Process Capability Indices for Non Normal Distributions

ABSTRACT

PROCESS CAPABILITY INDICES FOR NON NORMAL

DISTRIBUTIONS

Sulagna Das, M.S.

Department of Mathematical Sciences

Division of Statistics

Northern Illinois University, 2009

Dr Alan M. Polansky, Director

Process capability analysis is a statistical technique that is used to iden-

tify and reduce the variability of a manufacturing process in order to

produce items that meet certain specifications. Many different process

capability indices have been developed to measure the capability of a

manufacturing process. But they all have some drawbacks. The biggest

drawback is that they can be applied only for processes that are normally

distributed. This thesis makes an attempt to deal with the problem of

non-normality by developing an index based on quantiles. To measure

the accuracy of the estimates, confidence intervals have been computed

in four different ways. Finally, the thesis shows how these confidence

intervals work well only for large sample sizes using samples obtained by

the bootstrap method.

NORTHERN ILLINOIS UNIVERSITYDE KALB, ILLINOIS

AUGUST 2009

PROCESS CAPABILITY INDICES FOR NON NORMAL DISTRIBUTIONS

BY

SULAGNA DAS

c 2009 Sulagna Das

A THESIS SUBMITTED TO THE GRADUATE SCHOOL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE

MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICAL SCIENCES

Thesis Director:Dr Alan M. Polansky

UMI Number: 1468057

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ACKNOWLEDGMENTS

I would like to express my gratitude to my advisor, Prof. Alan Polansky, for

his guidance and help in writing my thesis. He introduced me to this topic and

I got interested right away. I am grateful to him for his continued support and

time inspite of his busy schedule. I also wish to thank all of my professors and

friends who offered their suggestions from time to time. Finally, I cannot forget the

incredible support of our office staff, without which it would have been difficult for

me to complete my academic program.

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Chapter

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Process Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Process Capability Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 The Cp Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 One-Sided Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 The Cpk Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.4 The Cpm Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Some Indices Robust to Non-Normality . . . . . . . . . . . . . . . . . 8

1.3.1 The Cpc Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Indices Based on Quantiles . . . . . . . . . . . . . . . . . . . . . . . 10

2. Confidence Intervals for the Cnpk Index . . . . . . . . . . . . . . . . . . 12

2.1 The Standard Bootstrap Confidence Interval . . . . . . . . . . . . . 13

2.2 The Percentile Method Bootstrap Confidence Interval . . . . . . 14

2.3 The Bootstrap-t Confidence Interval . . . . . . . . . . . . . . . . . . . . 14

2.4 The Hybrid Bootstrap Confidence Interval . . . . . . . . . . . . . . . 15

2.5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

iv

Chapter Page

2.5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3. R Program Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

LIST OF TABLES

Table Page

2.1. Simulation Results for samples from a Standard Normal Density . 20

2.2. Simulation Results for samples from a Skewed Unimodal Density . 21

2.3. Simulation Results for samples from a Strongly Skewed UnimodalDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4. Simulation Results for samples from a Kurtotic Unimodal Density 23

LIST OF FIGURES

Figure Page

2.1. Skewed Unimodal Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2. Strongly Skewed Unimodal Density . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3. Kurtotic Unimodal Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

CHAPTER 1

Introduction

1.1 Process Capability

Consider a manufacturing process. Even with the most well-designed manufac-

turing system a certain amount of inherent variability in the manufactured items

always exists. This inherent or natural variability is usually the effect of many small

unavoidable causes. An unavoidable cause is one that cannot be attributed to a

specific reason and occurs purely by chance. A process that is operating with only

unavoidable, or chance, causes of variation is said to be in statistical control. There

may be other kinds of variability in a manufacturing system that could be attributed

to a cause like operator error, defective raw materials, or improperly adjusted ma-

chines. Such variability is usually large compared to the natural variability in a

process and affects the performance of the manufacturing process. Such sources of

variability are referred to as assignable causes of variation. Statistical control can be

restored in a process that is not in control by detecting and eliminating assignable

causes in the process. Once a process is in control, one can then focus on the quality

of the manufactured items.

