Optimal parameter estimation of the two-parameter ... - Semantic Scholar
Applied Mathematics and Computation 167 (2005) 807–819
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Optimal parameter estimation of the two-parameter bathtub-shaped lifetime distribution based on a type II right censored sample Jong-Wuu Wu *, Chin-Chuan Wu, Mei-Huei Tsai Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan 25137, Republic of China
Abstract This paper proposes different types of exact (or approximate) confidence intervals and exact (or approximate) joint confidence regions for the parameters of the twoparameter lifetime distribution with bathtub-shaped or increasing failure rate (IFR) function based on a type II right censored sample. Moreover, we provide optimal criteria for finding a best exact (or approximate) confidence interval and a best exact (or approximate) joint confidence region among these estimations. Two examples are used to compare our proposed method to the existing approach of [Statistics & Probability Letters 49 (2000) 155]. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Bathtub-shaped; Failure rate function; Confidence interval; Joint confidence region; Censored sample
*
Corresponding author. E-mail address: [email protected] (J.-W. Wu).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.133
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1. Introduction The failure pattern of many products/systems (as electro-mechanical, electronic and mechanical products, etc) can be represented by the bathtub curve. It includes three phases: early failure (or burn-in) phase with a decreasing failure rate, normal use phase with an approximately constant failure rate, and wear-out phase with an increasing failure rate. In survival analysis, the lifetime of human beings exhibits this pattern. So, some probability distributions have been proposed to fit real life data with bathtub-shaped failure rates, such as Smith and Bain [13], Gaver and Acar [4], Hjorth [6], Leemis [8], Rajarshi and Rajarshi [12], Haupt and Schabe [5], Mudholkar and Srivastava [10], Mi [11], Xie and Lai [15], Chen [2,3], Wang [14], Lai et al. [7] and Xie et al. [16]. In this paper, we discuss that the two-parameter lifetime distribution with bathtub-shaped or increasing failure rate (IFR) function is proposed by Chen [3]. The cumulative distribution function (c.d.f.) of this distribution is xb
F ðxÞ ¼ 1 ekð1e Þ ;
ð1Þ
x > 0;
where k > 0 is the parameter but it does not affect the shape of failure rate function h(x) as in (2) and b > 0 is the shape parameter. The corresponding failure rate function of this distribution is b
hðxÞ ¼ kbxb1 ex ;
ð2Þ
x > 0:
Since b
h0 ðxÞ ¼ kbxb2 ex ðb 1 þ bxb Þ;
x > 0;
1=b
ð3Þ
1b b
. The h(x) has a bathtub shape when b < 1, achieving a minimum at x ¼ distribution has increasing failure rate function when b P 1. In Section 2, we extend the ideas of Mann et al. [9] and Chen [3] to develop different types of exact (or approximate) joint confidence regions for the parameters b and k of the two-parameter lifetime distribution with bathtub-shaped or increasing failure rate function based on a type II right censored sample. In Section 3, we provide an optimal criterion to find a best estimation according to the idea of the shortest tolerance length of an interval or minimal area of a region for the above censored sample. Section 4, we includes illustrative examples using both real failure data of electronic devices from Wang [14] (or Xie et al. [16]) and simulated data from Chen [3] to compare our proposed method and ChenÕs method and give a discussion of the results. Conclusions are given in Section 5. 2. Parameter estimation Let X(1), . . ., X(k) be the first k order statistics of a sample of size n from a population distribution with c.d.f. as in (1). It can be shown that
J.-W. Wu et al. / Appl. Math. Comput. 167 (2005) 807–819
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b b X X Y ð1Þ ¼ k e ð1Þ 1 ; . . . ; Y ðkÞ ¼ k e ðkÞ 1 are the first k order statistics of a sample of size n from a standard exponential distribution. By the property of the standard exponential distribution, we know that these random variables Z1 = nY(1) and Z(i) = (n i + 1) (Y(i) Y(i1)), i = 2, . . ., k are independent and identically distributed with the standard exponential distribution (also see Bickel and Doksum [1, p. 528]). Therefore, we use Z1, . . ., Zk to define Uj and Vj as following: " # k k X X Uj ¼ 2 Zi ¼ 2 Y ðiÞ þ ðn kÞY ðkÞ þ ðj nÞY ðjÞ ð4Þ i¼jþ1
i¼jþ1
and Vj ¼2
j X
" Zi ¼ 2
i¼1
j X
# Y ðiÞ þ ðn jÞY ðjÞ ;
ð5Þ
i¼1
where j = 1, 2, . . ., k 1. Then Uj and Vj are two independent variables and have a v2 distribution with 2(k j) and 2j degrees of freedom, respectively. By using Uj and Vj, we can also define other variables such as hP i k j Y þ ðn kÞY ðn jÞY ðiÞ ðkÞ ðjÞ i¼jþ1 U j =½2ðk jÞ Pj ¼ nj ðbÞ ¼ V j =ð2jÞ ðk jÞ i¼1 Y ðiÞ þ ðn jÞY ðjÞ hP b b b i X ðiÞ X X k j 1 ðn jÞ e ðjÞ 1 þ ðn kÞ e ðkÞ 1 i¼jþ1 e hP b b i ¼ X ðiÞ X j ðk jÞ 1 þ ðn jÞ e ðjÞ 1 i¼1 e and
" 1j ðb; kÞ ¼ U j þ V j ¼ 2 ¼ 2k
" k X
k X
# Y ðiÞ þ ðn kÞY ðkÞ
i¼1
e
X bðiÞ
# b X ðkÞ 1 þ ðn kÞ e 1 ;
i¼1
where j = 1, 2, . . ., k 1. Then, it is easy to see that nj(b) has an F distribution with 2(k j) and 2j degrees of freedom and 1j(b, k) has a v2 distribution with 2k degrees of freedom for j = 1, 2, . . ., k 1. Furthermore, these two random variables nj(b) and 1j(b, k) are independent for each j. In addition, by using the idea of Mann et al. [9, p. 171], we also shown that U 0j ¼ 2mðY ðkÞ Y ðjÞ Þ=v is approximately v2 distribution with 2m2/v, where m ¼ Pk Pk 0 1 1 i¼jþ1 niþ1 and v ¼ i¼jþ1 ðniþ1Þ2 , j = 1, 2, . . ., k 1. Moreover, U j and Vj are independent since they involve different independent standard exponential
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variates. Furthermore, by using U 0j and Vj, we can also define other variables such as U 0j =ð2m2 =vÞ jðY ðkÞ Y ðjÞ Þ ¼ Pj V j =ð2jÞ m i¼1 Y ðiÞ þ ðn jÞY ðjÞ b X Xb j e ðkÞ e ðjÞ b i ¼ hP b X ðiÞ X j m 1 þ ðn jÞ e ðjÞ 1 i¼1 e
n0j ðbÞ ¼
and " 10j ðb; kÞ
¼
U 0j
þ V j ¼ 2mðY ðkÞ Y ðjÞ Þ=v þ 2
j X
# Y ðiÞ þ ðn jÞY ðjÞ
i¼1
¼ 2mk e
X bðkÞ
e
X bðjÞ
" # j b b X X ðiÞ X ðjÞ e 1 þ ðn jÞ e 1 ; =v þ 2k i¼1
where j = 1, 2, . . ., k 1. Then, it is easy to see that n0j ðbÞ has a approximately F distribution with 2m2/v and 2j degrees of freedom and 10j ðb; kÞ has a approximately v2 distribution with 2(m2/v + j) degrees of freedom for j = 1, 2, . . ., k 1. Furthermore, these two random variables n0j ðbÞ and 10j ðb; kÞ are independent for each j. Before we prove the following theorems, some necessary lemmas based on the above definitions are given. Lemma 1. For all j = 1, 2, . . ., k 1 and t > 0, the equation, in b > 0, nj(b) = t has a unique solution. Proof. Since the function nj(b) is strictly increasing in b > 0 with nj(0) = 0 and nj(1) = 1, then the lemma follows. h Lemma 2. For all j = 1, 2, . . ., k 1 and t > 0, the equation, in b > 0, n0j ðbÞ ¼ t has a unique solution. Proof. Since the function n0j ðbÞ is strictly increasing in b > 0 with n0j ð0Þ ¼ 0 and n0j ð1Þ ¼ 1, then the lemma follows. h Therefore, by using Lemmas 1 and 2, we can obtain the exact (or approximate) confidence interval of b and joint region of b and k as the following theorems. Note that Fa(v1, v2) is the upper a critical value of the F distribution with v1 and v2 degrees of freedom and v2a ðdÞ is the upper a critical value of the v2 distribution with d degrees of freedom in this paper.
