Qual Quant (2009) 43:291–304 DOI 10.1007/s11135-007-9110-6 ORIGINAL PAPER
Implementing lifetime performance index for the pareto lifetime businesses of the service industries Ching-Wen Hong · Jong-Wuu Wu · Ching-Hsue Cheng
Received: 15 March 2006 / Accepted: 15 February 2007 / Published online: 30 May 2007 © Springer Science + Business Media B.V. 2007
Abstract Process capability analysis is an effective means of measuring process performance and potential capability. In the service industries, process capability indices (PCIs) are utilized to assess whether business quality meets the required level. Hence, the performance index C L is used as a means of measuring business performance, where L is the lower specification limit. In the technology of data transformation, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of C L based on the right type II censored sample from the pareto distribution. The UMVUE of C L is then utilized to develop a novel hypothesis testing procedure in the condition of known L. Finally, we give one practical example and the Monte Carlo simulation to assess the behavior of this test statistic for testing null hypothesis under given significance level. Moreover, the managers can then employ the new testing procedure to determine whether the business performance adheres to the required level. Keywords Business performance · Data transformation · Right type II censored sample · Pareto distribution · Uniformly minimum variance unbiased estimator · Monte Carlo simulation 1 Introduction Effectively managing and measuring the business operational process is widely seen as a means of ensuring business survival through reduced time to market, increased quality and reduced costs. Process capability analysis is an effective means of measuring process performance and potential capability. In the manufacturing industry, process capability indices are
C.-W. Hong · C.-H. Cheng Department of Information Management, National Yunlin University of Science and Technology, Yunlin 64002, Taiwan, R.O.C. J.-W. Wu (B) Department of Applied Mathematics, National Chiayi University, 300 Syuefu RD., Chiayi City 60004, Taiwan, R.O.C. e-mail:
[email protected]
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utilized to assess whether product quality meets the required level. For instance, Montgomery (1985) or Kane (1986) proposed the process capability index C L (or C P L ) for evaluating the lifetime performance of electronic components, where L is the lower specification limit, since the lifetime of electronic components exhibits the larger-the-better quality characteristic of time orientation. Tong et al. (2002) constructed a uniformly minimum variance unbiased estimator (UMVUE) of C L based on the complete sample from an exponential distribution. Moreover, the UMVUE of C L is then utilized to develop the hypothesis testing procedure. The purchasers can then employ the testing procedure to determine whether the lifetime of electronic components adheres to the required level. Manufacturers can also utilize this procedure to enhance process capability. In the service industries, process capability indices are also utilized to assess whether business quality meets the required level. Venkatraman and Ramanujam (1986) argued that business performance is a subset of the overall concept of organizational effectiveness. These researchers view the domain of business performance at three levels: financial, operational, and organizational effectiveness. “Performance measurement” is a very complex and often misunderstood term. This study evaluates business performance with business survival time. Hence, in this paper, the lifetime performance index C L is also utilized to measure business performance for lifetime of business with the pareto distribution. The pareto distribution was introduced by Pareto (1987) for the distribution of income. It has been found to have a wide variety of applications. The pareto distribution has played an important role in the investigations of city population sizes, stock prices fluctuations, income distributions, insurance risk, business failures, etc. Recently, it has also been recognized as a useful model for the analysis of lifetime data. In life testing experiments, the experimenter may not always be in a position to observe the life times of all the businesses (or items) put on test. This may be because of time limitation and/or other restrictions (such as money, material resources, mechanical or experimental difficulties, etc.) on data collection. Therefore, censored samples may arise in practice. In this paper, we consider the case of the right type II censoring. For the right type II censoring, only the first r lifetimes have been observed and the lifetimes for the remaining (n − r ) components are unobserved or missing. So, the main aim of this research will apply data transformation technology to construct the new hypothesis testing procedure with UMVUE of C L based on the right type II censored sample. The new testing procedure can be employed by managers to assess whether the business performance adheres to the required level in the condition of known L. The rest of this paper is organized as follows. Section 2, we introduce some properties of the lifetime performance index for lifetime of business with the pareto distribution based on the right type II censored sample. Moreover, we also discusses the relationship between the lifetime performance index and conforming rate. Section 3 then presents the UMVUE of the lifetime performance index and its statistical properties. Section 4 develops a new hypothesis testing procedure for the lifetime performance index. Section 5 discusses a Monte Carlo simulation of power function and gives a practical example to illustrate the use of the testing procedure. Concluding remarks are made in Sect. 6.
2 The lifetime performance index and the conforming rate Suppose that the lifetime (in years) of businesses may be modeled by a Pareto distribution. Let years denote the lifetime of such a business and X has the pareto distribution with the probability density function (p.d.f.) is
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f X (x; θ ) = θ x −(θ +1) , x ≥ 1, θ > 0.
293
(1)
By the transformation Y = ln(X ), and the distribution of Y is a exponential distribution, the p.d.f. of Y is f Y (y; θ ) = θ e−θ y ,
y > 0, θ > 0.