Process capability refers to how well a process is capable in producing items that

meet the product requirements or specifications. The statistical technique of identi-

fying and reducing process variability in order to produce items within specifications

2is called a process capability analysis. A process capability analysis is a formal study

that can be used to study the variability of a process. Such an analysis usually

focuses on the variation in parameters or quality characteristics of a product that

are required to meet certain specifications.

Specifications refer to the range of a quality characteristic of an item where

the item is useful or of acceptable quality. Specification limits are set for a man-

ufacturing process and are determined by Industrial Engineers. USL refers to the

upper specification limit and LSL refers to the lower specification limit for a single

univariate quality characteristic. Process capability analysis can be done without

specifications by simply describing the process variation. However the analysis is

much more meaningful when done in terms of specifications.

The extent or majority of the variation in a quality characteristic is defined as

the natural tolerance of a process. For a normally distributed quality characteristic

with mean and standard deviation the natural tolerance is the 6 interval in

the distribution around the process mean. This measure, in conjunction with the

specification limits, can be used as a measure of process capability.

When defining the natural tolerance, or natural variability, of a normally dis-

tributed product quality characteristic there are certain additional assumptions that

should be kept in mind. For example, the process is also required to be stable, or

in control. A stable process refers to one that does not exhibit changes in process

distribution with time.

3The normality of a quality characteristic can be verified by plotting a histogram,

using a normal quantile plot, or by using a Shapiro-Wilk test for normality. The

shape and spread of the histogram helps to determine if the distribution is approx-

imately normal. A histogram also gives an immediate and visual impression of the

process performance. The normal quantile plot and Shapiro-wilk test provide a

more formal statistical method for assessing normality. For small samples a his-

togram may not provide reliable results. In these cases a normal probability plot

or Shapiro-wilk test can be used as an alternative to the histogram as it produces

reasonable results for moderately small samples.

Some uses of process capability analysis are :

Predicting how well the process will hold to tolerances prescribed by the spec-ification limits. Process capability is often measured in terms of the natural

tolerance of the process compared to the range of the specification set. Hence,

process capability indicates how much of the process is within engineering

tolerances.

Selecting and modifying processes. This measure tells if a process is capableenough to meet specifications and hence helps in determining if a manufactur-

ing process requires modifications.

Constructing plans for process monitoring. A process capability analysis ofa process can help monitor the process and also give warning signs when a

process does not meet capability standards.

Selecting between competing vendors. A better manufacturing process canbe judged by comparing the relative manufacturing process capabilities of the

4competing vendors. See, for example, Chou [1], Tseng and Wu [14], Huang

and Lee [4], and Polansky [11,12].

1.2 Process Capability Indices

Process capability ratios (PCRs) express the capability of a process to manufac-

ture products that meet specifications. PCRs provide a convenient way of expressing

the capability of a process with a unit-less measure usually formed as a ratio of the

acceptable variability of the process to the actual variability in the process. Several

such measures of process capability have been proposed. A few of these proposals

are presented here. A complete overview of the process capability indices can be

found in Kotz and Johnson (1993).

1.2.1 The Cp Index

The most basic process capability index is the Cp index. Let be the process

standard deviation. The Cp index is defined as

Cp =USL LSL

6.

In practice when is not known it is replaced by an estimate such as the sample

standard deviation of some observed process data, an unbiased estimate such as Rd2

where R is the average range computed from an R-chart, or Sc4

where S is determined

from S-chart and d2 is a constant that changes with sample size. See Appendix 6

of Montgomery (2009). Therefore, an estimate of Cp is given by

Cp =USL LSL

6,

where is the estimated process standard deviation.

Practical interpretation of the Cp index is only valid when the quality character-

5istic has a normal distribution, the process is in statistical control, and the process

mean, , is centered between the upper and lower specification limits. That is, when

=USL + LSL

2.

This can be verified using a hypothesis test

H0 : =USL + LSL

2,

against

Ha : 6= USL + LSL2

.

These hypotheses can be tested using the standard t-test, under the assumption

that the quality characteristic approximately follows a normal distribution.

