Reliability Engineering and System Safety 25 (1989) 99-107

On Discrete Failure-Time Distributions

A b d u l K h a l i q u e

Statistics Department, College of Science, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia

(Received 17 July 1988; accepted 24 July 1988)

A B S T R A C T

Different classes of discrete life distributions are defined parallel to the continuous life distributions. Some implications known to be present in continuous life distributions are shown to exist among discrete life distributions. Some test statistics of geometric distribution property versus other classes o f failure-time distribution are proposed. Critical points of these statistics, based on simulations are presented. Powers of these tests against increasing failure rate ( IFR ) and decreasing failure rate ( D FR ) distributions are also calculated.

1 I N T R O D U C T I O N

There is vast statistical literature concerning continuous life distributions. However, it appears that in most practical s i tuations the data measured are discrete, as for example the number of days or number of cycles of a device before a failure. When the number of observed data values are large, cont inuous failure distribution is a convenient model to describe discrete data. But when the number of observed values are small a cont inuous distribution may not describe the data adequately. Hence, the ageing notions should be considered in context with the discrete distributions. Barlow & Proschan (1981) and Klefsjo (1982) define some concepts in discrete failure distributions. 1'2 If X is a discrete random variable with

99 Reliability Engineering and System Safety 0951-8320/89/$03.50 O 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

100 Abdul Khalique

probability mass function Px, the corresponding distribution function p~ = Pr[X < x] and the survival function

Px = Pr[X >_ x] = Pi

i = x

the various concepts of ageing can be defined as follows. The random variable X, its distribution function Px or survival function

Px is said to have:

(i) Increasing failure rate (IFR) i f p x / P ~ is increasing in x. (ii) Increasing failure rate average (IFRA) if (/5)1/~ is decreasing in x. (iii) New better than used (NBU) if P~+y <_ P~Py for x , y = 0, 1,2,. . . (iv) New better than used in expectation (NBUE) if

~ P x <

X ~ t

(v) Harmonic new better

~ Px <

X = t

(vi) New better than used

x = 0

than used in expectation (HNBUE) if

p 1 - for t = 0 , 1 , 2 . . . .

in average failure rate (NBAFR) if

P~ < (Pl) ~ for k = 0, 1, 2 . . . .

(vii) New better than used in failure rate (NBUFR) if

Px+ 1 ~-~ PxP1 for x -- 0, 1, 2,. . .

The dual classes DFR, DFRA, NWU, NWUE, H N W U E , N W A F R and N W U F R can be defined by reversing the monotonici ty or the inequality in these definitions. The first five classes in the case of continuous random variables have been well-investigated. The N B A F R class was introduced by Loh (1984) and the N B U F R by Deshpande et al. (1986) and A b o u a m m o h & Ahmed (1988). 3 - 5 For various implications among these and other classes in the case of continuous life distribution refer to A b o u a m m o h (1988). 6 Implications among discrete life distributions were expected to be similar to those in continuous life distributions. However, this is investigated in Section 2 and the following implications established:

IFR =~ IFRA =~ NBU ~ N B U F R => N B A F R

In Section 3, test statistics for these classes are proposed. These statistics

On Discrete Failure-Time Distributions 101

are based on the empirical evaluation of the basic definitions of various classes. Selected percentile points (based on simulation) of these statistics are also given. Section 4 consists of conclusions and remarks.

2 I M P L I C A T I O N S A M O N G T H E CLASSES

Increasing failure rate means px/Px increasing in x. I f the definitions in section 1 are assumed:

(i) I F R =¢-IFRA. Proof: If Px is IFR, then px/Px is increasing in x. Px is I F R A if the average

x

1 S p i

i = 1

is increasing in x. But x x x

1 V p i = I S P I - 1 - P i _ I V P i - I x /_..a P i x f f i x ~..a P i 1

i = 1 i = 1 i = 1

Hence for IFRA, x

x z . a Pi i = l

is increasing in x. But this is the ari thmetic mean of the sequence {Pi-1/P~}, i = 1, 2, . . . , x. And the increasing proper ty of ari thmetic mean implies the increasing proper ty of the geometric mean of the same sequence. Hence, (P0/Px) 1/x = (1/Px) 1/~ is increasing in x for ap~ which is IFRA. This yields that [Px] ~/x is decreasing in x for IFRA.