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Theorem 1. Let X(1), . . ., X(k) be a type II right censored sample with the c.d.f. as (1), where 2 6 k 6 n. Then for any 0 < a < 1, 1 n1 j ðF a=2 ð2ðk jÞ; 2jÞÞ < b < nj ðF 1a=2 ð2ðk jÞ; 2jÞÞ
ð6Þ
is a 1 a confidence interval for the parameter b for j = 1, 2, . . ., k 1, where for t > 0, n1 j ðtÞ is the solution of b for the equation hP b b b i X ðiÞ X X k j 1 ðn jÞ e ðjÞ 1 þ ðn kÞ e ðkÞ 1 i¼jþ1 e hP b b i ¼ t: X ðiÞ X j ðk jÞ 1 þ ðn jÞ e ðjÞ 1 i¼1 s e Proof. By using Lemma 1, we have 1 P fn1 j ðF 1a=2 ð2ðk jÞ; 2jÞÞ < b < nj ðF a=2 ð2ðk jÞ; 2jÞÞg ¼ P F 1a=2 ð2ðk jÞ; 2jÞ
b b b i X ðiÞ X ðjÞ X ðkÞ e 1 ðn jÞ e 1 þ ðn kÞ e 1 i¼jþ1 hP b b i < X ðiÞ X j ðk jÞ 1 þ ðn jÞ e ðjÞ 1 i¼1 e < F a=2 ð2ðk jÞ; 2jÞ ¼ 1 a; j
hP k
where j = 1, 2, . . ., k 1. This completes the proof.
h
By using Lemma 2, the proof of the following theorem is similar to that of Theorem 1 and hence is omitted. Theorem 2. Let X(1), . . ., X(k) be a right censored sample with the c.d.f. as (1), where 2 6 k 6 n. Then for any 0 < a < 1, 01 2 2 n01 j ðF a=2 ð2m =v; 2jÞÞ < b < nj ðF 1a=2 ð2m =v; 2jÞÞ
ð7Þ
is a 1 a approximate confidence interval for the parameter b for j = 1, 2, . . ., k 1, where m and v as the above definition and for t > 0, n01 j ðtÞ is the solution of b for the equation b X Xb j e ðkÞ e ðjÞ hP b b i ¼ t: X ðiÞ X ðjÞ j m e 1 þ ðn jÞ e 1 i¼1 By the same way, we can also prove the exact (or approximate) joint confidence regions for b and k in the Theorems 3 and 4.
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Theorem 3. Let X(1), . . ., X(k) be defined in Theorem 1. Then for any 0 < a < 1, the following inequalities determine a 1 a joint confidence region for b and k with j = 1, 2, . . ., k 1: 8 1 1 pffiffiffiffiffiffi n ðF ð1þpffiffiffiffiffiffi > 1aÞ=2 ð2ðk jÞ; 2jÞÞ < b < nj ðF ð1 1aÞ=2 ð2ðk jÞ; 2jÞÞ; > < j 2 v pffiffiffiffiffi ð2kÞ v2 pffiffiffiffiffi ð2kÞ 1aÞ=2 1aÞ=2 hP bð1þ b i < k < hP bð1 b i ; > > X X X X k k : ðiÞ ðkÞ ðiÞ ðkÞ 2
i¼1
1 þðnkÞ e
e
1
2
1 þðnkÞ e
e
i¼1
1
where n1 j ðtÞ is defined in Theorem 1. Proof. By using Lemma 1, we have ( 1 pffiffiffiffiffiffi pffiffiffiffiffiffi P n1 j ðF ð1þ 1aÞ=2 ð2ðk jÞ;2jÞÞ < b < nj ðF ð1 1aÞ=2 ð2ðk jÞ;2jÞÞ;
¼P
8 < :
k
i¼1
ffiffiffiffiffi
p ð2kÞ ð1þ 1aÞ=2 b X e ðiÞ 1 þðnkÞ
hP
2
v2
e
X
F ð1þpffiffiffiffiffiffi 1aÞ=2 ð2ðk jÞ;2jÞ
nj ðF ð1þpffiffiffiffiffiffi 1aÞ=2 ð2m =v; 2jÞÞ < b < nj ðF ð1 1aÞ=2 ð2m =v; 2jÞÞ; > > > > 2 2 v pffiffiffiffiffi ð2ðm =vþjÞÞ > > 1aÞ=2 > h b ð1þ P b b i < k < b X X X X j ðkÞ ðjÞ ðiÞ ðjÞ 2 m e
e
=vþ
i¼1
e
1 þðnjÞ e
1
> > > v2 pffiffiffiffiffi ð2ðm2 =vþjÞÞ > > 1aÞ=2 > h b ð1 P b b i ; < > b > : 2 m eX ðkÞ eX ðjÞ =vþ j eX ðiÞ 1 þðnjÞ eX ðjÞ 1 i¼1
where m, v and
n01 j ðtÞ
are defined in Theorem 2.
Remark. When j = k 1, the results of Theorems 2 and 4 are the same as the results of Theorems 1 and 3. In addition, when j = 1, the result of Theorems 1 and 3 can be reduced to the result of Chen [3].
