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A moment inequality for decreasing (increasing) mean residual life distributions with hypothesis testing application
Statistics & Probability Letters 57 (2002) 171–177
A moment inequality for decreasing (increasing) mean residual life distributions with hypothesis testing application S.E. Abu-Youssef Department of Statistics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Received October 2001; received in revised form November 2001
Abstract A moment inequality is derived for decreasing (increasing) mean residual life DMRL (IMRL) distributions. A new test statistic for testing exponentiality against DMRL (IMRL) is introduced based on this inequality. It is shown that the proposed test has high relative e0ciency for some commonly used alternative and also enjoys a good power. Critical values are tabulated for sample sizes n = 5(1)50. A set of real data is used as c 2002 Elsevier an example to elucidate the use of the proposed test statistic for practical reliability analysis. Science B.V. All rights reserved. Keywords: IMRL (DMRL); Exponentiality; Normality; E0ciency; Moments; Power functions
1. Introduction The mean residual life (MRL) function is useful in many areas including biometry, actuarial 9 science and reliability. ∞ Let F be a continuous life distribution with survival function F = 1 − F and 9 :nite mean = 0 F(x) d x. The MRL function is de:ned as ∞ 9 9 F(u) du= F(x): (1.1) m(x) = E[X − x|X ¿ x] = x
If m(x) is nonincreasing in x ¿ 0, then F is said to be a decreasing mean residual life (DMRL) distribution. The dual class, increasing mean residual life (IMRL) distribution, can be de:ned by replacing (nonincreasing) by (nondecreasing) in the de:nition of the DMRL class. The DMRL distribution was introduced by Bryson and Siddiqi (1969) and independently by Marshall and Proschan (1972). Several authors have proposed nonparametric procedures for testing exponentiality against monotonicity properties of the MRL function. Hollander and Proschan (1975) suggested a test for exponentiality against the alternative of decreasing and nonconstant MRL. Bergman and Klefsjo (1989) developed a family of test statistics intended for testing exponentiality against DMRL when the data is both complete and censored. Aly c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 0 4 5 - 7
172
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177
(1990) suggested a set of tests for testing monotonicity of MRL function among which one is for detecting DMRL. Ahmad (1992) proposed a new test procedure for testing exponentiality against DMRL alternative and showed that his test performs better than Hollander and Proschan’s test by calculating Pitman asymptotic relative e0ciencies for several alternatives. Lim and Park (1993) generalized Ahmad’s test to accommodate the situation where the data is incomplete due to random censoring. Lim and Park (1997) proposed a new family of test statistics for testing hypothesis H0 that the MRL is constant against the alternative hypothesis H1 that the MRL is decreasing (increasing) for the cases when the data is both complete and censored. They also calculated the asymptotic relative e0ciencies of their test statistics with respect to Hollander and Proschan’s test (1975) and Chen et al. (1983) for several alternatives. The thread that connects most work reported here is that a measure of departure from H0 , which is often some weighted function of F, is developed which is strictly positive under H1 and is zero under H0 . Then a sample version of this measure is used as test statistic and its properties are studied. In this spirit, the moment inequality developed in Section 2 can be used to construct test statistics for DMRL (IMRL). In Section 3, this test statistic is based on sample moments of aging distributions. This test statistic is simple to derive, and have exceptionally high e0ciencies and power for some of the well known alternatives relative to other tests. Monte Carlo null distribution critical points are obtained for sample sizes n = 5(1)50. An example using real data representing 40 patients suIering from blood cancer from one of the Ministry of Health hospitals in Saudi Arabia is given as an application.