In practical situations a common problem that is encountered is that the as-

sumption of normality is often violated. Since the capability index Cp uses 6 as

the natural tolerance, the index requires that the quality characteristic of the pro-

cess data follow normal distribution and hence a non-normal process data can lead

to erroneous results. That is, statements made about expected fallout or percentage

of non-conformity may be in error. Also Cp does not take into account where the

process mean is located relative to specifications. It simply measures the spread of

the specification relative to the 6 spread in process. An off-center process has lower

capability than a centered process in that it does not operate at the midpoint of the

interval between the specifications where the lowest proportion of non-conformity

would occur. Due to these reasons Cp process capability index is not considered a

process capability index that can be used in general situations.

61.2.2 One-Sided Indices

Often there are processes that have just either an upper or lower specification

limit. For example, strength often has just a lower specification limit, and time often

has just an upper specification limit. When a process has just an upper specification

limit, a measure of process capability is defined as

Cpu =USL

3.

When a process has just a lower specification limit a measure of process capability

is defined as

Cpl = LSL

3.

Estimates of Cpu and Cpl are obtained by replacing and by estimates and

respectively. The estimate of is usually the sample mean of an observed sample

of process data. The estimate of is the same as used for Cp. Some important

assumptions should be kept in mind. The quality characteristic should be normally

distributed and the process should be in statistical control.

1.2.3 The Cpk Index

The quantity Cpk is a process capability index defined by Kane (1986) that has

been defined to take into account some of the problems encountered with the Cp

index. The Cpk process capability index is the minimum of Cpu and Cpl. If Cp = Cpk

the process is centered at the midpoint of the specification set. But when Cp and

Cpk are not same the process is off-center. Hence, the Cpk index provides a better

7measure of process capability than Cp when the process is not centered. In general

Cpk is less than Cp . Note that there is a relation between Cp and Cpk given by

Cpk =

[1 |

USL+LSL2

|USLLSL

2

]Cp.

An estimate of Cpk is given by

Cpk =

[1 |

USL+LSL2

|USLLSL

2

]Cp.

where and Cp are specified above.

1.2.4 The Cpm Index

The Cpm index was developed to deal with the problems often encountered with

the Cp and Cpk process capability indices. The Cpk index was developed as an

alternative to Cp which does not work well for a process where the mean is not

centered between specification limits. Also the Cpk index has a limitation when

approaches zero. The Cpk index depends inversely on and hence becomes large as

decreases. A large value of Cpk gives no information about the relative location

of the mean in the interval LSL to USL.

The Cpm index was proposed by Chan, Cheng and Spiring (1988) as a better

indicator of process centering. This index is given by

Cpm =USL LSL

6,

where =

E(X T )2 = 2 + ( T )2, and is the target value for theprocess. Hence

Cpm =USL LSL

6

2 + ( T )2 =Cp1 + 2

,

8where

= T

.

It can be seen that as ( T ) , Cpm 0 whereas Cpk . A necessarycondition for Cpm 1 is that | T | < USLLSL6 . This means that if the targetvalue T is the midpoint of the specifications, a Cpm index of one or greater implies

that the mean lies within the middle third of the specification band.

To estimate the Cpm we usually use

Cpm =Cp

1 + V 2,

where

V = T

,

and Cp, and are as specified above.

1.3 Some Indices Robust to Non-Normality

Several nonparametric indices have been formulated to deal with the problem

of non-normal data. The most commonly used approach deals with the problem of

non-normality by transforming the data and specification limits. There are various

graphical and analytical approaches to selecting a transformation. See Polansky and

Kirmani (2003). A suitable transformation of the data to normal distribution can

be done to compute and interpret capability indices. A popular transformation is

taking the reciprocal of the original data. A skewed distribution responds well to

the square root of the original data. However a major disadvantage with the method

9of transformation is that it involves further calculations. Also it is seen that some

people may not be able to handle and interpret a transformed data. Hence this

method is often discouraged.

Another approach is to fit the observed process data to a family of distributions.

Indices specialized to the family of distributions are then computed to measure the

process capability. One needs to make sure that the parameter estimates are based

on a large enough sample to give reliable results. Also choice of the fitted distribu-

tion may not always offer the best fit.