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3. Optimal criteria In Section 2, we obtain different kinds of exact (or approximate) interval estimations of the same parameter according to j changes. In order to find out a better estimation, we consider choosing a exact (or approximate) confidence interval with the shortest tolerance length or a exact (or approximate) joint confidence region with the minimal area to be the optimal interval or region. Based on this idea, we provide the optimal criteria as following criteria. Criterion 1. Let lj1 and uj1 be the lower and upper confidence limit defined in Eq. (6), respectively. Then confidence interval ðl1 ; u1 Þ is the optimal confidence interval for all j, if its tolerance length L satisfies L ¼
min ðuj1 lj1 Þ:
ð8Þ
j¼1;2;...;k1
Criterion 2. Let lj2 and uj2 be the lower and upper confidence limit defined in Eq. (7), respectively. Then approximate confidence interval ðl2 ; u2 Þ is the optimal approximate confidence interval for all j, if its tolerance length L** satisfies L ¼
min ðuj2 lj2 Þ:
ð9Þ
j¼1;2;...;k1
~j1 be the lower and upper confidence limit of b Criterion 3. Let ~lj1 and u defined in a joint confidence region of Theorem 3, respectively. Also klj1 ðbÞ and kuj1 ðbÞ be the lower and upper confidence limit of k defined as The u orem 3, respectively. Then confidence region ð~l1 ; ~u1 ; kl 1 ðbÞ; k1 ðbÞÞ is the opti mal confidence region for all j, if its area A satisfies Z ~uj1 Z kuj1 ðbÞ A ¼ min dk db: ð10Þ j¼1;2;...;k1
klj1 ðbÞ
~lj1
uj2 be the lower and upper confidence limit of b Criterion 4. Let ~lj2 and ~ defined in a joint confidence region of Theorem 4, respectively. Also klj2 ðbÞ and kuj2 ðbÞ be the lower and upper confidence limit of k defined as Theorem 4, respectively. Then approximate confidence region u ð~l2 ; ~ u2 ; kl 2 ðbÞ; k2 ðbÞÞ is the optimal approximate confidence region for all j, if its area A** satisfies
A ¼
Z
uj2 ~
Z
kuj2 ðbÞ
min
j¼1;2;...;k1
dk db: ~lj2
klj2 ðbÞ
ð11Þ
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4. Illustrative examples In this section, we apply the proposed methods to one of practical data set and another simulated data set. Example 1 considered is the failure data of n = k = 18 electronic devices from Wang [14] (or Xie et al. [16]). Example 2 considered is the first k = 11 observations of a computer generated sample of size n = 15 from a population distribution defined by (1) with parameter k = 0.02 and b = 0.5 (see Chen [3]). Example 1 (Real life data). The complete sample of the failure data of n = k = 18 electronic devices from Wang [14] (or Xie et al. [16]) is given as following: 5; 11; 21; 31; 46; 75; 98; 122; 145; 165; 195; 224; 245; 293; 321; 330; 350; 420: Xie et al. [16] have been shown that the data set has a bathtub-shaped failure rate function. According to their assumption, we apply ChenÕs method (that is, j = 1 in Theorem 1) to get an interval length of 95% confidence interval for b is 0.177057095 (see Table 1). By using Theorem 1 and Criterion 1, we obtain that the interval length of 95% confidence interval for b is given in Table 1. From Table 1, we find that when j = 7, our proposed method obtains a shorter confidence interval length is 0.119291259. Moreover, we see that if j = 7, F 0:975 ð22; 14Þ ¼ 0:395494284
and
F 0:025 ð22; 14Þ ¼ 2:813940631;
Table 1 The length of 95% exact (or approximate) confidence interval for b j
By using Theorem 1
By using Theorem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 min(j)
0.177057095 0.145143463 0.131952392 0.126130859 0.122527588 0.119466248 0.119291259 0.120534240 0.122850586 0.126876465 0.131615478 0.138036376 0.148391357 0.160359131 0.179460327 0.223914063 0.331802979 0.119291259 (7)
0.150146942 0.122560089 0.113409912 0.110536773 0.109812378 0.111204743 0.113756836 0.117913697 0.122458740 0.128150329 0.135126342 0.143895996 0.154786865 0.170855469 0.194511718 0.235051269 0.331802979 0.109812378 (5)
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then (0.210587891, 0.329879150) is an optimal 95% confidence interval for the parameter b. Furthermore, from this result, we find that the data set has a bathtub-shaped failure rate function. Hence, this agrees with the conclusion stated in Xie et al. [16]. In addition, from Table 2, we can see that if j = 14, F pffiffiffiffiffiffi ð8; 28Þ ¼ 0:205950946 and F pffiffiffiffiffiffi ð8; 28Þ ¼ 3:085177303; ð1þ 0:95Þ=2
ð1 0:95Þ=2
v2ð1þpffiffiffiffiffiffi ð36Þ ¼ 19:73449165 0:95Þ=2
and
v2ð1pffiffiffiffiffiffi ð36Þ ¼ 57:57500499; 0:95Þ=2
then by Theorem 3 and Criterion 3 an optimal 95% joint confidence region for the parameters b and k is determined by the following inequalities: 8 < 0:217148438 < b < 0:401312988; 57:57500499 i h 19:73449165 i < k < hP : b 18 X : 2 P18 eX bðiÞ 18 e ðiÞ 18 2 i¼1
i¼1
Moreover, the area of the optimal 95% joint confidence region is 0.002331714. By using Theorem 2 and Criterion 2, we obtain that the interval length of 95% approximate confidence interval for b is also given in Table 1. From Table 1, we find that when j = 5, our proposed method obtains a shorter confidence interval length is 0.109812378. Moreover, we see that if j = 5, F 0:975 ð10:40; 10Þ ¼ 0:245974153
and
F 0:025 ð10:40; 10Þ ¼ 3:309474589;
Table 2 The area of 95% exact (or approximate) joint confidence region for (b, k) j
By using Theorem 3
By using Theorem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 min(j)
0.004002693 0.004582333 0.004509095 0.004776620 0.004647857 0.003589845 0.003318413 0.003104469 0.003056782 0.003264490 0.003030044 0.002932669 0.003529089 0.002331714 0.002497938 0.008274441 0.041296306 0.002331714 (14)
0.004647363 0.005053846 0.004775387 0.004852702 0.004586928 0.003575137 0.003319062 0.003143474 0.003129565 0.003305460 0.003224458 0.003281763 0.003890330 0.003551379 0.004720985 0.011328405 0.041296306 0.003129565 (9)
