A. M. Abouammoh* and G. R. Elkahlout**
King Saud University, Saudi Arabia

ABSTRACT
In this paper, the new or worse better than renewal used classes of life distributions are considered. Relations of these classes with other existing ageing criteria are presented. Closure properties under some reliability operations is studied. Test statistics of exponentiality for these classes are established under random censoring. Percentiles and the power estimates are simulated. The asymptotic normality of these tests is also tested.

1. INTRODUCTION
Different ageing criteria have been used to classify positive and negative ageing properties. For example increasing failure rate (IFR), New better than used (NBU), Decreasing mean remaining life (DMRL) and new better than used in expectation (NBUE) and their duals are the main existing ageing criteria; see Bryson and Siddiqui (1969) and Barlow and Proschan (1981).
Testing of exponentiality against classes of life distributions based on a randomly censored samples are investigated by many authors for example, Chen et al. (1983) proposed a test statistics for testing exponentiality versus NBU distribution. Gerlach(1987) generalized Ahmed’s (1975) test for testing IFR distribution in the case of complete sample to the case of randomly censored data. Gerlach (1989) studied the same problem for the class of IFRA. Abouammoh and Alsadi(1996) introduced a test statistic for testing NBRU class of life distributions in the case of complete sample.
For other studies on testing the classes of life distributions based on a random censored samples, see Kumazawa (1992) for testing IFR class of life distributions, Kumazawa (1987) for testing NBU class of life distributions, and Tiwari et al. (1989) for testing IFRA class of life distributions.
The random censoring samples motivated in the following: Let be a sequence of life lengths of n independent identically distributed random variables with distribution function F, and be a sequence of independent and identically distributed random variables with distribution function G and distributed independently of . Here, if we consider the right censoring, the available observations consist of the pairs , where and with I(.) as the indicator function. We have to make here an inference about F using .
We assume that and are stochastically independent. This kind of censoring is useful when the items involved are costly and the time element of obtaining the observations is important. This random censorship arises in medical applications. with animal studies or clinical trials. The life times are observed, but censoring occurs according to loss of follow-up, dropout or termination of the study.
The renewal process by considering a unit with life distribution F in operation. The unit is replaced upon failure by a sequence of mutually independent units, independent of the first failure unit and identically distributed with some distribution F. In the long run the residual life of the unit under operations is given by the stationary renewal survival
, (1.1)
where .
Ageing classes of life distributions based on comparison between new and used survival are shown to be important in constructing maintenance and replacement polices in reliability studies, see Barlow, Marshall and Proschan (1963).
In this paper classes of life distributions based on comparison of renewal used survival with its new parent survival is considered. Definitions and basic properties of these classes are investigated in section 2. In section 3, the test statistics of exponentiality as a null hypothesis against a specific ageing properties as alternatives are established under randomly censored samples. Percentiles of the test statistics are simulated. In section 4, power estimates for commonly used distributions in reliability are calculated and the asymptotic normality of the proposed statistic is tested via simulation.
2. BASIC RESULTS
Let T be a non-negative random variable with absolutely continuous distribution function , , such that , survival function , , and density function , .
Let be the corresponding renewal random variable with renewal survival function defined in (1.1) and density function . T is said to have a new better (worse) than renewal used property denoted by NBRU, (NWRU) if
, (2.1)
or, equivalently, if
. (2.2)
Relation (2.2) can be written as
, . (2.3)
This class has been defined by Abouammoh et al. (1993). Cao & Wang (1991) used the concept of partial ordering to introduce this ageing property as the concept of new better than used in convex ordering (NBUC).
It is noted that the random variable T is both NBRU and NWRU if and only if it is an exponential random variable. This property can be shown in the following example.
EXAMPLE 2.1
Let the random variable T be exponential distribution with survival function . In the present case, the relation (2.3) is true with strict equal sign and each side equals . Hence T is both NBRU and NWRU.
Let the random variable T be with NBRU and NWRU life variable at the same time. This is equivalent to stating that (2.3) be met with the inequality replaced by the equality. Differentiating the identity with respect to s, we get
The only distribution having this property is the exponential distribution. This is essentially the memoryless property; see Parzen (1962).