1.3.1 The Cpc Index

The Cpc index is another attempt to define capability in the case when the data

are not normally distributed, developed by Luceno (1996). The Cpc index is defined

as

Cpc =USL LSL

6

12piE | X T |

,

where T is the target value for the process which is often taken to be the midpoint

of the specification set, given by

T =USL + LSL

2,

and X is a random variable equal to the quality characteristic. The Cpc index can

be estimated by estimating E | X T | with

c =

ni=1 | Xi T |

n,

where X1, X2, . . . , Xn is a sample of process data. Therefore, an estimate of the Cpc

10

index is given by

Cpc =USL LSL

6

12pic

.

The denominator 6

pic2

is a more robust measure of natural tolerance than 6 is

when the quality characteristic data are non-normal.

1.3.2 Indices Based on Quantiles

Alternative process capability indices have been proposed that use a more uni-

versal measurement of the natural tolerance of a distribution. These measures are

usually based on quantiles of the process distribution. For these measures, the as-

sumption of normality is not required, but the indices may require large sample sizes

to obtain accurate estimates. For example, an alternative to the Cp index is given

by

Cpq =USL LSL

Q0.99865 Q0.00135 ,

where Qy = yth quantile of the process distribution.

Since for normal distribution Q0.00135 = 3 and Q0.99865 = +3, in the caseof a normally distributed data Cpq reduces to Cp. The Cpq index can be estimated

with

Cpq =USL LSL

Q0.99865 Q0.00135,

where Qy is the yth sample quantile. It is the value where y fraction of the data is

below that value in a given dataset.

Another similar capability index was developed based on this principal as an

alternative to the Cpk index, by McCormack et.al.(2000). The Cnpk index is defined

as

Cnpk = min(Cnpl, Cnpu) where

11

Cnpl =Q50 LSLQ50 Q0.5 ,

and

Cnpu =USLQ50Q99.5 Q50 .

An estimate of the Cnpk index is given by

Cnpk = min (Cnpl, Cnpu) where,

Cnpl =Q50 LSLQ50 Q0.5

,

and

Cnpu =USL Q50Q99.5 Q50

.

where Qy = yth sample quantile from a sample of observations from the process

distribution.

This thesis explores methods for computing four different approximate confidence

intervals for Cnpk. We will empirically determine how well they perform in terms of

capturing the true value of Cnpk. This is done using computer based simulations.

CHAPTER 2

Confidence Intervals for the Cnpk Index

As discussed in the previous chapter, it is clear that when the distribution of a

process deviates from normality, statements made about many of the process capa-

bility indices could be in error if the usual process capability indices such as Cp, Cpk

or Cpm are used. Hence, in this work we have elected to focus on the Cnpk process

capability index which does not require the assumption of normality. In order to

make useful statements about a manufacturing process when the true value of the

Cnpk index is not known, we wish to develop confidence intervals for the Cnpk index.

It is required that the sampling distribution of the capability index be determined

before computing statistics like a confidence interval, since confidence intervals are

required to account for the sample variation in the estimates of the capability index.

The sampling distribution of Cnpk is very complicated due to the fact that it is

a minimum of two functions that involve ratios of sample quantiles. Moreover, the

distribution of the sample quantiles depends on the population density f . For large

samples there is an asymptotic normal result for sample quantiles. Let 0 < p < 1. If

the distribution function of the process, F , possesses a density f in a neighborhood

of Qp and f is positive and continuous at Qp, then the distribution of the sample

quantile Qp has an approximate normal distribution with mean Qp and variance

p(1p)[nf2(Qp)]

when n is large. Therefore, one can note that the variance of Qp depends

on the unknown density f evaluated at the unknown quantile Qp. Since the dis-

13

tribution of f is not known, it is difficult to use this result in practice. Note that

even if the distribution F were known it would still be a difficult task to derive the

sampling distribution of Cnpk.

To deal with the problem of computing confidence intervals for Cnpk for non-

normal data, alternative methods were considered. These methods can estimate

the sampling distribution of Cnpk without having to specify the unknown density

f . These methods are based on the concept of bootstrap estimation developed by

Efron (1979). Four different types of bootstrap confidence intervals are considered.

