Article
Evaluation of Bootstrap Confidence Intervals Using a New NonNormal Process Capability Index
Gadde Srinivasa Rao 1, Mohammed Albassam 2 and Muhammad Aslam 2,
1 Department of Statistics, The University of Dodoma, Dodoma PO. Box. 259, Tanzania;
malbassamkau.edu.sa
Correspondence: aslamravianhotmail.com or magmuhammadkau.edu.sa
Received: 20 February 2019; Accepted: 28 March 2019; Published: 3 April 2019
Abstract: This paper assesses the bootstrap confidence intervals of a newly proposed process capability index PCI for Weibull distribution, using the logarithm of the analyzed data. These methods can be applied when the quality of interest has nonsymmetrical distribution. Bootstrap confidence intervals, which consist of standard bootstrap SB, percentile bootstrap PB, and bias corrected percentile bootstrap BCPB confidence interval are constructed for the proposed method. A Monte Carlo simulation study is used to determine the efficiency of newly proposed indexover the existing method by addressing the coverage probabilities and average widths. The outcome shows that the BCPB confidence interval is recommended. The methodology of the proposed index has been explained by using the real data of breaking stress of carbon fibers.
Keywords: nonnormal process capability index; bootstrap confidence intervals; average width; coverage probabilities; Weibull distribution
1. Introduction
Process capability indices are used to assess the given production process, or whether not to produce items according to the given specifications. The efficiency of these indices required that the process follows a normal distribution 13. However, the involvement of different noise factors, sometime may result in the nonnormal behaviors of the production process 4. In case of non normal processes, the normality based classical process capability index PCI may give unreliable and misleading results 5. Several methods have been proposed 1,37 to estimate PCIs and deal with nonnormality issue. However, until now, we could not rely ona single method that creates efficiently under all situations 4. Moreover, it is also argued that all methods showed a variable performance under different nonnormal distributions 3. Therefore, the evaluation of different non normal PCIs, under different distributions, is an important research area for practitioners and statisticians.
In this regard 4,the proposed nonnormal PCI for Weibull and Lognormal distribution, using the relationships among the parameters of the Weibull, Gumbel, and Lognormal distribution. Using different examples, they discussed the point estimation of proposed PCIs and concluded that the proposed PCIs could be used as standard indices. However, the applicability of these indices required more research. Furthermore, the point estimation of all indices was compared with there commended minimum value, in order to assess the capability of a particular process. Since PCIs are viewed as a random variable, so the comparison of the point estimation of indices with any fixed minimum value is criticized by many researchers 8,9. Therefore, it is more practical to provide confidence intervals
gaddesraogmail.com
2 Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia;
Symmetry 2019, 11, 484; doi:10.3390sym11040484 www.mdpi.comjournalsymmetry
Symmetry 2019, 11, 484 2 of 10
for PCIs or at least lower limits to ensure that the process will not be worse than the lower interval value 10.
Confidence intervals depict useful information about the parameter of a population, based on a random sample. The classical CIs are mostly used for obtaining exact limits, but there are some situations in which the exact CIs are unavailable. These situations attributed to the use of another conventional approach for constructing CIs. Likewise, point estimation, most of the work for construction of the CIs for PCIs are based on normality assumption. The construction of CIs for non normal PCIs, nonparametric technique called bootstrap is commonly used by 11. The bootstrap technique is a computerorientation statistical tool to estimate biases, standard errors, confidence intervals, and so forth, where estimates are more complicated 12. The idea of the bootstrapping method was introduced by Efron 13, and later on, Efron and Tibshirani 14, have proposed three types of bootstrap confidence intervals. In this paper, a bootstrap confidence interval for nonnormal PCIs for Weibull distribution is proposed.
Recently 4 there is research that shows the process capability indices for nonnormal distributions like Weibull and lognormal, whereas bootstrap confidence intervals have not been studied. Therefore, in the present research work we focused on bootstrap confidence intervals, which have been extensively employed to obtain confidence intervals for various PCIs. Various bootstrap estimation methods, for constructing confidence intervals, have been developed, namely, standard bootstrap SB, percentile bootstrap PB, and biascorrected percentile bootstrap BCPB.