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then (0.212167969, 0.321980347) is an optimal 95% approximate confidence interval for the parameter b. Furthermore, from this result, we also find that the data set has a bathtub-shaped failure rate function. Hence, this agrees with the conclusion stated in Xie et al. [16]. In addition, from Table 2, we can see that if j = 9, F pffiffiffiffiffiffi ð4:88; 18Þ ¼ 0:306392397 and ð1þ 0:95Þ=2
F ð1pffiffiffiffiffiffi 0:95Þ=2 ð4:88; 18Þ ¼ 3:58787930; ð22:88Þ ¼ 10:465426950 v2ð1þpffiffiffiffiffiffi 0:95Þ=2 v2ð1pffiffiffiffiffiffi ð22:88Þ 0:95Þ=2
and
¼ 40:58016026;
then by Theorem 4 and Criterion 4 an optimal 95% approximate joint confidence region for the parameters b and k is determined by the following inequalities: 8 < 0:209179688 < b < 0:350528687; 10:465426950 h i < k < hP b 40:58016026 i: b b X X X 9 : 2 P9 eX bðiÞ þ1:84eX bð18Þ þ7:16eX bð9Þ 18 e ðiÞ þ1:84e ð18Þ þ7:16e ð9Þ 18 2 i¼1
i¼1
Moreover, the area of the optimal 95% joint confidence region is 0.003129565. Example 2 (Simulated data). The following are the first k = 11 observations of a computer generated sample of size n = 15 from a population distribution defined by (1) with parameters k = 0.02 and b = 0.5 (also see Chen [3]): 0:29; 1:44; 8:38; 8:66; 10:20; 11:04; 13:44; 14:37; 17:05; 17:13 and 18:35: It was found by Chen [3] that (0.189628906, 0.621608498) is a 95% confidence interval with interval length 0.431979592 for the shape parameter b. By using Theorem 1 and Criterion 1, we obtain that the interval length of 95% confidence interval for b is given in Table 3. From Table 3, we find that when j = 3, our proposed method obtains a shorter confidence interval length is 0.326654174. Moreover, we see that if j = 3, F 0:975 ð16; 6Þ ¼ 0:29934465
and
F 0:025 ð16; 6Þ ¼ 5:243860453;
then (0.378259766, 0.704913940) is an optimal 95% confidence interval for the parameter b. In addition, from Table 4, we can see that if j = 3, F pffiffiffiffiffiffi ð16; 6Þ ¼ 0:251777713 and F pffiffiffiffiffiffi ð16; 6Þ ¼ 6:866053989; ð1þ 0:95Þ=2
v2ð1þpffiffiffiffiffiffi ð22Þ ¼ 9:882428630 0:95Þ=2
ð1 0:95Þ=2
and
v2ð1pffiffiffiffiffiffi ð22Þ ¼ 39:409864930; 0:95Þ=2
then by Theorem 3 and Criterion 3 an optimal 95% joint confidence region for the parameters b and k is determined by the following inequalities:
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Table 3 The length of 95% exact (or approximate) confidence interval for b j
By using Theorem 1
By using Theorem 2
1 2 3 4 5 6 7 8 9 10 min(j)
0.431979592 0.353967285 0.326654174 0.330638672 0.338140625 0.356469238 0.388075928 0.437386108 0.568963623 0.726945557 0.326654174 (3)
0.400400291 0.326712525 0.315468018 0.317179199 0.329073852 0.347750000 0.383925293 0.434970703 0.569601929 0.726945557 0.315468018 (3)
Table 4 The area of 95% exact (or approximate) joint confidence region for (b, k) j
By using Theorem 3
By using Theorem 4
1 2 3 4 5 6 7 8 9 10 min(j)
0.040874777 0.036363783 0.009172863 0.014829086 0.016317729 0.022498179 0.020748075 0.032519189 0.023784073 0.094975565 0.009172863 (3)
0.042209719 0.035504938 0.009171929 0.014143445 0.015691044 0.021150061 0.020098366 0.030979240 0.025998578 0.094975565 0.009171929 (3)
8 < 0:352310547 < b < 0:725291016; 9:882428630 h i < k < hP 39:409864930 b b b b X 11 X : P11 X ðiÞ X ð10Þ ðiÞ ð10Þ 2
i¼1
e
þ4e
15
2
i¼1
e
þ4e
i:
15
Moreover, the area of the optimal 95% joint confidence region is 0.009172863. By using Theorem 2 and Criterion 2, we obtain that the interval length of 95% approximate confidence interval for b is also given in Table 3. From Table 3, we find that when j = 3, our proposed method obtains a shorter confidence interval length is 0.315468018. Moreover, we see that if j = 3, F 0:975 ð14:72; 6Þ ¼ 0:290876237
and
F 0:025 ð14:72; 6Þ ¼ 5:276299285;
then (0.380255859, 0.695723877) is an optimal 95% approximate confidence interval for the parameter b. In addition, from Table 4, we can see that if j = 3, F pffiffiffiffiffiffi ð14:72; 6Þ ¼ 0:243358406 and ð1þ 0:95Þ=2
F ð1pffiffiffiffiffiffi 0:95Þ=2 ð14:72; 6Þ ¼ 6:91305235;
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v2ð1þpffiffiffiffiffiffi ð20:72Þ ¼ 9:038252397 0:95Þ=2 v2ð1pffiffiffiffiffiffi ð20:72Þ 0:95Þ=2
and
¼ 37:682762022;
then by Theorem 4 and Criterion 4 an optimal 95% approximate joint confidence region for the parameters b and k is determined by the following inequalities: 8 < 0:243358406 < b < 6:91305235; 9:038252397 h i < k < hP b 37:682762022 i: b b X X X 3 : 2 P3 eX bðiÞ þ7:45eX bð11Þ þ4:55eX bð3Þ 15 e ðiÞ þ7:45e ð11Þ þ4:55e ð3Þ 15 2 i¼1
i¼1
Moreover, the area of the optimal 95% joint confidence region is 0.009171929.
5. Conclusions This paper was aimed at proposing different types of exact (or approximate) confidence intervals for the parameter b and exact (or approximate) joint confidence regions for the parameters b and k of a population distribution with c.d.f. as (1). In Tables 1–4, we find that our proposed method is better than ChenÕs method based on our optimal criteria. Furthermore, we can also use nj(b) and n0j ðbÞ to test null hypothesis H0: b = b0. In future, one can also easily extend the method to establish different types of exact (or approximate) confidence intervals for the parameter b and exact (or approximate) joint confidence regions for the parameters b and k by using the multiply type II censored sample where multiply type II censored sample supposes that first r, last s and middle l observations are censored and the only observations available are Xr + 1 < < Xr + k and Xr + k + l + 1 < < Xn s.
Acknowledgment This research was partially supported by the National Science Council, R.O.C. (Plan No. NSC 92-2118-M-032-010).
References [1] P.J. Bickel, K.A. Doksum, 2nd ed.Mathematical Statistics: Basic Ideas and Selected Topics, Vol. 1, Prentice-Hall, Inc., 2001. [2] Z. Chen, Statistical inference about the shape parameter of the exponential power distribution, Statistical Papers 40 (1999) 459–468. [3] Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics & Probability Letters 49 (2000) 155–161.