In the following results it is shown that this class of life distributions contains the class of new better than used (NBU)distributions.
THEOREM 2.1
NBU  NBRU.
PROOF
Let F be a NBU distributions, i.e.
Integrating both sides of the inequality with respect to x over , yields
.This is equivalent ( in obvious notation ) to
, and .
This means that F is an NBRU.
The implications for the dual classes, i.e
.can be proved in parallel steps.
In the following we investigate the closure properties NBRU or NWRU classes of life distributions, under some reliability operations such as mixing, convolution and formation of coherent systems.
The class NBRU is closed under convolution, see Cao and Wang (1991) and it is closed under the formation of parallel system of identical components, see Hendi et al. (1993). Note that the class NWRU is not closed under convolution; this is shown by the following example:
EXAMPLE 2.2
Let . As this is the survival of an exponential distribution, we have it to be that corresponding to an NWRU. The two fold convolution of F with itself gives the distribution
, ,
or
,
which is the Gamma distribution with scale 1 and index 2. The failure rate in this case is a strictly increasing and equals to , .This implies that the convolution G is not an NWRU. Thus we have that NWRU is not closed under convolution.
The following example shows that the NBRU is not closed under mixing.
EXAMPLE 2.3
Consider F to be the scale mixture of the exponential distributions with the mixing distribution as exponential with mean 1. Clearly then
The exponential distribution is NBRU, while the mixture F(t) has a strictly decreasing failure rate implying that the mixture F is not NBRU.
The class NBRU is not necessarily closed under formation of series system as shown by the following example.
EXAMPLE 2.4
Consider a series system of two independent components with common survival function:
We need to show that the inequality (2.3) holds for F(.). The inequality is trivial if , and . For , constant. The LHS of (2.3) is decreasing in x, it
is sufficient to show that it holds for x=1 and .(Note that if x=1 and , we have and hence The LHS equal zero.) In this case we have
and
For ,both sides are equal and hence the inequality is trivially valid. For , we have to show that , or , which is obviously valid. Hence the distribution F has the NBRU property.
The life of the system does not follow NBRU, as can be shown by the following: Consider here
then
Denoting the probability distribution function of by G. Now let .Then
and
.This concludes the argument.
The NWRU is not closed under the formation of parallel system as shown by the following example.
EXAMPLE 2.5
Consider a parallel system of two independent and identically distributed components, each with survival function Then the life distribution with survival function of the system is given by
;
this has density function given by .Then the failure rate given by ,for , which is a strictly increasing function of t. Hence G is not an NWRU distribution, whereas F is an NWRU distribution.
The NWRU is not closed under the formation of coherent system of the form of 2-out-of-3 structure function. This can be shown through the following example.
EXAMPLE 2.6
Consider a 2-out-of-3 system of three independent and identically distributed components, , each with survival function , .The structure function of the system is given by
, .
Then the life distribution with survival function has density function and failure rate is given by , ,
which is a strictly increasing function of t. Hence H is not an NWRU distribution, but F is an NWRU distribution.
In example 2.5, we considered the case of a parallel system of two components, which is a coherent system of 1-out-of-2 structure function.
Hence for the general case of a coherent system of n independent and identically distributed components, , each with survival function , the failure rate of the system of k-out-of-n structure function is given by
which is a strictly increasing function of t for
Hence the NWRU class of life distributions is not closed under the formation of any coherent system of the form k-out-of-n. An aeroplane, which is capable of functioning if and only if at least two of its three engines are functioning, is an example of a 2-out-of-3 system. Other forms of coherent systems are found in Barlow and Proschan (1981).
3. TESTING EXPONENTIALITY VERSUS NBRU AND NWRU AGEING CRITERIA
In this section, we propose a test statistic for testing the exponentiality versus the NBRU ageing property. We consider the null hypothesis
against the alternative
is NBRU or NWRU, on the basis of random censored data .
Based on the definition of NBRU class, a test statistic can be defined by the parameter
. (3.1)
This measure of deviation from NBRU-ness property can be written as
(3.2)
Now the average deviation for the distribution F from exponentiality towards
NBRU can be expressed by
(3.3)
where F(s) and F(t) are weight functions.
Thus a natural test for testing H0 against H1 can be proposed by substituting the empirical distribution survival function in , to arrive at an asymptotically equivalent statistic referred to as .
Thus H0 is rejected in favour of H1 for small negative values of .
In the following, we will write (3.3) in a simpler form.
= . (3.4)
For simplicity, let