InSection2,wepresentedtheprocesscapabilityindex ,usingWeibulldistributionandthree methods of bootstrap confidence interval estimations. The simulation study, using the selected sample size, different shape, and scale parameters are presented in Section 3. In Section 4, a numerical example is provided to illustrate the use of process capability indices and some conclusions are given in Section 5.
2. Materials and Methods
Process capability index measures the performance of product, in such a way that the capability of the process relates to the specified tolerance requirements. The manufacture process is assumed to be normally distributed with meanand standard deviation;lower specification limit LSL,and upper specification limit USL, then the process capability index C is defined as:
, 3
1
Suppose, the process mean and standard deviation are unknown, then they are estimated from
thesampledata X1,X2,…,Xn,whichisdrawnfromthenormaldistribution.IfX,and S arethetrue
estimate of the unknown parameter and fora normal distribution. Then the sample process capability indexC can be defined as
,
3
2
If X, and S are the estimators of mean and standard deviation, when samples are drawn from normal distribution, these estimates happen to be extremely unstable. Clements 15 has proposed an alternative method to handling the process capability for nonnormal data. Based on Clements idea,
the nonnormal process capability index Cpk is defined as:
where is the pthquantile of the process, i.e., PX p, and p 0.00135, p 0.5
USL LSL
C Min p2 , p23
pk pppp
nonnormal capability index for the Weibull distribution, using X, and S as mean and standard deviation of the logarithm of the analyzed process, and although in the logarithm matrix, they are
3221
p p12
and p30.99865 of the analyzed distribution, respectively. Recently, the authors in15, proposed
Symmetry 2019, 11, 484 3 of 10 analogous to those of the normal distribution. Based on this fact, they proposed the Weibull process
capability index Cpkw is given by
C MinlnUSLw ,w lnLSL 4
pkw 3 3 w w
ln1 , ,is the EulerMascheroni constant ww
where
2.1. Bootstrap Confidence Intervals
0.577216.
Among various bootstrap methods to obtain the confidence intervals, a mainly used methods are the SB, PB, and the BCPB method 14. The bootstrap methods has been suggested by 16 in the following manner. Let x, x, x,xbe a random sample of size n drawn from any distribution of interest say . i.ex,x,x, x . Letrepresents the estimator of PCI saydefined in Equation 4. The bootstrap procedure required the following steps 1.
1. Obtained a bootstrap sample of size i.e.,x, x, xx from original sample by putting 1 as mass at each point.
2. LetX where 1 be the mth bootstrap sample, then mth bootstrap estimator ofis computed as
, , 5
3. We getresamples, for each sample a values of are obtained. Each of these would be estimateof .Fromthesetalltheseestimateswouldconstituteanempiricalbootstrapdistribution of , see 1.
In this study, we assumed 1000 bootstrap resamples. The construction of confidence intervals of the PCIusing bootstrap techniques are described as follows.
2.1.1. Standard Bootstrap SB Confidence Interval
The mean and standard deviation for B 1000 bootstrap estimates of are obtained as follows:
6
1
9 9 9
1000
6
7
8
Thus the SB 1100 confidence interval CIis:
where is obtained by using 1 quantiles of the standard normal distribution. 2.1.2. Percentile Bootstrap PB Confidence Interval
If ; i1,2,…,B be the ordered collection of bootstrap estimates ofthen 1100 confidence interval ofis calculated as:
, 9
For a 95 confidence intervalwith 1000, this would be:
, 10
Symmetry 2019, 11, 484 4 of 10 2.1.3. BiasCorrected Percentile Bootstrap BCPB Confidence Interval
A third method of bootstrap confidence interval, BCPB is based on correct the probable bias. The step by step procedure to obtain BCPB method as follows:
i. The probabilityis calculated using the ordered distribution ofas:
11 ii. Letandrepresents the cumulative and inverse cumulative distribution functions ofa
standard normal variable , then:
iii. The percentiles of the ordered distribution ofis obtained as:
2
Finally, the BCPB confidence interval is given as:
12
2
13 14
, 15
The performance of three bootstrap methods of confidence intervals are studied by using average width and coverage probabilities are defined as:
Coverage probability
16 17
Average width
.
whereandare1 confidence interval based on B1000 replicates.