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[4] D.P. Gaver, M. Acar, Analytical hazard representations for use in reliability, mortality, and simulation studies, Communications in Statistics B 8 (1979) 91–111. [5] E. Haupt, H. Schabe, A new model for a lifetime distribution with bathtub shaped failure rate, Mircoelectronics & Reliability 32 (1992) 633–639. [6] U. Hjorth, A reliability distribution with increasing, decreasing, and bathtub-shaped failure rate, Technometrics 22 (1980) 99–107. [7] C.D. Lai, M. Xie, D.N.P. Murthy, Bathtub-shaped failure rate life distributions, in: Handbook of statistics, Advances in Reliability, Vol. 20, Elsevier, London, 2001, pp. 69–104. [8] L.M. Leemis, Lifetime distribution identities, IEEE Transactions on Reliability 35 (1986) 170– 174. [9] N.R. Mann, R.E. Schafer, N.D. Singpurwalla, Methods for Statistical Analysis of Reliability and Lifetime Data, Wiley, New York, 1974. [10] G.S. Mudholkar, D.K. Srivastava, Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Transactions on Reliability 42 (1993) 299–302. [11] J. Mi, Bathtub Failure Rate and Upside-Down Bathtub Mean Residual Life, IEEE Transactions on Reliability 44 (1995) 388–391. [12] M.B. Rajarshi, S. Rajarshi, Bathtub distributions: A review, Communications in Statistics: Theory and Method 17 (1988) 2597–2621. [13] R.M. Smith, L.J. Bain, An exponential power life-testing distribution, Communications in Statistics B 4 (1975) 469–481. [14] F.K. Wang, A new model with bathtub-shaped failure rate using an additive Burr XII distribution, Reliability Engineering and System Safety 70 (2000) 305–312. [15] M. Xie, C.D. Lai, Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function, Reliability Engineering and System Safety 52 (1995) 87–93. [16] M. Xie, Y. Tang, T.N. Goh, A modified Weibull extension with bathtub-shaped failure rate function, Reliability Engineering and System Safety 76 (2002) 279–285.
www.elsevier.com/locate/amc
Optimal parameter estimation of the two-parameter bathtub-shaped lifetime distribution based on a type II right censored sample Jong-Wuu Wu *, Chin-Chuan Wu, Mei-Huei Tsai Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan 25137, Republic of China
Abstract This paper proposes different types of exact (or approximate) confidence intervals and exact (or approximate) joint confidence regions for the parameters of the twoparameter lifetime distribution with bathtub-shaped or increasing failure rate (IFR) function based on a type II right censored sample. Moreover, we provide optimal criteria for finding a best exact (or approximate) confidence interval and a best exact (or approximate) joint confidence region among these estimations. Two examples are used to compare our proposed method to the existing approach of [Statistics & Probability Letters 49 (2000) 155]. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Bathtub-shaped; Failure rate function; Confidence interval; Joint confidence region; Censored sample
*
Corresponding author. E-mail address: [email protected] (J.-W. Wu).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.133
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1. Introduction The failure pattern of many products/systems (as electro-mechanical, electronic and mechanical products, etc) can be represented by the bathtub curve. It includes three phases: early failure (or burn-in) phase with a decreasing failure rate, normal use phase with an approximately constant failure rate, and wear-out phase with an increasing failure rate. In survival analysis, the lifetime of human beings exhibits this pattern. So, some probability distributions have been proposed to fit real life data with bathtub-shaped failure rates, such as Smith and Bain [13], Gaver and Acar [4], Hjorth [6], Leemis [8], Rajarshi and Rajarshi [12], Haupt and Schabe [5], Mudholkar and Srivastava [10], Mi [11], Xie and Lai [15], Chen [2,3], Wang [14], Lai et al. [7] and Xie et al. [16]. In this paper, we discuss that the two-parameter lifetime distribution with bathtub-shaped or increasing failure rate (IFR) function is proposed by Chen [3]. The cumulative distribution function (c.d.f.) of this distribution is xb
F ðxÞ ¼ 1 ekð1e Þ ;
ð1Þ
x > 0;
where k > 0 is the parameter but it does not affect the shape of failure rate function h(x) as in (2) and b > 0 is the shape parameter. The corresponding failure rate function of this distribution is b
hðxÞ ¼ kbxb1 ex ;
ð2Þ
x > 0:
Since b
h0 ðxÞ ¼ kbxb2 ex ðb 1 þ bxb Þ;
x > 0;
1=b
ð3Þ
1b b
. The h(x) has a bathtub shape when b < 1, achieving a minimum at x ¼ distribution has increasing failure rate function when b P 1. In Section 2, we extend the ideas of Mann et al. [9] and Chen [3] to develop different types of exact (or approximate) joint confidence regions for the parameters b and k of the two-parameter lifetime distribution with bathtub-shaped or increasing failure rate function based on a type II right censored sample. In Section 3, we provide an optimal criterion to find a best estimation according to the idea of the shortest tolerance length of an interval or minimal area of a region for the above censored sample. Section 4, we includes illustrative examples using both real failure data of electronic devices from Wang [14] (or Xie et al. [16]) and simulated data from Chen [3] to compare our proposed method and ChenÕs method and give a discussion of the results. Conclusions are given in Section 5. 2. Parameter estimation Let X(1), . . ., X(k) be the first k order statistics of a sample of size n from a population distribution with c.d.f. as in (1). It can be shown that
J.-W. Wu et al. / Appl. Math. Comput. 167 (2005) 807–819
809
b b X X Y ð1Þ ¼ k e ð1Þ 1 ; . . . ; Y ðkÞ ¼ k e ðkÞ 1 are the first k order statistics of a sample of size n from a standard exponential distribution. By the property of the standard exponential distribution, we know that these random variables Z1 = nY(1) and Z(i) = (n i + 1) (Y(i) Y(i1)), i = 2, . . ., k are independent and identically distributed with the standard exponential distribution (also see Bickel and Doksum [1, p. 528]). Therefore, we use Z1, . . ., Zk to define Uj and Vj as following: " # k k X X Uj ¼ 2 Zi ¼ 2 Y ðiÞ þ ðn kÞY ðkÞ þ ðj nÞY ðjÞ ð4Þ i¼jþ1