where, on assuming that F is absolutely continuous with density f ,
and
.Now substituting the empirical estimates in and we get, in the case of randomly censored data, the estimates of and as follows
(3.5)
and
(3.6)
where,
,
,
and
Note that here is the Kaplan-Meier product limit estimator of , see Kaplan and Meier (1958). Also, and are defined earlier. For computational purposes can be written as
. (3.7)
Let
, for .
where is not necessary an order statistic. (3.7) can be written as,
or as
Here is considered as an actual observation, whether or not it is censored, and in this case is taken as one to avoid appearance of indefinite values in the empirical calculation of the statistic .
For a specific significance level and a sample size , we reject H0 in favour of H1, that F has the NBRU property, for small negative values of .In otherwords, we reject H0 in favour of H1 if the observed or calculated values of the statistic is less than or equal the tabulated value of the statistic for the same value of and .
For testing H0 against H1 that F has the NWRU property, we reject H0 for small negative values of - ,or for large positive values of .
The asymptotic normality of the sequence, such as , has been dealt with in the literature under the following crucial assumption:
for some .
where G and H are the distribution functions of the actual values of X and the censored values Y respectively. Here G and H are assumed to have support on .
The above sequence, or stochastic process, converges weakly as to a Gaussian process with zero mean, for this result and the details of the corresponding variance function see Gill(1983).
A simulation of a small sample is calculated. In practice, simulated percentiles for small samples are commonly used by applied statisticians and reliability analysts.
Based on equation (3.8), the lower percentiles in the 0.01, 0.05, and 0.10 regions and upper percentiles in the 0.90, 0.95, and 0.99 regions for the sample sizes n=10(2),30(5) and 50 are presented in table 1 for the NBRU and NWRU test statistics.
The percentiles are calculated by computer program using FORTRAN and NAG subroutines.
4. PROPERTIES OF THE TEST
4.1. POWER ESTIMATES
The power estimates of the test statistic are considered for the significance level =0.05 and for commonly used distributions in reliability modeling. These distributions are Gamma, Weibull, Pareto, and Rayleigh with the following survival functions and failure rates.
Gamma distribution:
(4.1)
Weibull distribution:
(4.2)
Pareto distribution:
(4.3)
Rayleigh distribution:
(4.4)
For further references about these distributions, one can refer to Feller(1966).
Table 2 contains the power estimates of test statistic with respect to these distributions. The estimates are based on the average of five runs each with 1000 simulated samples of sizes n=10,20,30,40 and 50 with significance level =0.05.
The power estimates of NBRU statistic for Gamma distribution increases more rapidly as  increases than the other distributions, namely, Weibull, Pareto, and Rayleigh. This indicates that the power estimate increases as the class tends to be more NBRU-ness. The same pattern for the power estimates can be noted, when the sample size increases.
This table shows that the percentage of ( i.e. the percentage of the uncensored random variables in the sample) is approximately fixed when the sample size increases for the same parameter and decreases as the parameter of the distribution increases for all the distributions, namely, Gamma, Weibull, Pareto and Rayleigh.This study shows that the percentage of censored values is dependent on the parameter values of the distribution more than the sample size .
The resulting estimates indicate that the proposed statistic is suitable, in particular, for different small sample sizes reliability applications.
4.2 ESTIMATED RISKS
The estimated risks (ER) of the statistic are given by ;
(4.5)
where is the mean value, Xi is the statistic value corresponding to the distribution under study, NBRU, and m is the sample size.
The estimated risks are summarized in Table 3 . We note that, as the sample size increases, the estimated risks decrease, and the mean value is increasing. More precisely, ER/n decreases as n increases. For a large sample size ER/n approaches to zero. This indicates that our test is a powerful one.
4.3 A TEST OF NORMALITY
The Kolmogorov-Smirnov (KS) test is applied to check how well the underlying statistic tends to normality with unspecified mean and variance. For testing normality let S be the empirical distribution function based on the random sample . The test statistics D is defined as the greatest vertical distance between standardized version of , denoted by , and S. Symbolically,
(4.6)
Here we utilize the modified Kolmogorov-Smirnov test of normality proposed by Lilliefors (1967) which accommodate the sample estimated normal parameters.
The D values are given in Table 4. By comparing the calculated KS value with the tabulated one, one accepts the hypothesis approach those to normality for the considered sample sizes.
Acknowledgment: The authors are grateful to professors D. N. Shanbhag of Sheffield University and R. Ahmed of Strathclyde University for their constructive comments on the earlier version of this paper.
Table 1: Critical values for statistic for testing NBRU and
NWRU ageing properties.
