(-t). where O 1. Now, as n+ co, F,,+F in law, F being the d.f. of the exponential distribution having mean ,K But eF,Xm+ 00 > dFXm, though, both the sequences {,u”} and {xon} here are bounded. However, the question of closure of the NBWUE class under weak convergence still remains open.

M. Mitra. S.K. Basu 1 Nonparametric family of life distributions

Remark 5.6. In view of Remark

5.2, issues dealt with in Lemma

4.2 and Theorem

391

4.3

are, in general, not relevant in the context of NBWUE distributions. However, proceeding as in the proofs of Theorem 4.1 and Proposition 4.4, we can prove the following

version

of the latter.

Theorem 5.7. Let F,, n= 1,2, . . . , be a sequence ofNBWUE(xon) means pL,. Suppose

(i) F,-+F

Zf{xon}

in law, where F is a continuous

m m lim x dF, (x) = xdF(x) n-a, s 0 s0

(ii)

is bounded

life distributions with

that

th en F is NBWUE;

From the above theorem, we observe weak convergence provided (5.1) holds.

d.5;

(< co).

(5.1)

otherwise, F is NBUE.

easily that the NBUE

class is closed under

References Barlow, R.E. and F. Proschan (1981). Statistical Theory ofReliability and Life Testing. To Begin With, Silver Spring, MD. Basu, S.K. and M.C. Bhattacharjee (1984). On weak convergence within the HNBUE family of life distributions. J. Appl. Prob. 21, 6544660. Despande, J.V. and R.P. Suresh (1990). Non-monotonic ageing. Stand. J. Statist. 17, 257-262. Guess, F., M. Hollander and F. Proschan (1986). Testing exponentially versus a trend change in mean residual life. Ann. Statist. 14, 1388-1398. Loeve, M. (1963). Probability Theory, 3rd ed. Van Nostrand, New York.