(2)
Clearly, a longer lifetime implies a better business performance. Hence, the lifetime is a larger-the-better type quality characteristic. The lifetime is generally required to exceed L years to both be economically profitable and investors. Montgomery (1985) developed a capability index C L for properly measuring the larger-the-better quality characteristic. C L is defined as follows: CL =
µ−L , σ
(3)
where the process mean µ, the process standard deviation σ , and L is the lower specification limit. To assess the business performance of businesses, C L can be defines as the lifetime performance index. UnderX has the pareto distribution and the logarithmic transformation Y = ln(X ), the distribution of Y is a exponential distribution. Hence, the p.d.f. of Y is f Y (y; θ ) = θ e−θ y , y > 0, θ > 0. Moreover, there are several important properties, as follows: •
The lifetime performance index C L can be rewritten as: CL =
•
µ−L 1/θ − L = = 1 − θ L , C L < 1, σ 1/θ
where√ the process mean µ = E(Y ) = 1/θ , the process standard deviation σ = VAR(Y ) = 1/θ , and L is the lower specification limit. The cumulative distribution function of Y is given by: FY (t) = 1 − e−θ t , t > 0.
•
(4)
(5)
The failure rate function r (t) is defined by: r (t) =
f Y (t) θ e−θ t = = θ, θ > 0. 1 − FY (t) 1 − (1 − e−θ t )
(6)
The important properties can be determined by using logarithmically transformed data Y = ln(X ). Since, the logarithmic transformation Y = ln(X ) is one-to-one and strictly increasing, so data set of X and transformed data set of Y have the same effect in assessing the business performance of businesses. Moreover, the logarithmic transformation Y = ln(X ) enables the calculation of important properties to be easy. When the mean 1/θ (> L), then the lifetime performance index C L > 0. From Eqs. 4 and 6, we can see that the larger the mean 1/θ , the smaller the failure rate and the lager the lifetime performance index C L . Therefore, the lifetime performance index C L reasonably and accurately represents the business performance of new businesses. Moreover, if the lifetime of a business X which Y = ln(X ) exceeds the lower specification limit L, then the business is defined as a conforming business. The ratio of conforming businesses is known as the conforming rate, and can be defined as: ∞ Pr = P(Y ≥ L) = θ e−θ y dy = e−θ L = eC L −1 , −∞ < C L < 1. (7) L
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Table 1 The lifetime performance index C L versus the conforming rate Pr CL
Pr
CL
Pr
CL
Pr
−∞
0.00000
0.125
0.41686
0.575
0.65377
−3.00
0.01832
0.150
0.42741
0.600
0.67032
−2.75
0.02352
0.175
0.43823
0.625
0.68729
−2.50
0.03020
0.200
0.44933
0.650
0.70469
−2.25
0.03877
0.225
0.46070
0.675
0.72253
−2.00
0.04979
0.250
0.47237
0.700
0.74082
−1.75
0.06393
0.275
0.48432
0.725
0.75957
−1.50
0.08209
0.300
0.49659
0.750
0.77880
−1.25
0.10540
0.325
0.50916
0.775
0.79852
−1.00
0.13534
0.350
0.52205
0.800
0.81873
−0.75
0.17377
0.375
0.53526
0.825
0.83946
−0.50
0.22313
0.400
0.54881
0.850
0.86071
−0.25 0.000
0.28650
0.425
0.56270
0.875
0.88250
0.36788
0.450
0.57695
0.900
0.90484
0.025
0.37719
0.475
0.59156
0.925
0.92774
0.050
0.38674
0.500
0.60653
0.950
0.95123
0.075
0.39657
0.525
0.62189
0.975
0.97531
0.100
0.40657
0.550
0.63763
1.000
1.00000
Obviously, a strictly increasing relationship exists between conforming rate Pr and the lifetime performance index C L . Table 1 lists various C L values and the corresponding conforming rates Pr . For the C L values which are not listed in Table 1, the conforming rate Pr can be obtained through interpolation. The conforming rate can be calculated by dividing the number of conforming businesses by the total number of businesses sampled. To accurately estimate the conforming rate, Montgomery (1985) suggested that the sample size must be large. However, a large sample size is usually not practical from the perspective of cost, since collecting the lifetime data of new businesses need many monies. In addition, a complete sample is also not practical due to time limitation and/or other restrictions (such as material resources, mechanical or experimental difficulties, etc.) on data collection. Since a one-to-one mathematical relationship exists between the conforming rate Pr and the lifetime performance index C L . Therefore, utilizing the one-to-one relationship between Pr and C L , lifetime performance index can be a flexible and effective tool, not only evaluating business performance, but also for estimating the conforming rate Pr .
3 UMVUE of lifetime performance index In lifetime testing experiments of businesses, the experimenter may not always be in a position to observe the lifetimes of all the items on test due to time limitation and/or other restrictions (such as money, material resources, mechanical or experimental difficulties, etc.) on data collection. Therefore, censored samples may arise in practice. In this paper, we consider the case of the right type II censoring.