2. Moment inequality In the spirit of the work of Ahmad (2001), we state and prove the following result: Theorem 2.1. If F is DMRL (IMRL); then (2) ¿ (6)
2 ; 2
(2.1)
where (r) = E[min(X1 ; X2 )]r . 9 Proof. Since F is DMRL; m(x) = (x)= F(x) is ↓ and so (d=d x)m(x) ¡ 0. i.e. 2 F9 (x) ¿ (x)f(x);
(2.2)
∞ 9 where (x) = x F(u) du. Multiplying both sides in (2.2) by x and integrating over (0; ∞), w.r.t. x ∞ ∞ 2 xF9 (x) d x ¿ x (x) dF(x): 0
0
(2.3)
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177
But the right-hand side of (2.3) is given by ∞ ∞ 2 x (x) dF(x) = − xF9 (x) d x − 0
0
=−
0
0
∞
2
xF9 (x) d x +
∞
173
(x) · (x) d x
2
(0) : 2
∞ 9 Substituting (0) = 0 F(x) d x = , in (2.4), we obtain ∞ ∞ 2 2 x (x) dF(x) = − xF9 (x) d x + : 2 0 0
(2.4)
(2.5)
Using (2.5), the inequality in (2.3) becomes ∞ 2 2 2 (2.6) xF9 (x) d x ¿ : 2 0 ∞ 2 Since 0 xr F9 (x) d x = E[min(X1 ; X2 )]r+1 =(r + 1), cf. Ahmad (2001), inequality (2.6) becomes (2) ¿ 2 =2. The result follows. 3. Applications to hypotheses testing 3.1. Testing against DMRL (IMRL) alternatives The test presented here depends on a sample X1 ; : : : ; Xn from a population with distribution F. We wish to test the null hypothesis H0 : F9 is exponential with mean against H1 : F9 is DMRL (IMRL) and not exponential. Using Theorem (2.1), we may use the following as a measure of departure from H0 in favor of H1 : 2 : 2 To make the test scale invariant, we use = (2) −
(3.1)
ˆn ˆn = 2 ; X9
(3.2)
where ˆn =
2 1 2 2 min(Xi ; Xj ) − Xi Xj : n(n − 1) i¡j 2
(3.3)
Note that, since (X1 ; X2 ) = min(X12 ; X22 ) − 12 X1 X2 ; ˆn is a classical U -statistic, cf. Lee (1990). Now, we state and prove the following:
(3.4)
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S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177
√ Theorem 3.1. As n → ∞; n(ˆn − ) is asymptotic normal with mean 0 and variance X1 2 −4 2 9 2 = var X1 F(X1 ) − X1 + u dF(u) : 0
Under H0 ; n = 0 and the variance 02 =
(3.5)
2 . 27
ˆ 2 have the same limiting distribution; we use √n(ˆ − ). Now this is Proof. Since ˆn and = asymptotically normal with mean 0 and variance 2 = Var[(X1 )]; where (X1 ) = 2E{(X1 ; X2 )|X1 }: But
X1 9 1 ) − X1 + (X1 ) = 2 X12 F(X u2 dF(u) 4 : 2 0
(3.6)
Hence (3.5) follows. Under H0 (X1 ) = −4(1 + X )e−X + 4 − X: Hence 02 =
2 : 27
√ To use the above test, calculate nˆn =02 and reject H0 if this exceeds the Z-value Z1− . We calculate, via Monte Carlo method, the empirical critical points of ˆn in (3.2) for samples. Table 1 gives the upper percentile points for 90%, 95%, 98% and 99% and the calculations are based on 5000 simulated samples n = 5(1)50. The power of the test statistics ˆn is calculated for the 95% percentile and summarized in Table 2 for the following three most commonly used alternatives (cf. Hollander and Proschan, 1975). (i) Linear failure rate: F9 1 (x) = exp(−x − 12 x2 ), ¿ 0, x ¿ 0 (ii) Makeham: F9 2 (x) = exp[ − x − (x + e−x − 1)], ¿ 0, x ¿ 0 (iii) Weibull: F9 3 (x) = exp[ − x ], ¿ 1, x ¿ 0. These distributions reduced to the exponential distribution for appropriate values of . Finally to assess how good this procedure is relative to others in the literature, we use the concept of “Pitman’s asymptotic e0ciency” (PAE). To do this we need to evaluate the PAE of the proposed test and compare it with the PAE of the tests V ∗ and Un presented by Hollander and Proschan (1975) and Ahmad (1992), respectively. Note that the test of Ahmad (1992) is the same as the test of T k presented by Lim and Park (1997) when k = 1. The PAE of () is given by d ()|=0 =02 : d