i¼jþ1
and Vj ¼2
j X
" Zi ¼ 2
i¼1
j X
# Y ðiÞ þ ðn jÞY ðjÞ ;
ð5Þ
i¼1
where j = 1, 2, . . ., k 1. Then Uj and Vj are two independent variables and have a v2 distribution with 2(k j) and 2j degrees of freedom, respectively. By using Uj and Vj, we can also define other variables such as hP i k j Y þ ðn kÞY ðn jÞY ðiÞ ðkÞ ðjÞ i¼jþ1 U j =½2ðk jÞ Pj ¼ nj ðbÞ ¼ V j =ð2jÞ ðk jÞ i¼1 Y ðiÞ þ ðn jÞY ðjÞ hP b b b i X ðiÞ X X k j 1 ðn jÞ e ðjÞ 1 þ ðn kÞ e ðkÞ 1 i¼jþ1 e hP b b i ¼ X ðiÞ X j ðk jÞ 1 þ ðn jÞ e ðjÞ 1 i¼1 e and
" 1j ðb; kÞ ¼ U j þ V j ¼ 2 ¼ 2k
" k X
k X
# Y ðiÞ þ ðn kÞY ðkÞ
i¼1
e
X bðiÞ
# b X ðkÞ 1 þ ðn kÞ e 1 ;
i¼1
where j = 1, 2, . . ., k 1. Then, it is easy to see that nj(b) has an F distribution with 2(k j) and 2j degrees of freedom and 1j(b, k) has a v2 distribution with 2k degrees of freedom for j = 1, 2, . . ., k 1. Furthermore, these two random variables nj(b) and 1j(b, k) are independent for each j. In addition, by using the idea of Mann et al. [9, p. 171], we also shown that U 0j ¼ 2mðY ðkÞ Y ðjÞ Þ=v is approximately v2 distribution with 2m2/v, where m ¼ Pk Pk 0 1 1 i¼jþ1 niþ1 and v ¼ i¼jþ1 ðniþ1Þ2 , j = 1, 2, . . ., k 1. Moreover, U j and Vj are independent since they involve different independent standard exponential
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variates. Furthermore, by using U 0j and Vj, we can also define other variables such as U 0j =ð2m2 =vÞ jðY ðkÞ Y ðjÞ Þ ¼ Pj V j =ð2jÞ m i¼1 Y ðiÞ þ ðn jÞY ðjÞ b X Xb j e ðkÞ e ðjÞ b i ¼ hP b X ðiÞ X j m 1 þ ðn jÞ e ðjÞ 1 i¼1 e
n0j ðbÞ ¼
and " 10j ðb; kÞ
¼
U 0j
þ V j ¼ 2mðY ðkÞ Y ðjÞ Þ=v þ 2
j X
# Y ðiÞ þ ðn jÞY ðjÞ
i¼1
¼ 2mk e
X bðkÞ
e
X bðjÞ
" # j b b X X ðiÞ X ðjÞ e 1 þ ðn jÞ e 1 ; =v þ 2k i¼1
where j = 1, 2, . . ., k 1. Then, it is easy to see that n0j ðbÞ has a approximately F distribution with 2m2/v and 2j degrees of freedom and 10j ðb; kÞ has a approximately v2 distribution with 2(m2/v + j) degrees of freedom for j = 1, 2, . . ., k 1. Furthermore, these two random variables n0j ðbÞ and 10j ðb; kÞ are independent for each j. Before we prove the following theorems, some necessary lemmas based on the above definitions are given. Lemma 1. For all j = 1, 2, . . ., k 1 and t > 0, the equation, in b > 0, nj(b) = t has a unique solution. Proof. Since the function nj(b) is strictly increasing in b > 0 with nj(0) = 0 and nj(1) = 1, then the lemma follows. h Lemma 2. For all j = 1, 2, . . ., k 1 and t > 0, the equation, in b > 0, n0j ðbÞ ¼ t has a unique solution. Proof. Since the function n0j ðbÞ is strictly increasing in b > 0 with n0j ð0Þ ¼ 0 and n0j ð1Þ ¼ 1, then the lemma follows. h Therefore, by using Lemmas 1 and 2, we can obtain the exact (or approximate) confidence interval of b and joint region of b and k as the following theorems. Note that Fa(v1, v2) is the upper a critical value of the F distribution with v1 and v2 degrees of freedom and v2a ðdÞ is the upper a critical value of the v2 distribution with d degrees of freedom in this paper.
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Theorem 1. Let X(1), . . ., X(k) be a type II right censored sample with the c.d.f. as (1), where 2 6 k 6 n. Then for any 0 < a < 1, 1 n1 j ðF a=2 ð2ðk jÞ; 2jÞÞ < b < nj ðF 1a=2 ð2ðk jÞ; 2jÞÞ
ð6Þ
is a 1 a confidence interval for the parameter b for j = 1, 2, . . ., k 1, where for t > 0, n1 j ðtÞ is the solution of b for the equation hP b b b i X ðiÞ X X k j 1 ðn jÞ e ðjÞ 1 þ ðn kÞ e ðkÞ 1 i¼jþ1 e hP b b i ¼ t: X ðiÞ X j ðk jÞ 1 þ ðn jÞ e ðjÞ 1 i¼1 s e Proof. By using Lemma 1, we have 1 P fn1 j ðF 1a=2 ð2ðk jÞ; 2jÞÞ < b < nj ðF a=2 ð2ðk jÞ; 2jÞÞg ¼ P F 1a=2 ð2ðk jÞ; 2jÞ
b b b i X ðiÞ X ðjÞ X ðkÞ e 1 ðn jÞ e 1 þ ðn kÞ e 1 i¼jþ1 hP b b i < X ðiÞ X j ðk jÞ 1 þ ðn jÞ e ðjÞ 1 i¼1 e < F a=2 ð2ðk jÞ; 2jÞ ¼ 1 a; j
hP k
where j = 1, 2, . . ., k 1. This completes the proof.
h
By using Lemma 2, the proof of the following theorem is similar to that of Theorem 1 and hence is omitted. Theorem 2. Let X(1), . . ., X(k) be a right censored sample with the c.d.f. as (1), where 2 6 k 6 n. Then for any 0 < a < 1, 01 2 2 n01 j ðF a=2 ð2m =v; 2jÞÞ < b < nj ðF 1a=2 ð2m =v; 2jÞÞ
ð7Þ
is a 1 a approximate confidence interval for the parameter b for j = 1, 2, . . ., k 1, where m and v as the above definition and for t > 0, n01 j ðtÞ is the solution of b for the equation b X Xb j e ðkÞ e ðjÞ hP b b i ¼ t: X ðiÞ X ðjÞ j m e 1 þ ðn jÞ e 1 i¼1 By the same way, we can also prove the exact (or approximate) joint confidence regions for b and k in the Theorems 3 and 4.
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Theorem 3. Let X(1), . . ., X(k) be defined in Theorem 1. Then for any 0 < a < 1, the following inequalities determine a 1 a joint confidence region for b and k with j = 1, 2, . . ., k 1: 8 1 1 pffiffiffiffiffiffi n ðF ð1þpffiffiffiffiffiffi > 1aÞ=2 ð2ðk jÞ; 2jÞÞ < b < nj ðF ð1 1aÞ=2 ð2ðk jÞ; 2jÞÞ; > < j 2 v pffiffiffiffiffi ð2kÞ v2 pffiffiffiffiffi ð2kÞ 1aÞ=2 1aÞ=2 hP bð1þ b i < k < hP bð1 b i ; > > X X X X k k : ðiÞ ðkÞ ðiÞ ðkÞ 2
i¼1
1 þðnkÞ e
e
1
2
1 þðnkÞ e
e
i¼1
1
where n1 j ðtÞ is defined in Theorem 1. Proof. By using Lemma 1, we have ( 1 pffiffiffiffiffiffi pffiffiffiffiffiffi P n1 j ðF ð1þ 1aÞ=2 ð2ðk jÞ;2jÞÞ < b < nj ðF ð1 1aÞ=2 ð2ðk jÞ;2jÞÞ;
¼P
8 < :
k
i¼1
ffiffiffiffiffi
p ð2kÞ ð1þ 1aÞ=2 b X e ðiÞ 1 þðnkÞ
hP
2
v2
e
X
F ð1þpffiffiffiffiffiffi 1aÞ=2 ð2ðk jÞ;2jÞ
nj ðF ð1þpffiffiffiffiffiffi 1aÞ=2 ð2m =v; 2jÞÞ < b < nj ðF ð1 1aÞ=2 ð2m =v; 2jÞÞ; > > > > 2 2 v pffiffiffiffiffi ð2ðm =vþjÞÞ > > 1aÞ=2 > h b ð1þ P b b i < k < b X X X X j ðkÞ ðjÞ ðiÞ ðjÞ 2 m e
e
=vþ
i¼1
e
1 þðnjÞ e
1
> > > v2 pffiffiffiffiffi ð2ðm2 =vþjÞÞ > > 1aÞ=2 > h b ð1 P b b i ; < > b > : 2 m eX ðkÞ eX ðjÞ =vþ j eX ðiÞ 1 þðnjÞ eX ðjÞ 1 i¼1
where m, v and
n01 j ðtÞ
are defined in Theorem 2.