2.1 The Standard Bootstrap Confidence Interval

Consider a random sample X1, X2, . . . , Xn from a process that follows some un-

known distribution F . To compute a standard bootstrap confidence interval, we

begin by simulating b resamples of size n from the empirical distribution of the

sample. These samples are selected, with replacement, from the observed random

sample X1, X2, . . . , Xn. Such samples are called resamples. For each resample, Cnpk

is computed. Suppose Cnpk(1), C

npk(2),. . . , C

npk(b) are the b sample estimates of pro-

cess capability index Cnpk computed on the resamples. Then the standard bootstrap

confidence interval for Cnpk is given by,

[Cnpk Z/2 SE(Cnpk), Cnpk + Z/2 SE(Cnpk)]

where

SE(Cnpk) =

1b 1

bi=1

(Cnpk(i) Cnpk)2,

14

and

Cnpk =1

b

bi=1

Cnpk(i).

2.2 The Percentile Method Bootstrap Confidence Interval

Consider a random sample X1, X2, . . . , Xn from a process that follows some un-

known distribution F . To compute the percentile method bootstrap confidence

interval, we begin by simulating b resamples of size n. These samples are selected,

with replacement, from the observed random sample X1, X2, . . . , Xn. On each re-

sample Cnpk is computed. Suppose C

npk(1), C

npk(2),. . . ,C

npk(b) are the b sample es-

timates of process capability index Cnpk computed on the resamples. Sort C

npk(1),

Cnpk(2),. . . ,C

npk(b) in ascending order. Let C

npk[1], C

npk[2],. . . ,C

npk[b] denote these

sorted values. A 100(1 )% bootstrap percentile method confidence interval forCnpk is then given by [C

npk[b( 2)], C

npk[b(12)]].

2.3 The Bootstrap-t Confidence Interval

Consider a random sample X1, X2, . . . , Xn that follows some unknown distribu-

tion F . To compute the bootstrap-t confidence interval, we begin by simulating b

resamples of size n. These samples are selected, with replacement, from the ob-

served random sample X1, X2, . . . , Xn. On each resample Cnpk is computed. Sup-

pose Cnpk(1), C

npk(2),. . . ,C

npk(b) are the b sample estimates of process capability index

Cnpk. A second iteration of bootstrap samples are then generated by resampling from

each of the b resamples generated above. Suppose c resamples are generated from

each of the b samples. Let Cnpk(1), C

npk(2),. . . ,C

npk(c) be the c sample estimates of

process capability index Cnpk generated from each of the b samples. Thus if c resam-

15

ples are generated for each of the b resamples, then cb values of Cnpk are computed.

The standard error of Cnpk is computed for each of the b bootstrap samples given

by,

SE(Cnpk) =

1c 1

ci=1

(Cnpk(i) C

npk(i))2,

where

C

npk(i) =1

c

ci=1

Cnpk(i).

This is followed by computing the measure T =C

npkCnpk

SE(Cnpk)for each of the b original

bootstrap resamples. Finally the b values of T are sorted in ascending order. These

are denoted as T [1], T [2], . . . , T [b]. A 100(1)% bootstrap-t confidence intervalfor Cnpk is then defined as,

[Cnpk T [b(1 2

)] SE(Cnpk), Cnpk T [b2

)] SE(Cnpk)],

where SE(Cnpk) is as computed previously.

2.4 The Hybrid Bootstrap Confidence Interval

Consider a random sample X1, X2, . . . , Xn that follows some unknown process

distribution F . We begin by simulating b resamples of size n. These samples are

selected, with replacement, from the observed random sample X1, X2, . . . , Xn. On

each resample Cnpk is computed. Suppose C

npk(1), C

npk(2),. . . ,C

npk(b) are the b sam-

ple estimates of process capability index processCnpk computed on the resamples.

We compute the measure H = Cnpk Cnpk for each of the b original bootstrap

16

samples. The b values of H are then sorted in ascending order. These are denoted

as H[1], H[2], . . . , H[b]. A 100(1 )% hybrid bootstrap confidence interval isthen given by [Cnpk H[b(1 2 )], Cnpk H[b2 ]].

