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Simulation Study 113
Case III. Neither endpoint known. For the case where neither endpoint is known,
four data percentiles have to be matched with the corresponding percentiles of the
standard normal distribution. The resulting equations for i = 1, 2, 3, 4,

x i − ˆε
z i = ˆγ + ˆη log ,
ˆ ε + ˆ λ − x i
are nonlinear and must be solved by numerical methods.

18
The algorithm developed by Hill et al. is used to match the first four moments of X
to the above distribution families. PCIs are calculated using equations (9.1) and (9.2).

9.2.7 Wright’s process capability index C s
7
Wright proposed a CPI, C s , that takes into account the skewness by incorporating an
5
additional skewness correction factor in the denominator of C pmk , and is defined as
min (USL − μ, μ − LSL) min (USL − μ, μ − LSL)
= ,
C s = 2
2 2 3 σ + |μ 3/σ|
3 σ + (μ − T) + |μ 3/σ|
where T = μ and μ 3 is the third central moment.

Some of the methods described above have been widely applied in industry, such
as probability plotting and Clements’ method; however, method such as the Box--
Cox transformation is relatively unknown to practitioners. It should be noted that
when the underlying distribution is normal, theoretically, all the above methods,
with the exception of the distribution-free method, should give the same result as
the conventional C p and C pk given in equations (9.1) and (9.2). Nevertheless, as they
use different statistics and/or different ways of estimating the associated statistics,
the resultant estimates for C p and C pk will exhibit some variability. In particular,
the variability of C p could be reduced with increasing sample size, as its sampling
2
distribution is χ n−1 under the normality assumption. However, the variability of C pk
can be quite significant for all reasonable sample sizes, as it also depends on the
variability in the process shift. 19

9.3 SIMULATION STUDY

9.3.1 Yardstick for comparison
The very fact that different comparison yardsticks may well lead to different con-
clusions emphasizes the need to identify a suitable yardstick for comparing the per-
formance of the above seven methods. In the literature, different researchers have
utilized widely different yardsticks in tackling the nonnormality problem of PCIs.
2
English and Taylor used fixed values of C p and C pk (equal to 1.0) for all their
simulation runs in investigating the robustness of PCIs to nonnormality. The basis for
ˆ
ˆ
theircomparisonwastheproportionofC p andC pk (estimatedfromsimulation)greater
than 1.0 for the normal distribution case. Deleryd 12 focused on the proportion of