3. Simulation Study
In this section, a simulation study is conducted to compare the performance of three methods of
bootstrap confidence intervals when the process follows Weibull distribution. In this simulation we
examine performance for sample sizes n10,15, 20, 25, 30, 35, 40 ; the scale parametric values
5.0,5.5,6.0; shape parametric combination values2.5,3.0,3.5 and are displayed in
Tables 13. The lower and upper specification limits are set as 1, and 29, respectively. We have used the Monte Carlo simulation approach in this article to evaluate the performance of the proposed SB, PB, and BCPB confidence intervals of . A detailed study of the average length and coverage probabilitiesoftheproposedSB,PB,andBCPBconfidenceintervalsfor isprovidedinTables1 3 for some selective choices of n, and. From Tables 13, we observed that the average widths decrease with the increase in sample size, in all the parametric combinations, as anticipated. The average widths increases for SB, PB, and BCPB asincreases for a fixed , moreover asvalue increases average widths also increases for SB, PB, and BCPB for fixed combinations of. Furthermore, noticed that the average coverage probabilities, in three methods confidence intervals in all the parametric combinations, achieved the nominal level. However, it was marginally less than 95 confidence level. Among three methods of bootstrap confidence intervals regarding the average widths, we noticed that BCPB PBSB, thus BCPB performed well with respect to average width. Furthermore, in case of the average coverage probability, SB shows better performance and BCPB closer competitor to the SB method. Therefore, we conclude that BCPB shows an overall better performance of the bootstrap confidence intervals for Weibull distribution.
Symmetry 2019, 11, 484 5 of 10 Table 1. The average widths 95 bootstrap confidence intervals and coverage probabilities of C
for Weibull distribution when 5.0.
n10 2.00
15 2.00 20 2.00 25 2.00 30 2.00 35 2.00 40 2.00 10 2.50 15 2.50 20 2.50 25 2.50 30 2.50 35 2.50 40 2.50 10 3.00 15 3.00 20 3.00 25 3.00 30 3.00 35 3.00 40 3.00 10 3.50 15 3.50 20 3.50 25 3.50 30 3.50 35 3.50 40 3.50
True
0.6866
0.8957
1.1049
1.3140
Average Widths
SB PB BCPB
Coverage Probabilities
1.4114 1.3825 1.0081 0.9958 0.8208 0.8129 0.7115 0.7054 0.6341 0.6291 0.5751 0.5709 0.5322 0.5288 1.7474 1.7088 1.2333 1.2175 0.9976 0.9877 0.8616 0.8544 0.7660 0.7600 0.6936 0.6882 0.6411 0.6372 2.0854 2.0376 1.4601 1.4409 1.1759 1.1642 1.0131 1.0046 0.8993 0.8923 0.8135 0.8070 0.7515 0.7467 2.4241 2.3668 1.6876 1.6646 1.3550 1.3413 1.1654 1.1552 1.0335 1.0254 0.9342 0.9269 0.8626 0.8569
1.1175 0.8708 0.7362 0.6518 0.5894 0.5399 0.5033 1.3516 1.0497 0.8853 0.7829 0.7073 0.6475 0.6034 1.5885 1.2302 1.0361 0.9157 0.8268 0.7567 0.7051 1.8262 1.4118 1.1881 1.0494 0.9474 0.8667 0.8074
SB PB 0.9672 0.8584 0.9538 0.8922 0.9554 0.9034 0.9528 0.9166 0.9512 0.9158 0.9520 0.9234 0.9528 0.9282 0.9682 0.8530 0.9556 0.8874 0.9558 0.8998 0.9530 0.9120 0.9512 0.9132 0.9518 0.9206 0.9538 0.9274 0.9704 0.8480 0.9574 0.8824 0.9574 0.8984 0.9538 0.9106 0.9514 0.9120 0.9510 0.9190 0.9538 0.9254 0.9726 0.8452 0.9584 0.8798 0.9580 0.8956 0.9538 0.9078 0.9516 0.9114 0.9508 0.9170 0.9530 0.9248
BCPB 0.9222 0.9306 0.9422 0.9430 0.9400 0.9430 0.9434 0.9196 0.9316 0.9412 0.9404 0.9408 0.9412 0.9432 0.9178 0.9318 0.9392 0.9400 0.9394 0.9390 0.9428 0.9178 0.9314 0.9374 0.9382 0.9390 0.9390 0.9426
Table 2. The average
for Weibull distribution when 5.5.