Remark. When j = k 1, the results of Theorems 2 and 4 are the same as the results of Theorems 1 and 3. In addition, when j = 1, the result of Theorems 1 and 3 can be reduced to the result of Chen [3].
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813
3. Optimal criteria In Section 2, we obtain different kinds of exact (or approximate) interval estimations of the same parameter according to j changes. In order to find out a better estimation, we consider choosing a exact (or approximate) confidence interval with the shortest tolerance length or a exact (or approximate) joint confidence region with the minimal area to be the optimal interval or region. Based on this idea, we provide the optimal criteria as following criteria. Criterion 1. Let lj1 and uj1 be the lower and upper confidence limit defined in Eq. (6), respectively. Then confidence interval ðl1 ; u1 Þ is the optimal confidence interval for all j, if its tolerance length L satisfies L ¼
min ðuj1 lj1 Þ:
ð8Þ
j¼1;2;...;k1
Criterion 2. Let lj2 and uj2 be the lower and upper confidence limit defined in Eq. (7), respectively. Then approximate confidence interval ðl2 ; u2 Þ is the optimal approximate confidence interval for all j, if its tolerance length L** satisfies L ¼
min ðuj2 lj2 Þ:
ð9Þ
j¼1;2;...;k1
~j1 be the lower and upper confidence limit of b Criterion 3. Let ~lj1 and u defined in a joint confidence region of Theorem 3, respectively. Also klj1 ðbÞ and kuj1 ðbÞ be the lower and upper confidence limit of k defined as The u orem 3, respectively. Then confidence region ð~l1 ; ~u1 ; kl 1 ðbÞ; k1 ðbÞÞ is the opti mal confidence region for all j, if its area A satisfies Z ~uj1 Z kuj1 ðbÞ A ¼ min dk db: ð10Þ j¼1;2;...;k1
klj1 ðbÞ
~lj1
uj2 be the lower and upper confidence limit of b Criterion 4. Let ~lj2 and ~ defined in a joint confidence region of Theorem 4, respectively. Also klj2 ðbÞ and kuj2 ðbÞ be the lower and upper confidence limit of k defined as Theorem 4, respectively. Then approximate confidence region u ð~l2 ; ~ u2 ; kl 2 ðbÞ; k2 ðbÞÞ is the optimal approximate confidence region for all j, if its area A** satisfies
A ¼
Z
uj2 ~
Z
kuj2 ðbÞ
min
j¼1;2;...;k1
dk db: ~lj2
klj2 ðbÞ
ð11Þ
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4. Illustrative examples In this section, we apply the proposed methods to one of practical data set and another simulated data set. Example 1 considered is the failure data of n = k = 18 electronic devices from Wang [14] (or Xie et al. [16]). Example 2 considered is the first k = 11 observations of a computer generated sample of size n = 15 from a population distribution defined by (1) with parameter k = 0.02 and b = 0.5 (see Chen [3]). Example 1 (Real life data). The complete sample of the failure data of n = k = 18 electronic devices from Wang [14] (or Xie et al. [16]) is given as following: 5; 11; 21; 31; 46; 75; 98; 122; 145; 165; 195; 224; 245; 293; 321; 330; 350; 420: Xie et al. [16] have been shown that the data set has a bathtub-shaped failure rate function. According to their assumption, we apply ChenÕs method (that is, j = 1 in Theorem 1) to get an interval length of 95% confidence interval for b is 0.177057095 (see Table 1). By using Theorem 1 and Criterion 1, we obtain that the interval length of 95% confidence interval for b is given in Table 1. From Table 1, we find that when j = 7, our proposed method obtains a shorter confidence interval length is 0.119291259. Moreover, we see that if j = 7, F 0:975 ð22; 14Þ ¼ 0:395494284
and
F 0:025 ð22; 14Þ ¼ 2:813940631;
Table 1 The length of 95% exact (or approximate) confidence interval for b j
By using Theorem 1
By using Theorem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 min(j)
0.177057095 0.145143463 0.131952392 0.126130859 0.122527588 0.119466248 0.119291259 0.120534240 0.122850586 0.126876465 0.131615478 0.138036376 0.148391357 0.160359131 0.179460327 0.223914063 0.331802979 0.119291259 (7)
0.150146942 0.122560089 0.113409912 0.110536773 0.109812378 0.111204743 0.113756836 0.117913697 0.122458740 0.128150329 0.135126342 0.143895996 0.154786865 0.170855469 0.194511718 0.235051269 0.331802979 0.109812378 (5)
J.-W. Wu et al. / Appl. Math. Comput. 167 (2005) 807–819
815
then (0.210587891, 0.329879150) is an optimal 95% confidence interval for the parameter b. Furthermore, from this result, we find that the data set has a bathtub-shaped failure rate function. Hence, this agrees with the conclusion stated in Xie et al. [16]. In addition, from Table 2, we can see that if j = 14, F pffiffiffiffiffiffi ð8; 28Þ ¼ 0:205950946 and F pffiffiffiffiffiffi ð8; 28Þ ¼ 3:085177303; ð1þ 0:95Þ=2
ð1 0:95Þ=2
v2ð1þpffiffiffiffiffiffi ð36Þ ¼ 19:73449165 0:95Þ=2
and
v2ð1pffiffiffiffiffiffi ð36Þ ¼ 57:57500499; 0:95Þ=2
then by Theorem 3 and Criterion 3 an optimal 95% joint confidence region for the parameters b and k is determined by the following inequalities: 8 < 0:217148438 < b < 0:401312988; 57:57500499 i h 19:73449165 i < k < hP : b 18 X : 2 P18 eX bðiÞ 18 e ðiÞ 18 2 i¼1
i¼1
Moreover, the area of the optimal 95% joint confidence region is 0.002331714. By using Theorem 2 and Criterion 2, we obtain that the interval length of 95% approximate confidence interval for b is also given in Table 1. From Table 1, we find that when j = 5, our proposed method obtains a shorter confidence interval length is 0.109812378. Moreover, we see that if j = 5, F 0:975 ð10:40; 10Þ ¼ 0:245974153
and
F 0:025 ð10:40; 10Þ ¼ 3:309474589;
Table 2 The area of 95% exact (or approximate) joint confidence region for (b, k) j
By using Theorem 3
By using Theorem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 min(j)
0.004002693 0.004582333 0.004509095 0.004776620 0.004647857 0.003589845 0.003318413 0.003104469 0.003056782 0.003264490 0.003030044 0.002932669 0.003529089 0.002331714 0.002497938 0.008274441 0.041296306 0.002331714 (14)