2.5 Simulation Study

A computer based simulation was developed to study the performance of the

four bootstrap confidence intervals introduced above. Using samples from a known

distributions, the four different bootstrap confidence intervals were computed and

their ability to capture the true parameter value was studied.

The algorithm is as follows. The sample size n, upper specification limit (USL)

and lower specification limit (LSL) were specified. For each distribution, USL and

LSL were selected to give a proportion non-conforming equal to 0.0027. The true

value of Cnpk was computed using the specified limits. A random sample of size n

was generated from the specified distribution. 90% confidence intervals were created

using the four methods on the generated sample. This operation was repeated 1000

times and each time it was determined if the true value of Cnpk was in each of the

intervals. The width of the intervals was also computed.

Normal mixtures were used in the study. Four different kinds of distributions

were used. They were the normal, skewed unimodal, strongly skewed unimodal

and kurtotic unimodal. The last three densities were studied by Marron and Wand

(1992). Density plots of the skewed unimodal, strongly skewed unimodal, and kur-

totic unimodal distributions are given in Figures 2.1-2.3.

17

The skewed unimodal density has the form

1

5(x) +

1

5(x; =

1

2, =

2

3) +

3

5(x; =

13

12, =

5

9),

where

(x; , ) = (2pi2)1

2 exp[12

(x )22

].

This density is plotted in Figure 2.1.

3 2 1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

x

Den

sity

Figure 2.1: Skewed Unimodal Density

18

The strongly skewed unimodal density has the form

18(x) + 1

8(x; = 1, = 2

3) + 1

8(x; = 5

3, = 4

9) + 1

8(x; = 19

9, =

827

) + 18(x; = 65

27, = 16

81) + 1

8(x; = 211

81, = 32

243) + 1

8(x; = 665

243, =

64729

) + 18(x; = 2059

729, = 128

2187)),

and is plotted in Figure 2.2.

3 2 1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

Den

sity

Figure 2.2: Strongly Skewed Unimodal Density

19

The Kurtotic Unimodal Density has the form

2

3(x) +

1

3(x; = 0, =

1

10),

and is plotted in Figure 2.3.

3 2 1 0 1 2 3

0.0

0.5

1.0

1.5

x

Den

sity

Figure 2.3: Kurtotic Unimodal Density

20

Table 2.1: Simulation Results for samples from a Standard Normal Density

n Method Coverage Length

25 Percentile 22.9% 0.890Hybrid 62.6% 0.890

Bootstrap-t 58.8% 1.085Standard 62.5% 1.0248









21

Table 2.2: Simulation Results for samples from a Skewed Unimodal Density












22

Table 2.3: Simulation Results for samples from a Strongly Skewed Unimodal Density












23

Table 2.4: Simulation Results for samples from a Kurtotic Unimodal Density












2.5.1 Conclusion

From the tables above it is clear that

None of the methods do well for small samples

All of the methods get better as the sample size increases

The Standard bootstrap method, seems to do the best regarding coverage.Hence it would be the recommended approach for confidence interval calcula-

tion of capability index Cnpk, for the distributions used in this study.

CHAPTER 3

R Program Code

Below is the R code that was used to perform the simulation.

cnpkf=function(x,lsl,usl){

x50=quantile(x,0.5)

x99.5=quantile(x,0.995)

x0.5=quantile(x,0.005)

cnpl=(x50-lsl)/(x50-x0.5)

cnpu=(usl-x50)/(x99.5-x50)

cnpk=min(cnpl,cnpu)

return(cnpk)

}

cnpkt=function(lsl,usl){

z50=qnorm(0.5)

z99.5=qnorm(0.995)

z0.5=qnorm(0.005)

cnpl=(z50-lsl)/(z50-z0.5)

cnpu=(usl-z50)/(z99.5-z50)

true.cnpk=min(cnpl,cnpu)

return(true.cnpk)

25

}

cnpkbootpm=function(x,lsl,usl,b)

{

coverage=matrix(0,1,4)

n=length(x)

cnpk=cnpkt(lsl,usl)