probabilities ofCoverage Probabilities
widths 95 bootstrap confidence intervals and coverage
n10 2.00
15 2.00 20 2.00 25 2.00 30 2.00 35 2.00 40 2.00 10 2.50 15 2.50 20 2.50 25 2.50 30 2.50 35 2.50 40 2.50 10 3.00 15 3.00 20 3.00 25 3.00 30 3.00 35 3.00
True
0.7361
0.9576
1.1792
Average Widths
SB PB BCPB
SB PB 0.9620 0.8576 0.9468 0.8914 0.9464 0.9030 0.9464 0.9152 0.9436 0.9146 0.9448 0.9232 0.9480 0.9272 0.9654 0.8522 0.9488 0.8854 0.9486 0.8998 0.9482 0.9114 0.9452 0.9130 0.9450 0.9190 0.9498 0.9270 0.9676 0.8474 0.9516 0.8820 0.9518 0.8970 0.9480 0.9092 0.9464 0.9106 0.9456 0.9178
BCPB 0.9202 0.9244 0.9354 0.9380 0.9350 0.9394 0.9410 0.9180 0.9264 0.9356 0.9372 0.9362 0.9390 0.9424 0.9160 0.9272 0.9336 0.9370 0.9364 0.9376
1.4091 1.3832 1.0138 1.0025 0.8329 0.8251 0.7264 0.7201 0.6510 0.6454 0.5933 0.5885 0.5512 0.5472 1.7492 1.7141 1.2442 1.2296 1.0157 1.0061 0.8826 0.8750 0.7890 0.7824 0.7179 0.7119 0.6661 0.6615 2.0925 2.0484 1.4770 1.4589 1.2005 1.1890 1.0405 1.0313 0.9286 0.9208 0.8438 0.8369
1.1300 0.8852 0.7529 0.6692 0.6073 0.5584 0.5221 1.3717 1.0711 0.9090 0.8070 0.7316 0.6721 0.6281 1.6163 1.2595 1.0670 0.9467 0.8575 0.7872
Symmetry 2019, 11, 484
40 3.00
10 3.50 15 3.50 20 3.50 25 3.50 30 3.50 35 3.50 40 3.50
6 of 10
1.4007
0.7823 0.7767 2.4375 2.3839 1.7111 1.6895 1.3865 1.3730 1.1994 1.1887 1.0691 1.0602 0.9707 0.9629 0.8994 0.8928
0.7355 0.9504 1.8629 0.9688 1.4488 0.9528 1.2264 0.9526 1.0869 0.9480 0.9846 0.9478 0.9033 0.9448 0.8437 0.9508
0.9244 0.8460 0.8790 0.8954 0.9076 0.9106 0.9164 0.9248
0.9422 0.9142 0.9264 0.9342 0.9362 0.9362 0.9376 0.9416
Table 3. The average
for Weibull distribution when 6.0.