(3.7)
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177 Table 1 Critical values for percentiles of ˆn n
90%
95%
98%
99%
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
0.1604 0.1420 0.1311 0.1155 0.1150 0.1117 0.1017 0.0977 0.0941 0.0896 0.0896 0.0861 0.0834 0.0806 0.0775 0.0771 0.0751 0.0734 0.0712 0.0699 0.678 0.0683 0.0666 0.0660 0.0628 0.0639 0.0590 0.0614 0.0579 0.0603 0.0596 0.0576 0.0567 0.0541 0.0534 0.0547 0.0550 0.0535 0.0534 0.0521 0.0519 0.0516 0.0509 0.0488 0.0494 0.0479
0.2035 0.1837 0.1645 0.1499 0.1432 0.1377 0.1339 0.1259 0.1198 0.1121 0.1146 0.1093 0.1093 0.1027 0.0970 0.0970 0.0967 0.0933 0.0896 0.0891 0.0853 0.0866 0.0854 0.0840 0.0805 0.0816 0.0776 0.0774 0.0745 0.0773 0.0743 0.0711 0.0727 0.0690 0.0681 0.0685 0.0687 0.0697 0.0683 0.0670 0.0657 0.0654 0.0659 0.0626 0.0615 0.0622
0.2488 0.2191 0.2010 0.1867 0.1772 0.1685 0.1665 0.1563 0.1480 0.1388 0.1407 0.1343 0.1372 0.1282 0.1215 0.1187 0.1197 0.1160 0.1114 0.1114 0.1067 0.1062 0.1057 0.1045 0.1015 0.0994 0.0964 0.0956 0.0931 0.0923 0.0910 0.0923 0.0912 0.0855 0.0842 0.0856 0.0878 0.0861 0.0835 0.0807 0.0840 0.0794 0.0821 0.0763 0.0795 0.0757
0.2713 0.2448 0.2193 0.2097 0.1952 0.1911 0.1909 0.1784 0.1631 0.1587 0.1564 0.1501 0.1527 0.1460 0.1391 0.1317 0.1339 0.1313 0.1270 0.1244 0.1221 0.1197 0.1168 0.1153 0.1142 0.1113 0.1067 0.1087 0.1023 0.1048 0.1016 0.1048 0.1017 0.0982 0.0968 0.0993 0.0965 0.0959 0.0934 0.0922 0.0955 0.0889 0.0929 0.0863 0.0891 0.0877
175
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S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177 Table 2 Power estimate of ˆn -statistic Distribution
Parameter
Sample size
n = 10
n = 20
n = 30
F1 (Linear failure rate)
2 3 4
0.483 0.528 0.562
0.682 0.734 0.777
0.801 0.848 0.882
F2 (Makeham)
2 3 4
0.393 0.431 0.461
0.498 0.584 0.638
0.574 0.667 0.732
F3 (Weibull)
2 3 4
0.820 0.994 1.000
0.974 1.000 1.000
0.997 1.000 1.000
Table 3 PAE for V ∗ , Un , ˆn Distribution F1 (Linear failure rate) F3 (Weibull)
PAE V∗
Un
ˆn
0.906
0.87039
0.9192
0.87
1.206
0.71
Two of the most commonly used alternatives (cf. Hollander and Proschan, 1975) are (i) Linear failure rate: F9 1 () = exp(−x − 12 x2 ), ¿ 0, x ¿ 0 (ii) Weibull: F9 2 () = exp[ − x ], ¿ 1, x ¿ 0. Direct calculations of PAE of V ∗ , Un and ˆn are summarized in Table 2. From Table 3, it appears that the test statistics ˆn is more e0cient than V ∗ and Un , for Linear failure rate F1 . 4. Numerical examples Consider the data in Abouammoh et al. (1994). These data represent 40 patients suIering from blood cancer from one of the Ministry of Health Hospitals in Saudi Arabia and the ordered life times (in days) are: 115, 181, 255, 418, 441, 461, 516, 739, 743, 789, 807, 865, 924, 983, 1024, 1062, 1063, 1165, 1191, 1222, 1222, 1251, 1277, 1290, 1357, 1369, 1408, 1455, 1478, 1549, 1578, 1578, 15999, 1603, 1605, 1696, 1735, 1799, 1815, 1852.