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295
With type II censoring, the value of r is chosen before the data are collected, and the data consist of the r smallest lifetimes in a random samples X 1 , X 2 , . . . , X n from the pareto distribution with p.d.f. f X (x; θ ) = θ x −(θ +1) , x ≥ 1, θ > 0, and X (1) < X (2) < · · · < X (r ) are the corresponding right type II censored sample. Since the joint p.d.f. of X (1) , X (2) , . . . , X (r ) −(θ +1) r −θ n−r n! (x(r is (n−r )! , r ≤ n, so the joint p.d.f. can be rewritten as i=1 θ x (i) )) (8) P(x, θ ) = {exp[c(θ )W (x) + d(θ ) + S(x)]}I A (x), n! where c(θ ) = −θ , W (x) = ri=1 ln x(i) +(n −r ) ln x(r ) , d(θ ) = r ln θ , S(x) = ln (n−r )! − r r ln x , A = [1, ∞), x = (x , x , . . . , x ). Hence, the joint p.d.f. belongs (i) (1) (2) (r ) i=1 i=1 to the one parameter exponential family of distributions. By using Theorem 5.6. of Lehmann (1983) (also see Hogg et al. 2005, p. 391), W (X ) = ri=1 ln X (i) + (n − r ) ln X (r ) is a complete and sufficient statistic for θ . In addition, by using Theorem 4.1.1. and Cor2 . Moreover, we also ollary 4.1.1 of Lawless (2003) we also obtain that 2θ W (X ) ∼ χ(2r ) (r −1)L show that the estimator Cˆ L = 1 − r is an unbiased estimator of C L , where i=1
Y(i) +(n−r )Y(r)
Y(i) = ln X (i) , i = 1, 2, . . . , r . Further, by using Theorem of Lehmann and Scheff’e (1950) ˆ (also see Hogg et al. 2005, p. 387), the estimator r C L is a UMVUE of C L . 2 Let Z = Cˆ L = 1 − (r −1)L , where W = i=1 Y(i) + (n − r )Y(r ) , U = 2θ W ∼ χ(2r ) . W r−1 −u/2 The p.d.f. of U is f (u) = u e , u > 0. By using the change of variables U = 2θ (r −1)L U
and the Jacobian is ddUZ =
(r )2r 2θ (r −1)L , (1−Z )2
1−Z
we obtain that the p.d.f. of Z is given by θ (r−1)L
θ r (r − 1)r L r e− 1−z f Z (z) = , z < 1, θ > 0. (1 − z)r +1 (r )
(9)
Meanwhile, the kth moment of Cˆ L is (r − 1)L k ] W 2θ (r − 1)L k 2 = E[1 − ] , where 2θ W ∼ χ(2r ), 2θ W k (r − i) = Cik (−1)i [2θ (r − 1)L]i E(2θ W )−i , where E(2θ W )−i = , (r )2i
E(Cˆ L )k = E[1 −
i=0
k Cik (−1)i [(r − 1)L]i θ i (r − i) . = (r )
(10)
i=0
By the kth moment of Cˆ L , the expectation value and variance of Cˆ L can be obtained as: E Cˆ L =
1 Ci1 (−1)i [(r − 1)L]i θ i (r − i) (r ) i=0
= 1 − θL
(11)
and Var(Cˆ L ) = E(Cˆ L )2 − (E Cˆ L )2 θ2 L2 = , r > 2. r −2
(12)
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4 Testing procedure for the lifetime performance index Construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. Assuming that the required index value of lifetime performance is larger than c, where c denotes the target value, the null hypothesis H0 : C L ≤ c and the alternative hypothesis H1 : C L > c are constructed. By using Cˆ L , the
UMVUE of C L as ˆ the test statistic, the rejection region can be expressed as C L > C0 . Given the specified significance level α, the critical value C0 can be calculated as follows: P(Cˆ L > C0 |C L = c) = α, 1−c (r − 1)L > C0 θ = ) = α, ⇒ P(1 − W L 2(r − 1)Lθ 1−c ⇒ P(2θ W > |θ = ) = α, 1 − C0 L 2(r − 1)Lθ 1−c ) = 1 − α, ⇒ P(2θ W ≤ |θ = 1 − C0 L 2(r − 1)(1 − c) ⇒ P(2θ W ≤ ) = 1 − α, 1 − C0
(13)
2 . From Eq. 13, utilizing CHIINV(1 − α, 2r ) function which represents where 2θ W ∼ χ(2r ) 2 , then the lower 1-α percentile of χ(2r )
2(r − 1)(1 − c) = CHIINV(1 − α, 2r ) 1 − C0 is obtained. Thus, the following critical value can be derived: C0 = 1 −
2(r − 1)(1 − c) , CHIINV(1 − α, 2r )
(14)
where c, α, and r denote the target value, the specified significance level and the observed number, respectively. Moreover, we also find that C0 is independent of n. Tables 4 and 5 list the critical values C0 for r = 2(1)60 and c = 0.1(0.1)0.9 at α = 0.01 and α = 0.05 (see Appendix). When the 2r degrees of freedom is rather large, one may obtain approximations to the CHIINV(1 − α, 2r ) function, as 3 1 1 −1 ∼ CHIINV(1 − α, 2r ) = 2r (1 − α) + 1 − , 9r 9r where −1 (1 − α) is the lower 1-α percentage point of the standard normal distribution (see Johnson et al. 1994). The critical values C0 can be rewritten as C0 = 1 −
2r
2(r − 1)(1 − c) 1 −1 1 9r (1 − α) + 1 − 9r
3 , where r > 60.