This implies that [Px]l/x>_ [Py]~/Y for x < y (1) (ii) I F R A =:- NBU.

Proof: For px to be NBU,

P~ + y < P~Pr

Assume, for the case of contradict ion, that p~ is I F R A but

P~ + r > PxPr

>_ PyI~y/y = ply +x/y = ~ x + ~ ) / ~

o r 1/y ( e x + y ) l / ( x + Y ) > - - y

102 Abdul Khalique

(iii)

(iv)

since x + y > y, this means that (1) is contradicted orPx ~/x is increasing in x. Hence we conclude that I F R A = ~ N B U .

N B U ~ N B U F R . Proof: By definition, p~ is NBU if

For y = 1

~ p ~ is N B U F R .

N B U F R ~ N B A F R .

P~+y < PxP r

Px+ 1 < PxP1

Proof: F rom (ii) Px is N B U F R if,

Px <- P:,-IPl < P~Px-2 < P~ i.e. if

Px ___ (P,)~

which means that p~ is N B A F R . Hence we have shown that

I F R =~ I F R A =~ N B U =~ N B U F R =~ N B A F R .

3 TEST STATISTICS

In continuous life distributions, the failure rates of various distributions are measured with respect to the failure rate of exponential distribution which has a constant failure rate. In the discrete distribution, geometric distribution is the equivalent of the exponential distribution and has the constant failure rate. The probabil i ty mass function of a geometric random variable X, is given by

Px=Pq~-I, x = 1,2, . . .

here p is the probabil i ty of success in a single trial and q = 1 - p . The failure rate of other discrete failure distributions may be measured

with respect to the geometric distribution, e.g. the positive departures of failure rate from the geometric failure rate indicating increasing failure rate. R a n d o m samples from geometric distribution were generated and a statistic for a particular value of the sample calculated. The average or sum was then taken to find the test statistic as follows:

(i) I F R class implies px/Px increasing in x

Px < Px + 1 or /~x - P=~+ 1

On Discrete Failure-Time Distributions 103

summing over the whole sample, this gives n - 2

x = l

Substituting the estimates of various quantities and averaging over all points, the test statistic T~ is given as

n - 2

x = l

(ii) IFRA means that [Px] ux decreasing in x

1 or l_ log [/~x] < log (Px-x)

x - x - 1

or 1 log Px - - - 1 log Px- 1 < 0 x x - 1 -

Substituting the estimates and summing over the sample values, the test statistic is given as

n- -1

x = 2

(iii) NBU class means Px+y < PxPy On substituting the estimates and summing over x and y, we obtain

the test statistic as

x = l y = t

x+y<_n

(iv) NBAFR class implies that Px -<

or log Px -< x log P1

Again writing the estimates and summing over x, gives

n- -1

= )1, [log/~x _ x logPl l0~ T, x = |

104

(v)

Abdul Khalique

NBUFR class implies that Px+ x < PxPI

Writing the estimates and summing over x, obtains n--I

T5 = ~ [;x+l -- ;x;1]Px x=l

4 SIMULATED CRITICAL POINTS

The critical points for the test statistics in Section 3 were estimated by simulating random samples of various sizes from geometric distribution with p = 0.5. The percentile points were first calculated from 1000 samples of a given size and then the reported percentiles were obtained by averaging over five such replications. The critical points for different statistics are given in Tables 1-5.