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177
Using Eq. (3.2), the value of the test statistic, based on the above data, is ˆn = 0:2166. This value leads to the acceptance of H1 at the signi:cance level = 0:95 (see Table 1). The data therefore has DMRL property. Acknowledgements The author greatly appreciates the referees for their constructive comments. References Abouammoh, A.M., Abdulghani, S.A., Qamber, I.S., 1994. On partial ordering and testing of new better than renewal used classes. Reliability Engg. Systems Safety 43, 37–41. Ahmad, I.A., 2001. Moments inequalities of aging families of distributions with hypotheses testing applications. J. Statist. Plann. Inference 92 (1–2), 121–132. Ahmad, I.A., 1992. A new test for mean residual life time. Biometrika 79, 416–419. Aly, E.A.A., 1990. Test for monotonicity properties of the mean residual life function. Scand. J. Statist. 17, 184–200. Bergman, B., Klefsjo, B., 1989. A family of test statistics for detecting monotonic mean residual life function. J. Statist. Plann. Inference 21, 161–178. Bryson, M.C., Siddiqi, M.M., 1969. Some criteria of aging. J. Amer. Statist. Assoc. 64, 1472–1483. Chen, Y.Y., Hollander, M., Langberg, N.A., 1983. Tests for monotonic mean residual life using randomly censored data. Biometrics 39, 127–199. Hollander, M., Proschan, F., 1975. Test for mean residual life. Biometrika 62, 585–593. Lee, A.J., 1990. U -statistics. Marcel Dekker, New York. Lim, J.H., Park, D.H., 1993. Tests for the DMRL using censored data. J. Nonparametric Statist. 3, 167–173. Lim, J.H., Park, D.H., 1997. A family of tests statistics of DMRL (IMRL) alternatives. J. Nonparametric Statist. 8, 293–305. Marshall, A.W., Proschan, F., 1972. Classes of distributions applicable in replacement policies with renewal theory implications. Proc. 6th Berkeley Symp. Math. Statist. Probab. 1, 395–415.
A moment inequality for decreasing (increasing) mean residual life distributions with hypothesis testing application S.E. Abu-Youssef Department of Statistics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Received October 2001; received in revised form November 2001
Abstract A moment inequality is derived for decreasing (increasing) mean residual life DMRL (IMRL) distributions. A new test statistic for testing exponentiality against DMRL (IMRL) is introduced based on this inequality. It is shown that the proposed test has high relative e0ciency for some commonly used alternative and also enjoys a good power. Critical values are tabulated for sample sizes n = 5(1)50. A set of real data is used as c 2002 Elsevier an example to elucidate the use of the proposed test statistic for practical reliability analysis. Science B.V. All rights reserved. Keywords: IMRL (DMRL); Exponentiality; Normality; E0ciency; Moments; Power functions
1. Introduction The mean residual life (MRL) function is useful in many areas including biometry, actuarial 9 science and reliability. ∞ Let F be a continuous life distribution with survival function F = 1 − F and 9 :nite mean = 0 F(x) d x. The MRL function is de:ned as ∞ 9 9 F(u) du= F(x): (1.1) m(x) = E[X − x|X ¿ x] = x