(15)
The proposed testing procedure about C L can be organized as follows: Step 1 Let the transformation of Y(i) = ln X (i) , i = 1, 2, . . . , r for the right type II censored sample X (1) , X (2) , . . . , X (r ) .
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297
Step 2 Determine the lower lifetime limit L for businesses and performance index value c, then the testing null hypothesis H0 : C L ≤ c and the alternative hypothesis H1 : C L > c is constructed. Step 3 Specify a significance level α. Step 4 Calculate the value of test statistic Cˆ L = 1 − r (r −1)L . i=1
Y(i) +(n−r )Y(r)
Step 5 Obtain the critical value C0 from Tables 4 or 5, according to the target value c, the observed number r and the significance level α. Step 6 The decision rule of statistical test is provided as follows: If Cˆ L > C0 , it is concluded that the lifetime performance index of business operation meets the required level. Based on the proposed testing procedure, the business performance of businesses is easy to assess. One practical example of the proposed testing procedure is given in the Sect. 6, and this numerical example illustrates the use of the testing procedure.
5 The Monte Carlo simulation of power function and a practical example First, the power of a statistical test is the probability of correctly rejecting a false null hypothesis. The procedure used in practice is to limit the probability of type I error to some preassigned level α (usually 0.01 or 0.05) that is small and to maximize the power of a statistical test. The null hypothesis H0 : C L ≤ c and the alternative hypothesis H1 : C L > c are constructed. The power of a statistical test is derived as follows: Under type II censoring,
we get a size α test with the rejection region 2(r −1)(1−c) ˆ C L > 1 − CHIINV(1−α,2r ) , for the observed number r and sample size n (r ≤ n). The power P(c1 ) of the test at this point C L = c1 (> c) is then 2(r − 1)(1 − c) P(c1 ) = P Cˆ L > 1 − |C L = c1 CHIINV(1 − α, 2r ) (r − 1)L 2(r − 1)(1 − c) 1 − c1 = P 1− >1− |θ = W CHIINV(1 − α, 2r ) L 1 − c1 CHIINV(1 − α, 2r )θ L |θ = = P 2θ W > 1−c L (1 − c1 )CHIINV(1 − α, 2r ) , (16) = P 2θ W > 1−c 2 . where 2θ W ∼ χ(2r ) In addition, we will also report the results of a Monte Carlo simulation for the power P(c1 ). We considered α = 0.05 and (n,r ) = (10, 5), (10, 10), (20, 15), and (20, 20), c = 0.2, c1 = 0.3(0.1)0.9, L = 1.5. For each case, we estimate a power via the following Steps 1–2:
Step 1 Given c, c1 , L , α, r , and n, where c < c1 < 1 and r ≤ n. The value of θ is calculated by the equation C L = 1 − θ L = c1 , C L < 1. The generation of random sample data Y1 , Y2 , . . . , Yn is by the uniform distribution U (0, 1). (d) By the transformation of X i = (1 − Yi )−1/θ , i = 1, . . . , n, (X 1 , X 2 , . . . , X n ) is a random sample from the pareto distribution with p.d.f. f X (x; θ ) = θ x −(θ +1) , x ≥ 1, θ > 0.