TABLE 1 Critical Values for IFR Statistic

n Lower percentiles Upper percentiles 0"01 0"05 0"10 0"10 0"05 0"01

10 - 1 . 7 0 8 6 - 0 - 9 0 0 0 - 0 . 4 3 7 6 1.2567 1.8948 3.2600 15 - 2 . 1 9 8 1 - 0 - 7 6 0 4 - 0 . 3 8 0 6 0.8942 1.303 3 2.473 3 20 - 1.554 8 - 0-696 7 - 0 - 4 0 9 6 0-710 3 0.997 4 1.790 7 25 - 1.401 5 - 0.627 7 - 0-382 1 0.601 7 0.857 9 1.577 9 30 - 1 . 3 1 1 9 - 0 . 5 7 0 0 - 0 . 3 5 4 8 0.5466 0.7566 1.3276 35 - 1 . 2 9 4 2 - 0 . 5 5 3 2 - 0 . 3 3 4 8 0.4736 0.6501 1.0483 40 - 1.242 5 - 0 . 4 8 5 2 - 0 . 3 1 2 9 0.4242 0.591 8 0.983 1 45 - 1.125 7 - 0 . 4 7 2 1 - 0 . 3 0 5 8 0.387 7 0.5354 0.903 1 50 - 0 . 9 6 3 8 - 0 . 4 3 6 4 - 0 - 2 8 3 1 0.3580 0.4849 0.7875

TABLE 2 Critical Values for IFRA Statistic

n Lower percentiles Upper percentiles 0.01 0.05 0.10 0.10 0.05 0.01

15 - 0 ' 4 8 7 0 - 0 ' 2 6 6 8 - 0 - 2 0 3 1 0-1234 0"1472 0"2732 20 - 0 ' 3 7 4 6 --0-2175 - 0 ' 1 6 1 4 0-1125 1"1527 0-2191 25 - 0 ' 3 1 3 9 --0-1883 - 0 ' 1 3 6 5 0-1017 0"1406 0"2125 30 - 0 - 2 6 8 7 --0"173 6 - 0 ' 1 2 2 3 0"0954 0"127 5 0"206 1 35 - 0 " 2 2 7 9 --0"1439 - 0 ' 1 0 9 7 0"0900 0"1187 0"1923 40 - 0 ' 2 2 1 0 - 0 - 1 3 8 5 --0"0994 0-0843 0"1097 0 '1819 45 - 0 " 2 1 0 8 - 0 - 1 2 2 5 - 0 " 0 9 2 4 0"0809 0-110 1 0 '1680 50 - 0 " 1 8 6 2 -0"116 1 - 0 - 0 8 6 8 0"0784 0-102 1 0"151 6

On Discrete Failure-Time Distributions

TABLE 3 Critical Values for NBU Statistic

105

n Lower percentiles Upper percentiles 0"01 0"05 0"10 0"10 0"05 0"01

15 -0"234 5 -0"169 1 -0-141 2 0-153 7 0"212 3 0"329 3 20 -0"2114 -0"1539 -0-1242 0"1377 0"1886 0'2915 25 -0"1977 -0"1425 -0-1138 0"1239 0"1663 0'2502 30 -0"1843 -0"1339 -0"1079 0"1152 0"1521 0'2348 35 -0"1748 -0-125 7 -0.101 2 0-104 1 0"1409 0'2062 40 -0"1696 -0"1190 -0-0947 0"0958 0"1297 0'2003 45 -0"1585 -0-1131 -0"0897 0"0926 0"1234 0"1834 50 -0-1537 -0"1073 -0"0849 0'0882 0-1153 0'1795

TABLE 4 Critical Values for N B U F R Statistic

n Lower percentiles Upper percentiles 0"01 0"05 0"10 O" 10 0"05 0.01

15 -0 .1230 -0-091 1 -0-0728 0.0745 0.101 1 0.1365 20 -0 .1144 -0-0833 -0-0665 0-0666 0.0869 0-121 1 25 -0 .1099 -0 .0768 -0 .0613 0-0608 0-0767 0-1027 30 -0 .0994 -0.071 1 -0 .0574 0-0565 0.0713 0-0992 35 -0-0910 -0-0674 -0-053 1 0-051 0 0.063 5 0"088 0 40 -0 .0899 -0-0626 -0-0494 0-0480 0.0613 0.0868 45 -0 .0854 -0-0614 -0 .0486 0.0454 0.0570 0-0801 50 -0 .0804 -0-0575 -0 .0456 0.0424 0.0537 0.0742