If m(x) is nonincreasing in x ¿ 0, then F is said to be a decreasing mean residual life (DMRL) distribution. The dual class, increasing mean residual life (IMRL) distribution, can be de:ned by replacing (nonincreasing) by (nondecreasing) in the de:nition of the DMRL class. The DMRL distribution was introduced by Bryson and Siddiqi (1969) and independently by Marshall and Proschan (1972). Several authors have proposed nonparametric procedures for testing exponentiality against monotonicity properties of the MRL function. Hollander and Proschan (1975) suggested a test for exponentiality against the alternative of decreasing and nonconstant MRL. Bergman and Klefsjo (1989) developed a family of test statistics intended for testing exponentiality against DMRL when the data is both complete and censored. Aly c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 0 4 5 - 7
172
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177
(1990) suggested a set of tests for testing monotonicity of MRL function among which one is for detecting DMRL. Ahmad (1992) proposed a new test procedure for testing exponentiality against DMRL alternative and showed that his test performs better than Hollander and Proschan’s test by calculating Pitman asymptotic relative e0ciencies for several alternatives. Lim and Park (1993) generalized Ahmad’s test to accommodate the situation where the data is incomplete due to random censoring. Lim and Park (1997) proposed a new family of test statistics for testing hypothesis H0 that the MRL is constant against the alternative hypothesis H1 that the MRL is decreasing (increasing) for the cases when the data is both complete and censored. They also calculated the asymptotic relative e0ciencies of their test statistics with respect to Hollander and Proschan’s test (1975) and Chen et al. (1983) for several alternatives. The thread that connects most work reported here is that a measure of departure from H0 , which is often some weighted function of F, is developed which is strictly positive under H1 and is zero under H0 . Then a sample version of this measure is used as test statistic and its properties are studied. In this spirit, the moment inequality developed in Section 2 can be used to construct test statistics for DMRL (IMRL). In Section 3, this test statistic is based on sample moments of aging distributions. This test statistic is simple to derive, and have exceptionally high e0ciencies and power for some of the well known alternatives relative to other tests. Monte Carlo null distribution critical points are obtained for sample sizes n = 5(1)50. An example using real data representing 40 patients suIering from blood cancer from one of the Ministry of Health hospitals in Saudi Arabia is given as an application.
2. Moment inequality In the spirit of the work of Ahmad (2001), we state and prove the following result: Theorem 2.1. If F is DMRL (IMRL); then (2) ¿ (6)
2 ; 2
(2.1)
where (r) = E[min(X1 ; X2 )]r . 9 Proof. Since F is DMRL; m(x) = (x)= F(x) is ↓ and so (d=d x)m(x) ¡ 0. i.e. 2 F9 (x) ¿ (x)f(x);
(2.2)
∞ 9 where (x) = x F(u) du. Multiplying both sides in (2.2) by x and integrating over (0; ∞), w.r.t. x ∞ ∞ 2 xF9 (x) d x ¿ x (x) dF(x): 0
0
(2.3)
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177
But the right-hand side of (2.3) is given by ∞ ∞ 2 x (x) dF(x) = − xF9 (x) d x − 0
0
=−
0
0
∞
2
xF9 (x) d x +
∞
173
(x) · (x) d x
2
(0) : 2
∞ 9 Substituting (0) = 0 F(x) d x = , in (2.4), we obtain ∞ ∞ 2 2 x (x) dF(x) = − xF9 (x) d x + : 2 0 0
(2.4)
(2.5)
Using (2.5), the inequality in (2.3) becomes ∞ 2 2 2 (2.6) xF9 (x) d x ¿ : 2 0 ∞ 2 Since 0 xr F9 (x) d x = E[min(X1 ; X2 )]r+1 =(r + 1), cf. Ahmad (2001), inequality (2.6) becomes (2) ¿ 2 =2. The result follows. 3. Applications to hypotheses testing 3.1. Testing against DMRL (IMRL) alternatives The test presented here depends on a sample X1 ; : : : ; Xn from a population with distribution F. We wish to test the null hypothesis H0 : F9 is exponential with mean against H1 : F9 is DMRL (IMRL) and not exponential. Using Theorem (2.1), we may use the following as a measure of departure from H0 in favor of H1 : 2 : 2 To make the test scale invariant, we use = (2) −