T1=matrix(0,b,1)

H1=matrix(0,b,1)

T1S=matrix(0,b,1)

H1S=matrix(0,b,1)

cnpkstar=matrix(0,b,1)

cnpkstars=matrix(0,b,1)

cnpks=matrix(0,b,1)

sampleSD=sd(x)

sigma=min(sampleSD,IQR(x)/1.349)

h=1.587*sigma*n^(-1/3)

cnpkhat=cnpkf(x,lsl,usl)

for(i in 1:b)

{

xstar=sample(x,n,replace=T)

cnpkstar[i]=cnpkf(xstar,lsl,usl)

cnpkstar1=matrix(0,b,1)

for(j in 1:100)

{

26

xstar1=sample(xstar,n,replace=T)

cnpkstar1[j]=cnpkf(xstar1,lsl,usl)

}

std=sd(cnpkstar1)

T1[i]=(cnpkstar[i]-cnpk)/std

H1[i]=cnpkstar[i]-cnpk

}

cnpks=sort(cnpkstar)

se=sd(cnpkstar)

# standard bootstrap------------------------

BS.CL=cnpkhat-1.96*(se)

BS.CU=cnpkhat+1.96*(se)

if ((cnpk>=BS.CL)&&(cnpk=BSP.CL)&&(cnpk

27

tsort=sort(T1)

d1=b*.95

d2=b*.05

l1=tsort[as.integer(d1)]

u1=tsort[as.integer(d2)]

BST.CL=cnpkhat-se*l1

BST.CU=cnpkhat-se*u1

if ((cnpk>=BST.CL)&&(cnpk=BSH.CL)&&(cnpk

28

cnpksim=function(n,iter,lsl,usl)

{

covmat=matrix(0,iter,4)

b=1000

for (i in 1:iter)

{

x=rnorm(n,0,1)

covmat[i, ]=cnpkbootpm(x,lsl,usl,b)

}

return (covmat)

}

REFERENCES

[1] Chou, Y.-M. (1994). Selecting a better supplier by testing process capabilityindices. Quality Engineering, 6, 427-438.

[2] Chan, L.K., Cheng, S.W. and Spiring, F.A. (1988). A new measure ofprocess capability, Cpm. Journal of Quality Technology, 20, 160-175.

[3] Efron, B. (1979). Bootstrap methods: Another look at the jackknife. TheAnnals of Statistics, bf 7, 1-26.

[4] Huang, D.-Y., and Lee, R.F.(1995). Selecting the largest capability indexfrom several quality control processes. Journel of Statistical Planning andInference, 46, 335-346.

[5] Kane, V. E. (1986). Process capability indices. Journal of Quality Technol-ogy, 18, 41-52.

[6] Kotz, S., and Johnson, N.L. (1993). Process Capability Indices. Chapmanand Hall, London.

[7] Luceno, A. (1996). A process capability ratio with reliable confidenceintervals. Communications in Statistics, Simulation and Computation, 25,235-246.

[8] Mc Cormack, D.W., Harris, I.R., Horwitz, A.M. and Spagon, P.D.(2000).Capability indices for non-normal data. Quality Engineering, 12, 489-495.

[9] Montgomery, D.C. (2009). Introduction to Statistical Quality Control. SixthEdition.

[10] Marron, J.S., and Wand M. P. (1992). Exact mean integrated squared error.The annals of Statistics, 20, 712-736.

[11] Polansky, A.M.(2003). Supplier selection based on bootstrap confidenceregions of process capability indices. International Journel of Reliability,Quality and Safety Engineering, 10, 1-14.

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[12] Polansky, A.M.(2006). Permutation methods for comparing process capa-bility indices. Journal of Quality Technology, 38, 254-266.

[13] Polansky, A.M. and Kirmani, S.N.U.A (2003). Quantifying the capabilityof industrial processes. Handbook of Statistics, Volume 22. Elsevier Science.625-656.

[14] Tseng, S.-T., and Wu, T.-Y. (1991). Selecting the best manufacturing pro-cess. Journal of Quality Technology, 23, 53-62.

Documents

Process Capability Indices for Non Normal Distributions