(a) (b) (c)
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C.-W. Hong et al. 1 0.9 0.8 0.7
power
0.6 0.5 0.4 0.3 0.2
The power of n=10 r=5 The simulaion power of n=10 r=5 The power of n=10 r=10 The simulaion power of n=10 r=10
0.1 0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c1
Fig. 1 Power function of the significance level 0.05 test
The ranking of a random sample is X (1) < X (2) < . . . < X (n) . And only the first r observations X (1) < X (2) < . . . < X (r ) are used. . (f) The value of Cˆ L is calculated by Cˆ L = 1 − r ln X(r −1)L +(n−r ) ln X
(e)
i=1
(g)
(i)
(r)
2(r −1)(1−c) If Cˆ L > C0 then Count = 1, else Count = 0, where C0 = 1 − CHIINV . (1−α,2r )
Step 2 (a) The Step 1 is repeated with 1,000 times. ˆ 1 ) = Total count . (b) The estimation of power P(c1 ) is P(c 1,000 Based on 100 estimations of power P(c1 ), say Pˆ1 (c1 ), Pˆ2 (c1 ), . . . , Pˆ100 (c1 ), we com100 Pˆi (c1 ) i=1 ˆ 1) = pute the simulation power P(c and the sample mean square error (SMSE) 100 100
( Pˆi (c1 )−P(c1 ))2 , 100
where P(c1 ) can be calculated by (16). ˆ 1 ) with P(c1 ) are showed The results of power simulation for comparison between P(c in the Figs. 1–2 and Tables 2–3 for given (n, r ) = (10, 5), (10, 10), (20, 15), and (20, 20), c = 0.2, c1 = 0.3(0.1)0.9, L = 1.5, α = 0.05. The results of simulation. Hence, from Figs. 1–2 and based on c = 0.2 and α = 0.05, it appears clear that (a) for fixed c1 and n, as the ˆ 1 ) and power P(c1 ) increase except observed number r increases the simulation power P(c SMSE =
i=1
ˆ 1 ) and power P(c1 ) increase when c1 is c1 = 0.2; (b) for fixed n, the simulation power P(c ˆ 1 ) and power P(c1 ) for (n, r ) = (20, 20) increases; (c) for fixed c1 , the simulation power P(c (or (20, 15)) have higher values than (n,r ) = (10, 10) (or (10, 5)). From Tables 2–3, we find ˆ 1 ) close to power P(c1 ) for any value of c1 ; (e) all of that (d) all of the simulation power P(c the SMSE are enough small and the scope of SMSE is between 0.00000 and 0.00033. Second, we propose the new hypothesis testing procedure to a practical data set. Example considered is the failure data of n = 15, r = 10 businesses from Nigm and Hamdy (1987) or Wong (1998). Example The censored sample of the failure data of n = 15, r = 10 businesses from Nigm and Hamdy (1987) or Wong (1998) is given as following: 1.01, 1.05, 1.08, 1.14, 1.28, 1.30, 1.33, 1.43, 1.59, 1.62.
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Implementing Index for the Pareto Lifetime Businesses of the Service Industries
299
1 0.9 0.8 0.7
power
0.6 0.5 0.4 0.3 The power of n=20 r=15 The simulaion power of n=20 r=15 The power of n=20 r=20 The simulaion power of n=20 r=20
0.2 0.1 0 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c1
Fig. 2 Power function of the significance level 0.05 test ˆ 1 ), and SMSE for the different n and r Table 2 The values of P(c1 ), P(c n = 10 r = 5 α = 0.05
n = 10 r = 10 α = 0.05
c1
P(c1 )
ˆ 1) P(c
SMSE
P(c1 )
ˆ 1) P(c
SMSE
0.2
0.05000
0.04955
0.00004
0.05000
0.04999
0.00005
0.3
0.09910
0.09851
0.00011
0.12218
0.12167
0.00007
0.4
0.18565
0.18477
0.00013
0.26224
0.26336
0.00015
0.5
0.32414
0.32397
0.00025
0.48118
0.48163
0.00027
0.6
0.51760
0.51502
0.00033
0.73474
0.73375
0.00025
0.7
0.73811
0.73835
0.00021
0.92348
0.92303
0.00009
0.8
0.91760
0.91799
0.00006
0.99280
0.99262
0.00001
0.9
0.99361
0.99375
0.00001
0.99996
0.99998
0.00000
ˆ 1 ), and SMSE for the different n and r Table 3 The values of P(c1 ), P(c n = 20 r = 15 α = 0.05
n = 20 r = 20 α = 0.05
c1
P(c1 )
ˆ 1) P(c
SMSE
P(c1 )
ˆ 1) P(c
SMSE
0.2
0.05000
0.05037
0.00004
0.05000
0.05001
0.00005
0.3
0.14216
0.14416
0.00010
0.16060
0.16372
0.00012
0.4
0.32998
0.33157
0.00023
0.39177
0.39221
0.00029
0.5
0.60441
0.60395
0.00025
0.70097
0.69983
0.00025
0.6
0.85814
0.85792
0.00011
0.92594
0.92617
0.00007
0.7
0.97895
0.97948
0.00002
0.99446
0.99428
0.00001
0.8
0.99943
0.99943
0.00000
0.99996
0.99995
0.00000
0.9
1.00000
1.00000
0.00000
1.00000
1.00000
0.00000
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C.-W. Hong et al.
Moreover, the data denote the length of time in years for which a business operates until failure. Wong (1998) have been shown that the data set has a Pareto distribution with p.d.f. as (1) based on confidence interval. In addition, we can also show that the data set has a Pareto distribution with p.d.f. as (1) by using least squares method (see Lawless 1982) and the goodness of fit test (see Pettitt and Stephens 1976; D’Agostino and Stephens (1986). Next, we also state the proposed testing procedure about C L as following: Step 1 The operational lifetimes data of businesses{X (i) ,i = 1, 2, . . . , 10} = {1.01, 1.05, 1.08, 1.14, 1.28, 1.30, 1.33, 1.43, 1.59, 1.62} and we take the transformation of Y(i) = ln X (i) , i = 1, 2, . . . , 10. Step 2 The lower lifetime limit L is assumed to be 0.053. To deal with the business managers’ concerns regarding operational performance, the conforming rate Pr of operational performances is required to exceed 80%. Referring to Table 1, the C L value operational performances is required to exceed 0.80. Thus, the performance index value is set at c = 0.80. The testing hypothesis H0 : C L ≤ 0.80 v.s. H1 : C L > 0.80 is constructed. Step 3 Specify a significance level α = 0.05. Step 4 Calculate the value of test statistic Cˆ L = 1 −
(10 − 1) × 0.053 = 0.900. 2.365 + (15 − 10) × 0.482
Step 5 Obtain the critical value C0 = 0.885 from Table 5, according to c = 0.80, r = 10, and the significance level α = 0.05. Step 6 Because of Cˆ L = 0.900 > C0 = 0.885, so we reject to the null hypothesis H0 : C L ≤ 0.80. Thus, we can conclude that the lifetime performance index of business operation does meet the required level.