-----------------------------------------------------------------------------
n 0.010 0.050 0.100 0.900 0.950 0.990
------------------------------------------------------------------------------
10 -3.1457 -1.9310 -1.4567 0.7251 1.1764 2.2745
12 -2.7025 -1.6203 -1.2422 0.8980 1.3059 2.3555
14 -2.4597 -1.4717 -1.1170 0.9558 1.4362 2.6770
16 -2.0969 -1.3157 -0.9816 1.1010 1.6027 2.8588
18 -2.0062 -1.2031 -0.8924 1.1615 1.6451 2.9567
20 -1.7232 -1.1249 -0.8085 1.2522 1.6715 2.8280
22 -1.6796 -1.0047 -0.7420 1.2844 1.8103 2.9974
24 -1.4614 -0.9286 -0.6464 1.3456 1.8761 3.2730
26 -1.3858 -0.8170 -0.5815 1.4630 1.9602 3.5735
28 -1.2508 -0.7610 -0.5356 1.5441 2.1226 3.3913
30 -1.1563 -0.6970 -0.4633 1.5070 1.9842 3.3249
35 -1.0217 -0.5807 -0.3538 1.6309 2.1884 3.6401
40 -0.8643 -0.4260 -0.2355 1.7301 2.2746 3.5854
45 -0.7075 -0.3482 -0.1432 1.8070 2.3649 3.5055
50 -0.6091 -0.2636 -0.0834 1.8730 2.3875 3.7362
------------------------------------------------------------------------------
Table 2 : Power estimates for statistic for testing NBRU
and NWRU ageing properties.
-----------------------------------------------------------------------------------
DISTRIBUTION PARAMETER SAMPLE SIZE
-----------------------------------------------------------------------------------
10 20 30 40 50
-----------------------------------------------------------------------------------
F (Gamma) 2 0.0090 0.0120 0.0140 0.0160 0.0210
percentage of Delta=1 66 66 67 66 66
F (Gamma) 3 0.1620 0.2230 0.2710 0.2720 0.3470
percentage of Delta=1 54 54 54 54 54
F (Gamma) 4 0.5620 0.6130 0.6520 0.6650 0.6910
percentage of Delta=1 44 45 44 44 45
F (Gamma) 5 0.8340 0.8550 0.8620 0.9540 1.0000
percentage of Delta=1 36 36 36 36 36
F (Gamma) 6 0.9990 1.0000 1.0000 1.0000 1.0000
percentage of Delta=1 30 30 30 30 30
-----------------------------------------------------------------------------------
F (Weibull) 0.25 0.0800 0.1200 0.1800 0.3000 0.4600
percentage of Delta=1 36 39 42 40 41
F (Weibull) 0.50 0.0000 0.0200 0.0800 0.1000 0.1800
percentage of Delta=1 44 43 42 42 44
F (Weibull) 2.00 0.0000 0.0000 0.0000 0.0000 0.0000
percentage of Delta=1 53 53 55 56 55
F (Weibull) 3.00 0.1100 0.1200 0.1300 0.1400 0.1600
percentage of Delta=1 42 46 45 45 44
F (Weibull) 4.00 0.1500 0.1900 0.2500 0.2800 0.3500
percentage of Delta=1 39 38 39 39 39
F (Weibull) 5.00 0.1700 0.2300 0.4700 0.5300 0.6000
percentage of Delta=1 34 33 33 34 33
F (Weibull) 6.00 0.2600 0.4700 0.6400 0.7200 0.8200
percentage of Delta=1 27 27 29 29 29
-----------------------------------------------------------------------------------
F (Pareto) 0.25 0.0930 0.1180 0.0940 0.0820 0.0870
percentage of Delta=1 87 87 87 87 87
F (Pareto) 0.50 0.2170 0.2380 0.2500 0.2160 0.2280
percentage of Delta=1 85 84 84 84 84
F (Pareto) 2.00 0.3850 0.5050 0.5690 0.5820 0.6180
percentage of Delta=1 67 67 67 67 67
F (Pareto) 3.00 0.3250 0.4360 0.4590 0.5060 0.5300
percentage of Delta=1 58 59 59 59 58
F (Pareto) 4.00 0.2750 0.3210 0.3030 0.3410 0.3710
percentage of Delta=1 52 52 52 52 52
F (Pareto) 5.00 0.2290 0.2190 0.2270 0.2000 0.2000
percentage of Delta=1 47 47 47 47 47
F (Pareto) 6.00 0.1560 0.1230 0.1160 0.1070 0.1270
percentage of Delta=1 43 42 42 42 43
-----------------------------------------------------------------------------------
F (Rayleigh) 0.0130 0.1210 0.1470 0.1530 0.1600
percentage of Delta=1 62 62 62 62 62
-----------------------------------------------------------------------------------
Table 3 : Mean and estimated risks for statistic for testing
NBRU and NWRU ageing properties.
-------------------------------------------------------------------------------------------
n MEAN ER ER/n n MEAN ER ER/n
-------------------------------------------------------------------------------------------
10 -0.3224 0.9917 0.0992
11 -0.2701 0.9288 0.0844
12 -0.1833 0.8738 0.0728
13 -0.1998 0.6843 0.0526
14 -0.0581 0.9319 0.0666
15 -0.0471 0.7758 0.0517
16 0.0073 0.8968 0.0561
17 0.0562 0.8446 0.0497
18 0.0746 0.7587 0.0422
19 0.1111 0.7865 0.0414
20 0.1817 0.8775 0.0439
21 0.2421 0.8050 0.0383
22 0.2143 0.7624 0.0347
23 0.2301 0.7054 0.0307
24 0.3036 0.8189 0.0341
25 0.3366 0.8395 0.0336
26 0.3269 0.7370 0.0283
27 0.3426 0.7107 0.0263
28 0.4701 0.9417 0.0336
29 0.4626 0.8955 0.0309
30 0.4372 0.7959 0.0265
----------------------------------------
Table 4 : Test of normality for statistic for testing NBRU
and NWRU ageing properties.
--------------------------------------------------------------------------
n D n D n D
---------------------------------------------------------------------------
10 0.0630 28 0.1053 46 0.0996
11 0.0735 29 0.0863 47 0.1016
12 0.0753 30 0.0684 48 0.0887
13 0.0774 31 0.0830 49 0.1132
14 0.0694 32 0.1053 50 0.0709
15 0.0782 33 0.0937 51 0.1056
16 0.0792 34 0.1058 52 0.0935
17 0.0794 35 0.0876 53 0.0886
18 0.0700 36 0.1044 54 0.0832
19 0.0807 37 0.1109 55 0.1106
20 0.0674 38 0.0944 56 0.1080
21 0.0813 39 0.0815 57 0.1043
22 0.0825 40 0.1047 58 0.1047
23 0.0817 41 0.0783 59 0.1023
24 0.0890 42 0.0791 60 0.1063
25 0.0851 43 0.0825 61 0.1094
26 0.0706 44 0.1030 62 0.1188
27 0.0902 45 0.0995
---------------------------------------------------------------------------
REFERENCES
Abouammoh, A. M. , Ahmed, A. N & Barry, A. M. (1993): Shock models and testing for the renewal mean remaining life classes. Microelectronics and Reli., 33, 5, 729-740.