(3.1)
ˆn ˆn = 2 ; X9
(3.2)
where ˆn =
2 1 2 2 min(Xi ; Xj ) − Xi Xj : n(n − 1) i¡j 2
(3.3)
Note that, since (X1 ; X2 ) = min(X12 ; X22 ) − 12 X1 X2 ; ˆn is a classical U -statistic, cf. Lee (1990). Now, we state and prove the following:
(3.4)
174
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177
√ Theorem 3.1. As n → ∞; n(ˆn − ) is asymptotic normal with mean 0 and variance X1 2 −4 2 9 2 = var X1 F(X1 ) − X1 + u dF(u) : 0
Under H0 ; n = 0 and the variance 02 =
(3.5)
2 . 27
ˆ 2 have the same limiting distribution; we use √n(ˆ − ). Now this is Proof. Since ˆn and = asymptotically normal with mean 0 and variance 2 = Var[(X1 )]; where (X1 ) = 2E{(X1 ; X2 )|X1 }: But
X1 9 1 ) − X1 + (X1 ) = 2 X12 F(X u2 dF(u) 4 : 2 0
(3.6)
Hence (3.5) follows. Under H0 (X1 ) = −4(1 + X )e−X + 4 − X: Hence 02 =
2 : 27
√ To use the above test, calculate nˆn =02 and reject H0 if this exceeds the Z-value Z1− . We calculate, via Monte Carlo method, the empirical critical points of ˆn in (3.2) for samples. Table 1 gives the upper percentile points for 90%, 95%, 98% and 99% and the calculations are based on 5000 simulated samples n = 5(1)50. The power of the test statistics ˆn is calculated for the 95% percentile and summarized in Table 2 for the following three most commonly used alternatives (cf. Hollander and Proschan, 1975). (i) Linear failure rate: F9 1 (x) = exp(−x − 12 x2 ), ¿ 0, x ¿ 0 (ii) Makeham: F9 2 (x) = exp[ − x − (x + e−x − 1)], ¿ 0, x ¿ 0 (iii) Weibull: F9 3 (x) = exp[ − x ], ¿ 1, x ¿ 0. These distributions reduced to the exponential distribution for appropriate values of . Finally to assess how good this procedure is relative to others in the literature, we use the concept of “Pitman’s asymptotic e0ciency” (PAE). To do this we need to evaluate the PAE of the proposed test and compare it with the PAE of the tests V ∗ and Un presented by Hollander and Proschan (1975) and Ahmad (1992), respectively. Note that the test of Ahmad (1992) is the same as the test of T k presented by Lim and Park (1997) when k = 1. The PAE of () is given by d ()|=0 =02 : d
(3.7)
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177 Table 1 Critical values for percentiles of ˆn n
90%
95%
98%
99%
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
0.1604 0.1420 0.1311 0.1155 0.1150 0.1117 0.1017 0.0977 0.0941 0.0896 0.0896 0.0861 0.0834 0.0806 0.0775 0.0771 0.0751 0.0734 0.0712 0.0699 0.678 0.0683 0.0666 0.0660 0.0628 0.0639 0.0590 0.0614 0.0579 0.0603 0.0596 0.0576 0.0567 0.0541 0.0534 0.0547 0.0550 0.0535 0.0534 0.0521 0.0519 0.0516 0.0509 0.0488 0.0494 0.0479
0.2035 0.1837 0.1645 0.1499 0.1432 0.1377 0.1339 0.1259 0.1198 0.1121 0.1146 0.1093 0.1093 0.1027 0.0970 0.0970 0.0967 0.0933 0.0896 0.0891 0.0853 0.0866 0.0854 0.0840 0.0805 0.0816 0.0776 0.0774 0.0745 0.0773 0.0743 0.0711 0.0727 0.0690 0.0681 0.0685 0.0687 0.0697 0.0683 0.0670 0.0657 0.0654 0.0659 0.0626 0.0615 0.0622
0.2488 0.2191 0.2010 0.1867 0.1772 0.1685 0.1665 0.1563 0.1480 0.1388 0.1407 0.1343 0.1372 0.1282 0.1215 0.1187 0.1197 0.1160 0.1114 0.1114 0.1067 0.1062 0.1057 0.1045 0.1015 0.0994 0.0964 0.0956 0.0931 0.0923 0.0910 0.0923 0.0912 0.0855 0.0842 0.0856 0.0878 0.0861 0.0835 0.0807 0.0840 0.0794 0.0821 0.0763 0.0795 0.0757