6 Concluding remarks Process capability indices are widely employed by manufacturers to assess the performance and potential of their processes. Moreover, in life testing experiments, the experimenter may not always be in a position to observe the life times of all the businesses (or items) put on test. This may be because of time limitation and/or other restrictions (such as money, material resources, mechanical or experimental difficulties, etc.) on data collection. Therefore, censored samples may arise in practice. So, we consider the case of the right type II censoring, and in the technology of data transformation, our study constructs a UMVUE of C L , and develops a new testing procedure for the performance index of businesses with the pareto distribution based on the right type II censored sample. The proposed testing procedure is easily applied and can effectively evaluate whether the business performance meets requirements. In addition, this study provides a table of the lifetime performance index with its corresponding conforming rate. Hence, for any specified conforming rate, a corresponding C L can be obtained, and the hypotheses of the proposed testing procedure can also be expressed in terms of the conforming rate under L is known limit. Acknowledgements We thank the editor for his helpful comments, which has led to a substantial improvement in the paper. This research was partially supported by the National Science Council, Republic Of China (Plan No. NSC 95 - 2118 - M - 415 – 001).
123
Implementing Index for the Pareto Lifetime Businesses of the Service Industries
301
Appendix Table 4 Critical value C0 for r = 2(1)60 and c = 0.1(0.1)0.9 at α = 0.01 r
c = 0.1
c = 0.2
c = 0.3
c = 0.4
c = 0.5
c = 0.6
c = 0.7
c = 0.8
c = 0.9
2
0.864
0.879
0.895
0.910
0.925
0.940
0.955
0.970
0.985
3
0.786
0.810
0.833
0.857
0.881
0.905
0.929
0.952
0.976
4
0.731
0.761
0.791
0.821
0.851
0.881
0.910
0.940
0.970
5
0.690
0.724
0.759
0.793
0.828
0.862
0.897
0.931
0.966 0.962
6
0.657
0.695
0.733
0.771
0.809
0.847
0.886
0.924
7
0.629
0.671
0.712
0.753
0.794
0.835
0.876
0.918
0.959
8
0.606
0.650
0.694
0.737
0.781
0.825
0.869
0.912
0.956
9
0.586
0.632
0.678
0.724
0.770
0.816
0.862
0.908
0.954
10
0.569
0.617
0.665
0.713
0.760
0.808
0.856
0.904
0.952 0.950
11
0.553
0.603
0.653
0.702
0.752
0.801
0.851
0.901
12
0.539
0.591
0.642
0.693
0.744
0.795
0.846
0.898
0.949
13
0.527
0.579
0.632
0.684
0.737
0.790
0.842
0.895
0.947
14
0.515
0.569
0.623
0.677
0.731
0.785
0.838
0.892
0.946
15
0.505
0.560
0.615
0.670
0.725
0.780
0.835
0.890
0.945
16
0.495
0.551
0.607
0.663
0.720
0.776
0.832
0.888
0.944
17
0.486
0.543
0.600
0.658
0.715
0.772
0.829
0.886
0.943
18
0.478
0.536
0.594
0.652
0.710
0.768
0.826
0.884
0.942
19
0.470
0.529
0.588
0.647
0.706
0.765
0.823
0.882
0.941
20
0.463
0.523
0.582
0.642
0.702
0.761
0.821
0.881
0.940
21
0.456
0.517
0.577
0.637
0.698
0.758
0.819
0.879
0.940 0.939
22
0.450
0.511
0.572
0.633
0.694
0.755
0.817
0.878
23
0.444
0.506
0.567
0.629
0.691
0.753
0.815
0.876
0.938
24
0.438
0.501
0.563
0.625
0.688
0.750
0.813
0.875
0.938
25
0.433
0.496
0.559
0.622
0.685
0.748
0.811
0.874
0.937
26
0.428
0.491
0.555
0.618
0.682
0.746
0.809
0.873
0.936
27
0.423
0.487
0.551
0.615
0.679
0.743
0.808
0.872
0.936
28
0.418
0.483
0.547
0.612
0.677
0.741
0.806
0.871
0.935
29
0.414
0.479
0.544
0.609
0.674
0.739
0.805
0.870
0.935
30
0.409
0.475
0.541
0.606
0.672
0.737
0.803
0.869
0.934
31
0.405
0.471
0.537
0.604
0.670
0.736
0.802
0.868
0.934
32
0.401
0.468
0.534
0.601
0.667
0.734
0.800
0.867
0.933
33
0.398
0.465
0.532
0.598
0.665
0.732
0.799
0.866
0.933
34
0.394
0.461
0.529
0.596
0.663
0.731
0.798
0.865
0.933
35
0.391
0.458
0.526
0.594
0.661
0.729
0.797
0.865
0.932
36
0.387
0.455
0.523
0.592
0.660
0.728
0.796
0.864
0.932
37
0.384
0.452
0.521
0.589
0.658
0.726
0.795
0.863
0.932 0.931
38
0.381
0.450
0.519
0.587
0.656
0.725
0.794
0.862
39
0.378
0.447
0.516
0.585
0.654
0.724
0.793
0.862
0.931
40
0.375
0.444
0.514
0.583
0.653
0.722
0.792
0.861
0.931
41
0.372
0.442
0.512
0.581
0.651
0.721
0.791
0.860
0.930
123
302
C.-W. Hong et al.