probabilities ofCoverage Probabilities
widths 95 bootstrap confidence intervals and coverage
n10 2.00
15 2.00 20 2.00 25 2.00 30 2.00 35 2.00 40 2.00 10 2.50 15 2.50 20 2.50 25 2.50 30 2.50 35 2.50 40 2.50 10 3.00 15 3.00 20 3.00 25 3.00 30 3.00 35 3.00 40 3.00 10 3.50 15 3.50 20 3.50 25 3.50 30 3.50 35 3.50 40 3.50
4. Illustrative Examples
True
0.7813
1.0142
1.2470
1.4798
Average Widths
SB PB BCPB
SB PB 0.9630 0.8610 0.9448 0.8924 0.9434 0.9040 0.9418 0.9160 0.9374 0.9146 0.9378 0.9220 0.9402 0.9284 0.9656 0.8584 0.9478 0.9018 0.9476 0.9018 0.9412 0.9126 0.9402 0.9142 0.9394 0.9198 0.9424 0.9272 0.9686 0.8550 0.9514 0.8858 0.9496 0.9006 0.9436 0.9106 0.9414 0.9142 0.9402 0.9180 0.9428 0.9252 0.9702 0.8548 0.9540 0.8854 0.9504 0.9002 0.9460 0.9098 0.9420 0.9144 0.9420 0.9194 0.9448 0.9240
BCPB 0.9204 0.9246 0.9320 0.9302 0.9292 0.9286 0.9324 0.9186 0.9282 0.9326 0.9328 0.9298 0.9316 0.9322 0.9188 0.9298 0.9334 0.9344 0.9302 0.9338 0.9332 0.9174 0.9300 0.9332 0.9338 0.9352 0.9340 0.9366
1.3752 1.3523 0.9894 0.9811 0.8162 0.8101 0.7141 0.7085 0.6430 0.6380 0.5894 0.5845 0.5504 0.5461 1.7010 1.6702 1.2077 1.1971 0.9886 0.9816 0.8608 0.8545 0.7727 0.7673 0.7068 0.7016 0.6590 0.6544 2.0275 1.9882 1.4260 1.4129 1.1603 1.1524 1.0064 0.9993 0.9010 0.8952 0.8225 0.8173 0.7657 0.7610 2.3538 2.3066 1.6434 1.6276 1.3308 1.3220 1.1504 1.1430 1.0274 1.0215 0.9362 0.9309 0.8700 0.8653
1.1176 0.8771 0.7484 0.6664 0.6065 0.5597 0.5252 1.3518 1.0573 0.9000 0.7997 0.7268 0.6701 0.6282 1.5862 1.2372 1.0509 0.9322 0.8462 0.7795 0.7300 1.8190 1.4158 1.2002 1.0629 0.8866 0.8866 0.8298
In this section, the following real data set corresponds to an uncensored data set from 17 on breaking the stress of carbon fibers in Gba is considered and describe the proposed methods of
CNpk confidence intervals:
0.39, 0.81, 0.85, 0.98, 1.08, 1.12, 1.17, 1.18, 1.22, 1.25, 1.36, 1.41, 1.47, 1.57, 1.57, 1.59, 1.59, 1.61, 1.61, 1.69, 1.69, 1.71, 1.73, 1.80, 1.84, 1.84, 1.87, 1.89, 1.92, 2.00, 2.03, 2.03, 2.05, 2.12, 2.17, 2.17, 2.17, 2.35, 2.38, 2.41, 2.43, 2.48, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.76, 2.77, 2.79, 2.81, 2.81, 2.82, 2.83, 2.85, 2.87, 2.88, 2.93, 2.95, 2.96, 2.97, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.51, 3.56, 3.60, 3.65, 3.68, 3.68, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90, 4.91, 5.08, 5.56.
The goodness of fit for our model by plotting the superimposed for the data shows that the Weibull distribution is a good fit. This is shown in Figure 1andthe goodness of fit is emphasized with QQ plot, displayed in the following Figure 1. The maximum likelihood estimates of the two
Symmetry 2019, 11, 484
7 of 10
parameters of Weibull distribution for the breaking stress of carbon fibres are2.7928 and
2.9435, and by using the KolmogorovSmirnov test, we found that the maximum distance between the data and the fitted of the Weibull distribution is 0.06 with pvalue is 0.8586. Therefore, the Weibull distribution provides a reasonable fit for the breaking stress of carbon fibres. The process capability of Weibull distribution, using breaking stress of carbon fibresis, displayed in Figure 2. If LSL0.5 and USL9.5 thenusing Equation 4 is 1.0005 whereasusing Equation 3 is 0.90297. Hence our proposed method of process capability index performs better than the traditional process capability index.
Figure 1.Density plot and QQ plot of the fitted Weibull distribution for the breaking stress of carbon fibres.