TABLE 5 Critical Values of NBAFR

n Lower percentiles Upper percentiles 0"01 0"05 0"10 0"10 0"05 0"01

15 -0"8093 -0"5206 -0 ' 3956 0-2203 0'2994 0"4510 20 -0"7107 -0"481 7 -0-3786 0-2200 0'3032 0"4543 25 -0"6554 --0"4339 -0"3435 0"2130 0-2834 0"4224 30 -0 ' 5934 --0"4183 -0"3204 0"2082 0-2748 0"4114 35 -0 ' 5524 --0"3859 -0"3023 0"1943 0"2586 0'3739 40 -0 '5243 --0"3599 -0"2842 0"1880 0"2527 0-3622 45 - 0"497 9 -- 0"3510 - 0"275 6 0-178 7 0-245 7 0-356 2 50 -0-4783 -0"3269 -0"2606 0-1722 0-2280 0'3387

106 Abdul Khalique

5 P O W E R E S T I M A T E S O F T E S T S T A T I S T I C S

Using a level o f significance ~ = 0.05, the var ious test statistics were s imula ted for which the critical poin ts are given in Sect ion 4. The powers were ca lcula ted against discrete Weibull , Po isson and negat ive b inomial a l ternat ives for sample sizes 20 and 30. The p robab i l i ty mass func t ions for these d is t r ibut ions are given respectively, by

p l (x ) = q(X 1)~ ,,~ - q , x = l , 2 , . . . , f l > 0 , 0 < q < l

exp - "/P R2Ix) -- X! ' X = O, 1 . . . . ; # > 0

P 3 ( X ) = ( x ~ ) p ' ( p - 1) x, x = 0, 1, . . . ; ~ > 0 ; 0 < p < l

Here discrete Weibul l is I F R for fl > 1 and D F R for fl < 1, Poisson is I F R and negat ive b inomia l is I F R for ~ > 1 and D F R for ~ < 1. Table 6 below presents the est imates o f power for var ious tests against geomet r ic fai lure rate.

TABLE 6 Power Estimates of Various Tests Based on 2000 Samples of Sizes 10 and 20 with ct = 0.05

Distribution Sample Statistic size T 1 T 2 T 3 T 4 T 5

f l (q = 0.5, fl = 0.5) 20 0.249 0.000 0.005 0.001 0.002 30 0.471 0.000 0.849 0.492 0.733

./1 (q = 0.5, ~= 1.5) 20 0.144 0.000 0.015 0.089 0.255 30 0.560 0-567 0.068 0.055 0.060

.[2(/a = 2) 20 0.480 0.598 0.403 0.982 0.122 30 0.652 0.841 0.335 0.926 0.044

J3 (p = 0.5, ct = 0.5) 20 0.629 0.628 0.416 0.615 0.320 30 0.776 0.776 0.265 0.355 0.174

J3 (p = 0.5, ~t = 1.5) 20 0.758 0.912 0.807 0-876 0.366 30 0.873 0-991 0.895 0-998 0.650

6 C O M M E N T S

Almos t the ent i re reliabil i ty l i terature, except a few papers , deals with con t i nuous life dis t r ibut ions. F o r very large samples, the con t inuous a p p r o x i m a t i o n is na tu ra l bu t for small samples, it makes m o r e sense to cons ider the da t a as discrete and to eva lua te the reliabil i ty diagnost ics in con tex t with discre te-norms. This has been a t t emp ted in this paper.

On Discrete Failure-Time Distributions 107

Due to the small number of nonzero elements in small samples, sample sizes of less than 15 have not been considered. The test statistics obtained are usually not quite simple and therefore their asymptotic distributions seem difficult to obtain. The estimates of powers in Table 6 show the promising nature of the test statistics. In most of the cases the power is more than 50%.

R E F E R E N C E S

1. Barlow, R. E. & Proschan, F. (1981). Statistical Theory of Reliability and Life Testing Probability Models. To Begin With, Silver Spring, MD, p. 98.

2. Klefsjo, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval research logistic quarterly, 29, 331-44.

3. Loh, W. Y. (1984). A new generalization of the class of NBU distributions. IEEE Trans. Reliability, R-33, 419-22.

4. Deshpande, J. V., Kochar, S. C. & Singh, H. (1986). Positive ageing. J. Appl. Prob., 23, 748-58.

5. Abouammoh, A. M. & Ahmed, A. N. (1988). The new better than used failure rate class of life distributions. Adv. Appl. Prob., 20, 237-40.

6. Abouammoh, A. M. (1988). On the criteria of the mean remaining life. Statist. and Prob. Lettres, 6, (4), 205-11.