Table 4 continued r
c = 0.1
c = 0.2
c = 0.3
c = 0.4
c = 0.5
c = 0.6
c = 0.7
c = 0.8
c = 0.9
42
0.370
0.440
0.510
0.580
0.650
0.720
0.790
0.860
0.930
43
0.367
0.437
0.508
0.578
0.648
0.719
0.789
0.859
0.930
44
0.364
0.435
0.506
0.576
0.647
0.717
0.788
0.859
0.929
45
0.362
0.433
0.504
0.575
0.645
0.716
0.787
0.858
0.929 0.929
46
0.359
0.431
0.502
0.573
0.644
0.715
0.786
0.858
47
0.357
0.429
0.500
0.571
0.643
0.714
0.786
0.857
0.929
48
0.355
0.427
0.498
0.570
0.642
0.713
0.785
0.857
0.928
49
0.353
0.425
0.497
0.568
0.640
0.712
0.784
0.856
0.928
50
0.351
0.423
0.495
0.567
0.639
0.711
0.784
0.856
0.928 0.928
51
0.348
0.421
0.493
0.566
0.638
0.710
0.783
0.855
52
0.346
0.419
0.492
0.564
0.637
0.710
0.782
0.855
0.927
53
0.344
0.417
0.490
0.563
0.636
0.709
0.781
0.854
0.927
54
0.343
0.416
0.489
0.562
0.635
0.708
0.781
0.854
0.927
55
0.341
0.414
0.487
0.560
0.634
0.707
0.780
0.853
0.927
56
0.339
0.412
0.486
0.559
0.633
0.706
0.780
0.853
0.927
57
0.337
0.411
0.484
0.558
0.632
0.705
0.779
0.853
0.926
58
0.335
0.409
0.483
0.557
0.631
0.705
0.778
0.852
0.926
59
0.334
0.408
0.482
0.556
0.630
0.704
0.778
0.852
0.926
60
0.332
0.406
0.480
0.555
0.629
0.703
0.777
0.852
0.926
When r > 60, critical value C0 = 1 −
2r (
2r (1−c) 1 −1 1 3 9r (1−α)+1− 9r )
, where −1 (1 − α) is the lower 1-α
percentage point of the standard normal distribution Table 5 Critical value C0 for r = 2(1)60 and c = 0.1(0.1)0.9 at α = 0.05 r
c = 0.1
c = 0.2
c = 0.3
c = 0.4
c = 0.5
c = 0.6
c = 0.7
c = 0.8
c = 0.9
2
0.810
0.831
0.852
0.874
0.895
0.916
0.937
0.958
0.979
3
0.714
0.746
0.778
0.809
0.841
0.873
0.905
0.936
0.968
4
0.652
0.690
0.729
0.768
0.807
0.845
0.884
0.923
0.961
5
0.607
0.650
0.694
0.738
0.782
0.825
0.869
0.913
0.956 0.952
6
0.572
0.620
0.667
0.715
0.762
0.810
0.857
0.905
7
0.544
0.595
0.645
0.696
0.747
0.797
0.848
0.899
0.949
8
0.521
0.574
0.627
0.681
0.734
0.787
0.840
0.894
0.947
9
0.501
0.557
0.612
0.667
0.723
0.778
0.834
0.889
0.945
10
0.484
0.542
0.599
0.656
0.713
0.771
0.828
0.885
0.943 0.941
11
0.469
0.528
0.587
0.646
0.705
0.764
0.823
0.882
12
0.456
0.517
0.577
0.638
0.698
0.758
0.819
0.879
0.940
13
0.445
0.506
0.568
0.630
0.691
0.753
0.815
0.877
0.938
14
0.434
0.497
0.560
0.623
0.686
0.748
0.811
0.874
0.937
15
0.424
0.488
0.552
0.616
0.680
0.744
0.808
0.872
0.936
16
0.416
0.480
0.545
0.610
0.675
0.740
0.805
0.870
0.935
17
0.407
0.473
0.539
0.605
0.671
0.737
0.802
0.868
0.934
123
Implementing Index for the Pareto Lifetime Businesses of the Service Industries
303
Table 5 continued r
c = 0.1
c = 0.2
c = 0.3
c = 0.4
c = 0.5
c = 0.6
c = 0.7
c = 0.8
c = 0.9
18
0.400
0.467
0.533
0.600
0.667
0.733
0.800
0.867
0.933
19
0.393
0.461
0.528
0.595
0.663
0.730
0.798
0.865