LSL USL
Process Data LSL 0.1
Target
USL 6 Sample Mean 2.6214 Sample N 100 Shape 2.79283 Scale 2.94369
Observed Performance PPMLSL 0.00 PPMUSL 0.00 PPM Total 0.00
Overall Capability Pp 1.07 PPL 1.08 PPU 1.07 Ppk 1.07
Exp. Overall Performance
PPMLSL PPMUSL PPM Total
79.00 671.18 750.18
0123456
Figure 2.Capability of Weibull distribution for proposed method using breaking stress of carbon fibers.
5. Comparison of Proposed Index with the Existing
Theperformanceoftheproposedprocesscapabilityindex hasbeenmadewiththeexisting process capability indexproposed by Clements 15, using the coverage probabilities and average widths. The values of coverage probabilities and average widths are compared using
Symmetry 2019, 11, 484 8 of 10 breaking stress of carbon fibres data, which was fitted for Weibull distribution with parameters are
2.792827 and 2.943466. The Bootstrap confidence intervals and widths of the,and C
are given in Table 4 for the given example. A numerical example shows that the width of class intervalsareconsideredlargeintraditional Cpk methodascomparedtothebootstrapapproachfor
. Moreover, between three Bootstrap methods, BCPB sows better performance than other two methods of Bootstrapping, the simulation results also show the same. Therefore, the proposed process capability index comparatively better than traditional process capability index in case Weibull distribution.
Table 4. Bootstrap confidence intervals and widths of the newand Clements traditional Cpk Weibull Distribution
pk
Methods
SB
PB BCPB
Methods
SB
PB BCPB
Bootstrap New Confidence intervals
C pkw
Widths
Widths
Traditional Cpk Confidence Intervals
Traditional Cpk Confidence Intervals
Widths 0.2692 0.2631 0.2485
Widths 0.3194 0.3074 0.3225
0.5654, 0.7513 0.5948, 0.7829 0.5805, 0.7575
0.1859 0.1881 0.1770
1.0322, 1.3014 0.8974, 1.1605 0.9251, 1.1736
Exponential Weibull Distribution
Bootstrap Cpkew Confidence Intervals
0.5672, 0.8748 0.5953, 0.9056 0.5739, 0.8816
0.3076 0.3103 0.3077
0.7204, 1.0398 0.6633, 0.9707 0.6364, 0.9589
Furthermore, the proposed new process capability index and traditional process capability index are also compared with exponential Weibull distribution, which was first introduced by Mudholkar and Srivastava18 as an extension of the Weibull family. The probability density function of
exponential Weibull distribution with scale parameterand shape parameters as follows:
1fxx ex 1ex;x0
and
is given
18 19
EWD EWD1
PX pppln1 p
The proposed process capability indexis given below: USLEWD EWD LSL
CMinp2,p2pkew EWD EWD EWD EWD
20 The breaking stress of carbon fibres data is fitted to exponential Weibull distribution with
1
pppp
parameters are2.6879,2.4161 and1.3097. The Bootstrap confidence intervals and widths of
the,and Cpk are given in Table 4 for the exponential Weibull distribution for this data set. The results shows that the proposed process capability index for Weibull distribution show lower
confidence width than the exponential Weibull distribution.
6. Conclusions
In this article, we developed an interval estimation of newly proposed nonnormal process capability index for Weibull distribution using the logarithm of the analyzed data. A comparison is made between SB, PB, and BCPB confidence intervals and their coverage probabilities using simulation studies. A confidence interval is better performed if the average width is low and with
3221
Symmetry 2019, 11, 484 9 of 10
higher coverage probability. Compared to Clements 15, the nonnormal process capability index, the proposed method gives the smaller widths of confidence intervals. Therefore, we conclude thatthe proposed procedure has been very powerful and enriching the process capability index. Using a Monte Carlo simulation technique, we evaluated the performance of the proposed bootstrap confidence intervals with respect to the coverage probabilities and average widths of each confidence interval. The results indicated that the basiccorrected percentile bootstrap BCPB confidence interval showed better coverage probability and smaller widths, as compared to other two confidence intervals, for the proposed and traditional method. Hence it is concluded that the BCPB interval is more suitable than the other two methods. The methodology was illustrated using areal data set of breaking stress of carbon fibres and depicted a similar pattern, as observed using simulation studies. Our future research work may include the study of process capability indices for more nonnormal distributions using the transformation of variables. The proposed method can be extended for abroad family of distributions as future research.