Abouammoh, A. M. & Alsadi, M. (1996): on testing of new better than renewal used class. Arab J. of Math. Sciences, 2, 99-120.

Ahmed, I. A. (1975): A nonparametric test of the monotonicity of a failure rate function . Comm. Statist.,4, 10, 967-974.

Barlow, R. E. & Proschan, F. (1981): Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Reinhart and Winston Inc., New York.

Barlow, R. E., Marshall, A. W. & Proschan, F. (1963): Properties of probability distributions with monotone hazard rate. Ann. Math. Statist., 34, 375-389.

Bryson, M. C. & Siddiqui, M. M. (1969): Some criteria for ageing. JASA, 64, 1472-1483.

Cao, J. & Wang, Y. (1991): The NBUC and NWUC classes of life distributions. J. Appl. Prob., 28, 473-479.

Chen, Y. Y. , Hollander, M. & Langberg, N. A. (1983): Testing whether new is better than used with randomly censored data. Ann. Statist., 11, 1, 267-274.

Feller, W. (1966): An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley & Sons, New York.

Gerlach, B. (1987): Testing exponentiality against failure rate with randomly censored data. Statist., 18, 2, 275-286.

Gerlach, B. (1989): Tests for increasing failure rate average with randomly right censored data. Statist., 20, 2, 287-295.

Gill, R. (1983): Large sample behaviour of the product limit estimator on the whole line. Ann. Statist.,11,1,49-58.

Hendi, M. I. Mashhour, A. F. and Montasser, M. A. (1993): Closure of the NBUC class under formation of parallel systems. J. Appl. Prob., 30, 975-978.

Kaplan, E. L. & Meier, P. (1958): Nonparametric estimation from incomplete observations. JASA, 53, 457-481.

Kumazawa, Y. (1987): On testing whether new is better than used using randomly censored data. Ann. Statist., 15, 1, 420-426.

Kumazawa, Y. (1992): Test for increasing failure rate with randomly censored data. Statist., 23, 17-25.

Lilliefors, H. W. (1967) : On the Kolmogorov-Smirnov test for normality with mean and variance unknown. JASA, 62, 399-402.

Parzan (1962): Stochastic Processes, Holdem-Day Inc., San Francisco.

Tiwari, R. C., Jammalamadaka, F.R. & Zalkikar, J. N. (1989): Testing in IFRA distribution with censored data. Statist., 20, 2, 279-286.