0.2713 0.2448 0.2193 0.2097 0.1952 0.1911 0.1909 0.1784 0.1631 0.1587 0.1564 0.1501 0.1527 0.1460 0.1391 0.1317 0.1339 0.1313 0.1270 0.1244 0.1221 0.1197 0.1168 0.1153 0.1142 0.1113 0.1067 0.1087 0.1023 0.1048 0.1016 0.1048 0.1017 0.0982 0.0968 0.0993 0.0965 0.0959 0.0934 0.0922 0.0955 0.0889 0.0929 0.0863 0.0891 0.0877
175
176
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177 Table 2 Power estimate of ˆn -statistic Distribution
Parameter
Sample size
n = 10
n = 20
n = 30
F1 (Linear failure rate)
2 3 4
0.483 0.528 0.562
0.682 0.734 0.777
0.801 0.848 0.882
F2 (Makeham)
2 3 4
0.393 0.431 0.461
0.498 0.584 0.638
0.574 0.667 0.732
F3 (Weibull)
2 3 4
0.820 0.994 1.000
0.974 1.000 1.000
0.997 1.000 1.000
Table 3 PAE for V ∗ , Un , ˆn Distribution F1 (Linear failure rate) F3 (Weibull)
PAE V∗
Un
ˆn
0.906
0.87039
0.9192
0.87
1.206
0.71
Two of the most commonly used alternatives (cf. Hollander and Proschan, 1975) are (i) Linear failure rate: F9 1 () = exp(−x − 12 x2 ), ¿ 0, x ¿ 0 (ii) Weibull: F9 2 () = exp[ − x ], ¿ 1, x ¿ 0. Direct calculations of PAE of V ∗ , Un and ˆn are summarized in Table 2. From Table 3, it appears that the test statistics ˆn is more e0cient than V ∗ and Un , for Linear failure rate F1 . 4. Numerical examples Consider the data in Abouammoh et al. (1994). These data represent 40 patients suIering from blood cancer from one of the Ministry of Health Hospitals in Saudi Arabia and the ordered life times (in days) are: 115, 181, 255, 418, 441, 461, 516, 739, 743, 789, 807, 865, 924, 983, 1024, 1062, 1063, 1165, 1191, 1222, 1222, 1251, 1277, 1290, 1357, 1369, 1408, 1455, 1478, 1549, 1578, 1578, 15999, 1603, 1605, 1696, 1735, 1799, 1815, 1852.
S.E. Abu-Youssef / Statistics & Probability Letters 57 (2002) 171–177
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Using Eq. (3.2), the value of the test statistic, based on the above data, is ˆn = 0:2166. This value leads to the acceptance of H1 at the signi:cance level = 0:95 (see Table 1). The data therefore has DMRL property. Acknowledgements The author greatly appreciates the referees for their constructive comments. References Abouammoh, A.M., Abdulghani, S.A., Qamber, I.S., 1994. On partial ordering and testing of new better than renewal used classes. Reliability Engg. Systems Safety 43, 37–41. Ahmad, I.A., 2001. Moments inequalities of aging families of distributions with hypotheses testing applications. J. Statist. Plann. Inference 92 (1–2), 121–132. Ahmad, I.A., 1992. A new test for mean residual life time. Biometrika 79, 416–419. Aly, E.A.A., 1990. Test for monotonicity properties of the mean residual life function. Scand. J. Statist. 17, 184–200. Bergman, B., Klefsjo, B., 1989. A family of test statistics for detecting monotonic mean residual life function. J. Statist. Plann. Inference 21, 161–178. Bryson, M.C., Siddiqi, M.M., 1969. Some criteria of aging. J. Amer. Statist. Assoc. 64, 1472–1483. Chen, Y.Y., Hollander, M., Langberg, N.A., 1983. Tests for monotonic mean residual life using randomly censored data. Biometrics 39, 127–199. Hollander, M., Proschan, F., 1975. Test for mean residual life. Biometrika 62, 585–593. Lee, A.J., 1990. U -statistics. Marcel Dekker, New York. Lim, J.H., Park, D.H., 1993. Tests for the DMRL using censored data. J. Nonparametric Statist. 3, 167–173. Lim, J.H., Park, D.H., 1997. A family of tests statistics of DMRL (IMRL) alternatives. J. Nonparametric Statist. 8, 293–305. Marshall, A.W., Proschan, F., 1972. Classes of distributions applicable in replacement policies with renewal theory implications. Proc. 6th Berkeley Symp. Math. Statist. Probab. 1, 395–415.