Author Contributions: Conceived and designed the experiments, G.S.R., M.A., M.A.; performed the experiments, M.A.; analyzed the data, M.A.; contributed reagentsmaterialsanalysis tools, M.A.; wrote the paper, M.A., M.A.
Funding: This article was funded by the Deanship of Scientific Research DSR at King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.
Acknowledgments: The authors are deeply thankful to the editor and reviewers for their valuable suggestions to improve the quality of this manuscript.
Conflicts of Interest: The authors declare no conflict of interest. References
1. Kashif, M.; Aslam, M.; AlMarshadi, A.H.; Jun, C.H. Capability Indices for NonNormal Distribution Using Ginis Mean Difference as Measure of Variability. IEEE Access 2016, 4, 73227330.
2. Panichkitkosolkul, W. Confidence Intervals for the Process Capability Index Cp Based on Confidence Intervals for Variance under NonNormality. Malays. J. Math. Sci. 2016, 10, 101115.
3. Senvar, O.; Sennaroglu, B. Comparing performances of clements, boxcox, Johnson methods with weibull distributions for assessing process capability. J. Ind. Eng. Manag. 2016, 9, 634.
4. PinaMonarrez, M.R.; OrtizYanez, J.F.; RodriguezBorbon, M.I. Nonnormal Capability Indices for the Weibull and Lognormal Distributions. Qual. Reliab. Eng. Int. 2016, 32, 13211329.
5. Senvar, O.;Kahraman, C. Type2 fuzzy process capability indices for nonnormal processes. J. Intell. Fuzzy Syst. 2014, 27, 769781.
6. Besseris, G. Robust process capability performance: An interpretation of key indices from a nonparametric viewpoint. TQM J. 2014, 26, 445462.
7. Sennaroglu, B.; Senvar, O. Performance comparison of boxcox transformation and weighted variance methods with weibull distribution. J. Aeronaut. Space Technol.2015, 8, 4955.
8. Chou, Y.M.;Own, D.B.; Salvador, A.; Borrego, A.A. Lower Confidence Limits on Process Capability Indices. J. Qual. Technol. 1990, 22, 223229.
9. Porter, L.J.; Oakland, J.S. Process capability indicesAn overview of theory and practice. Qual. Reliab. Eng. Int. 1991, 7, 437448.
10. Wu, H. Development of a Weighted Variance Approach to Modify Cerrent Process Capability Indices. The University of Alabama Huntsville: Huntsville, AL, USA, 1998.
11. Balamurali, S. Bootstrap confidence limits for the process capability index Cpmk. Int. J.Qual. Eng. Technol. 2012, 3, 7990.
12. Panichkitkosolkul, W.;Saothayanun, L. Bootstrap confidence intervals for the process capability index under halflogistic distribution. Maejo Int. J. Sci. Technol. 2012, 6, 272281.
13. Efron, B. Bootstrap methods: Another look for the Jackknife.Ann. Stat. 1979, 7, 126.
14. Efron, B.; Tibshirani, R. Bootstrap methods for standard errors, confidence intervals, and other measures
of statistical accuracy. Stat. Sci. 1986, 1, 5477.
15. Clements, J.A. Process capability calculations for nonnormal distributions. Qual. Prog. 1989, 22, 9597.
Symmetry 2019, 11, 484 10 of 10
16. Tosasukul, J.;Budsaba, K.; Volodin, A. Dependent bootstrap confidence intervals for a population mean. Thail. Stat. 2009, 7, 4351.
17. Nichols, M.D.;Padgett, W.J. A bootstrap control chart for Weibull percentiles. Qual. Reliab. Eng. Int. 2006, 22, 141151.
18. Mudholkar, G.S.;Srivastava, D.K. Exponentiated Weibull family for analyzing bathtub failurerate data. IEEE Trans. Reliab. 1993, 42, 299302.
2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution CC BY license http:creativecommons.orglicensesby4.0.
Reviews
There are no reviews yet.