0.933
20
0.387
0.455
0.523
0.591
0.659
0.727
0.796
0.864
0.932
21
0.381
0.449
0.518
0.587
0.656
0.725
0.794
0.862
0.931 0.931
22
0.375
0.444
0.514
0.583
0.653
0.722
0.792
0.861
23
0.370
0.440
0.510
0.580
0.650
0.720
0.790
0.860
0.930
24
0.365
0.435
0.506
0.576
0.647
0.718
0.788
0.859
0.929
25
0.360
0.431
0.502
0.573
0.644
0.716
0.787
0.858
0.929
26
0.356
0.427
0.499
0.570
0.642
0.714
0.785
0.857
0.928 0.928
27
0.351
0.423
0.496
0.568
0.640
0.712
0.784
0.856
28
0.347
0.420
0.492
0.565
0.637
0.710
0.782
0.855
0.927
29
0.344
0.416
0.489
0.562
0.635
0.708
0.781
0.854
0.927
30
0.340
0.413
0.487
0.560
0.633
0.707
0.780
0.853
0.927
31
0.336
0.410
0.484
0.558
0.631
0.705
0.779
0.853
0.926
32
0.333
0.407
0.481
0.555
0.630
0.704
0.778
0.852
0.926
33
0.330
0.404
0.479
0.553
0.628
0.702
0.777
0.851
0.926
34
0.327
0.402
0.476
0.551
0.626
0.701
0.776
0.850
0.925
35
0.324
0.399
0.474
0.549
0.624
0.700
0.775
0.850
0.925
36
0.321
0.397
0.472
0.547
0.623
0.698
0.774
0.849
0.925
37
0.318
0.394
0.470
0.546
0.621
0.697
0.773
0.849
0.924 0.924
38
0.316
0.392
0.468
0.544
0.620
0.696
0.772
0.848
39
0.313
0.390
0.466
0.542
0.619
0.695
0.771
0.847
0.924
40
0.311
0.388
0.464
0.541
0.617
0.694
0.770
0.847
0.923
41
0.309
0.385
0.462
0.539
0.616
0.693
0.770
0.846
0.923
42
0.306
0.383
0.461
0.538
0.615
0.692
0.769
0.846
0.923 0.923
43
0.304
0.381
0.459
0.536
0.613
0.691
0.768
0.845
44
0.302
0.380
0.457
0.535
0.612
0.690
0.767
0.845
0.922
45
0.300
0.378
0.456
0.533
0.611
0.689
0.767
0.844
0.922
46
0.298
0.376
0.454
0.532
0.610
0.688
0.766
0.844
0.922
47
0.296
0.374
0.453
0.531
0.609
0.687
0.765
0.844
0.922
48
0.294
0.373
0.451
0.529
0.608
0.686
0.765
0.843
0.922
49
0.292
0.371
0.450
0.528
0.607
0.686
0.764
0.843
0.921
50
0.291
0.369
0.448
0.527
0.606
0.685
0.764
0.842
0.921
51
0.289
0.368
0.447
0.526
0.605
0.684
0.763
0.842
0.921
52
0.287
0.366
0.446
0.525
0.604
0.683
0.762
0.842
0.921
53
0.286
0.365
0.444
0.524
0.603
0.683
0.762
0.841
0.921 0.920
54
0.284
0.364
0.443
0.523
0.602
0.682
0.761
0.841
55
0.283
0.362
0.442
0.522
0.601
0.681
0.761
0.841
0.920
56
0.281
0.361
0.441
0.521
0.601
0.680
0.760
0.840
0.920
57
0.280
0.360
0.440
0.520
0.600
0.680
0.760
0.840
0.920
58
0.278
0.358
0.439
0.519
0.599
0.679
0.759
0.840
0.920
123
304
C.-W. Hong et al.
Table 5 continued r
c = 0.1
c = 0.2
c = 0.3
c = 0.4
c = 0.5
c = 0.6
c = 0.7
c = 0.8
c = 0.9
59
0.277
0.357
0.437
0.518
0.598
0.679
0.759
0.839
0.920
60
0.275
0.356
0.436
0.517
0.597
0.678
0.758
0.839
0.919
When r > 60, critical value C0 = 1 −
2r (
2r (1−c) 1 −1 1 3 9r (1−α)+1− 9r )
, where −1 (1 − α) is the lower 1-α
percentage point of the standard normal distribution
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