0.004647363 0.005053846 0.004775387 0.004852702 0.004586928 0.003575137 0.003319062 0.003143474 0.003129565 0.003305460 0.003224458 0.003281763 0.003890330 0.003551379 0.004720985 0.011328405 0.041296306 0.003129565 (9)
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then (0.212167969, 0.321980347) is an optimal 95% approximate confidence interval for the parameter b. Furthermore, from this result, we also find that the data set has a bathtub-shaped failure rate function. Hence, this agrees with the conclusion stated in Xie et al. [16]. In addition, from Table 2, we can see that if j = 9, F pffiffiffiffiffiffi ð4:88; 18Þ ¼ 0:306392397 and ð1þ 0:95Þ=2
F ð1pffiffiffiffiffiffi 0:95Þ=2 ð4:88; 18Þ ¼ 3:58787930; ð22:88Þ ¼ 10:465426950 v2ð1þpffiffiffiffiffiffi 0:95Þ=2 v2ð1pffiffiffiffiffiffi ð22:88Þ 0:95Þ=2
and
¼ 40:58016026;
then by Theorem 4 and Criterion 4 an optimal 95% approximate joint confidence region for the parameters b and k is determined by the following inequalities: 8 < 0:209179688 < b < 0:350528687; 10:465426950 h i < k < hP b 40:58016026 i: b b X X X 9 : 2 P9 eX bðiÞ þ1:84eX bð18Þ þ7:16eX bð9Þ 18 e ðiÞ þ1:84e ð18Þ þ7:16e ð9Þ 18 2 i¼1
i¼1
Moreover, the area of the optimal 95% joint confidence region is 0.003129565. Example 2 (Simulated data). The following are the first k = 11 observations of a computer generated sample of size n = 15 from a population distribution defined by (1) with parameters k = 0.02 and b = 0.5 (also see Chen [3]): 0:29; 1:44; 8:38; 8:66; 10:20; 11:04; 13:44; 14:37; 17:05; 17:13 and 18:35: It was found by Chen [3] that (0.189628906, 0.621608498) is a 95% confidence interval with interval length 0.431979592 for the shape parameter b. By using Theorem 1 and Criterion 1, we obtain that the interval length of 95% confidence interval for b is given in Table 3. From Table 3, we find that when j = 3, our proposed method obtains a shorter confidence interval length is 0.326654174. Moreover, we see that if j = 3, F 0:975 ð16; 6Þ ¼ 0:29934465
and
F 0:025 ð16; 6Þ ¼ 5:243860453;
then (0.378259766, 0.704913940) is an optimal 95% confidence interval for the parameter b. In addition, from Table 4, we can see that if j = 3, F pffiffiffiffiffiffi ð16; 6Þ ¼ 0:251777713 and F pffiffiffiffiffiffi ð16; 6Þ ¼ 6:866053989; ð1þ 0:95Þ=2
v2ð1þpffiffiffiffiffiffi ð22Þ ¼ 9:882428630 0:95Þ=2
ð1 0:95Þ=2
and
v2ð1pffiffiffiffiffiffi ð22Þ ¼ 39:409864930; 0:95Þ=2
then by Theorem 3 and Criterion 3 an optimal 95% joint confidence region for the parameters b and k is determined by the following inequalities:
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817
Table 3 The length of 95% exact (or approximate) confidence interval for b j
By using Theorem 1
By using Theorem 2
1 2 3 4 5 6 7 8 9 10 min(j)
0.431979592 0.353967285 0.326654174 0.330638672 0.338140625 0.356469238 0.388075928 0.437386108 0.568963623 0.726945557 0.326654174 (3)
0.400400291 0.326712525 0.315468018 0.317179199 0.329073852 0.347750000 0.383925293 0.434970703 0.569601929 0.726945557 0.315468018 (3)
Table 4 The area of 95% exact (or approximate) joint confidence region for (b, k) j
By using Theorem 3
By using Theorem 4
1 2 3 4 5 6 7 8 9 10 min(j)
0.040874777 0.036363783 0.009172863 0.014829086 0.016317729 0.022498179 0.020748075 0.032519189 0.023784073 0.094975565 0.009172863 (3)
0.042209719 0.035504938 0.009171929 0.014143445 0.015691044 0.021150061 0.020098366 0.030979240 0.025998578 0.094975565 0.009171929 (3)
8 < 0:352310547 < b < 0:725291016; 9:882428630 h i < k < hP 39:409864930 b b b b X 11 X : P11 X ðiÞ X ð10Þ ðiÞ ð10Þ 2
i¼1
e
þ4e
15
2
i¼1
e
þ4e
i:
15
Moreover, the area of the optimal 95% joint confidence region is 0.009172863. By using Theorem 2 and Criterion 2, we obtain that the interval length of 95% approximate confidence interval for b is also given in Table 3. From Table 3, we find that when j = 3, our proposed method obtains a shorter confidence interval length is 0.315468018. Moreover, we see that if j = 3, F 0:975 ð14:72; 6Þ ¼ 0:290876237
and
F 0:025 ð14:72; 6Þ ¼ 5:276299285;
then (0.380255859, 0.695723877) is an optimal 95% approximate confidence interval for the parameter b. In addition, from Table 4, we can see that if j = 3, F pffiffiffiffiffiffi ð14:72; 6Þ ¼ 0:243358406 and ð1þ 0:95Þ=2
F ð1pffiffiffiffiffiffi 0:95Þ=2 ð14:72; 6Þ ¼ 6:91305235;
818
J.-W. Wu et al. / Appl. Math. Comput. 167 (2005) 807–819
v2ð1þpffiffiffiffiffiffi ð20:72Þ ¼ 9:038252397 0:95Þ=2 v2ð1pffiffiffiffiffiffi ð20:72Þ 0:95Þ=2
and
¼ 37:682762022;
then by Theorem 4 and Criterion 4 an optimal 95% approximate joint confidence region for the parameters b and k is determined by the following inequalities: 8 < 0:243358406 < b < 6:91305235; 9:038252397 h i < k < hP b 37:682762022 i: b b X X X 3 : 2 P3 eX bðiÞ þ7:45eX bð11Þ þ4:55eX bð3Þ 15 e ðiÞ þ7:45e ð11Þ þ4:55e ð3Þ 15 2 i¼1
i¼1
Moreover, the area of the optimal 95% joint confidence region is 0.009171929.
5. Conclusions This paper was aimed at proposing different types of exact (or approximate) confidence intervals for the parameter b and exact (or approximate) joint confidence regions for the parameters b and k of a population distribution with c.d.f. as (1). In Tables 1–4, we find that our proposed method is better than ChenÕs method based on our optimal criteria. Furthermore, we can also use nj(b) and n0j ðbÞ to test null hypothesis H0: b = b0. In future, one can also easily extend the method to establish different types of exact (or approximate) confidence intervals for the parameter b and exact (or approximate) joint confidence regions for the parameters b and k by using the multiply type II censored sample where multiply type II censored sample supposes that first r, last s and middle l observations are censored and the only observations available are Xr + 1 < < Xr + k and Xr + k + l + 1 < < Xn s.
Acknowledgment This research was partially supported by the National Science Council, R.O.C. (Plan No. NSC 92-2118-M-032-010).
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