To investigate the properties and performance of the proposed chart, we discuss the method and supporting material for development of the chart here.
2.1. Inverse Maxwell Distribution and Its Properties
Assume a random variable
is continuous and holds the assumptions of Maxwell distribution with a single scale parameter
. According to this consideration, the probability density function (PDF) and cumulative distribution function (CDF) of Maxwell distribution can respectively be expressed as
where
As in [
10], if we consider
where
is a Maxwell random variable, then the generated random variable
R can be declared as an inverse Maxwell random variable. By adopting inverse transformation, we can derive the PDF of this distribution from Maxwell distribution with scale parameter
as
Note that [
26] uses another version of the inverse Maxwell distribution. The version in [
26] can be obtained from this version by applying the transformation
From (4) above, the CDF of the inverse Maxwell distribution can be obtained as
Note that the integration of Equation (4) over the entire range is one. That is,
. Since Equation (4) is simultaneously nonnegative valued, it is verified as a PDF. From
Figure 1 and
Figure 2, we can state that PDF and CDF of the inverse Maxwell distribution are, respectively, positively skewed and monotonically increasing.
From the literature, several properties of inverse Maxwell distribution have been studied, but some important properties like entropy and Fisher information have yet to be provided. Moreover, as can be found in later application sections of this paper, it is important to discuss raw moments, as they provide the basis for theoretical development of control charts. So, in
Appendix A, these properties are provided.
2.2. Derivation of the Control Chart
Shewhart control chart is one sophisticated tool to detect the variation of scale parameter in process monitoring. Although the Shewhart control chart is originally used to monitor normally distributed process, we propose a new control chart for evaluating non-normally distributed processes. In this study, the inverse Maxwell distribution is considered, and the suggested chart is control chart. For this, we will introduce a pivotal quantity and define an estimate of named . The control chart is developed under the probability limits and -sigma limits. In case of probability limits, the lower probability limits and upper probability limits are denoted by LPL and UPL. Similarly for the second case, the lower control limit, the and the are denoted by , and . To define and or , we need to estimate the quantile or mean and variance of respectively. The procedure is described in the following paragraphs.
From the PDF that is given in Equation (4), we let,
, and by simplification we get
. Then the Jacobian of transformation becomes
. Now, the distribution of
can be written as
Finally, we can write this in a more familiar form as follows:
The above expressed equation is a gamma density where the shape and scale parameters are and respectively. Symbolically, we can write .
Now, according to the additive property of the gamma distribution, we can write the sum of independent and identically distributed function of inverse Maxwell variates,
. In addition, remember the MLE of scale parameter derived in Equation (A4) in
Appendix A and let
. As
is estimated from the sample observations in different time points, the values of
are not the same across time, it is thus as random variable. So, we can write
Finally, we can consider
as a pivotal quantity, and the PDF of
follows a gamma distribution with parameters
and
. Here,
is a gamma distributed random variable, so its mean is
Thus, we can write
. Therefore,
Equation (7) states that is an unbiased estimator of. Following the above procedure, first we find the variance of , and from this we can calculate the variance of.
The variance of
is
. This means that
. Hence,
The CDF of the pivotal quantity,
follows gamma distribution with parameters
and
, i.e.,
, where
denote the incomplete gamma function. So, the
quantile can be derived as
Now, the probability limits for
can be written as
These can also be presented as
where
and
. In
Table 1, we provide several estimated values of these coefficients from the gamma distribution for various combinations of false alarm rate
and sample size
.
In process monitoring, we usually face two situations, and these cases are when the desired parameter
is known or unknown. When dealing with known
, the limits will be defined as
We will estimate
when
is unknown and use the estimated value for evaluating the process
where
is the estimated value of
and is calculated from the estimates of
attained from each of the sample batches over time, and finally taking their average.
Adopting the two moments of
found in Equations (7) and (8),
-sigma limits of
are presented as
where
and
.
Table 2 below contains the values of the
L coefficient calculated from the gamma quantiles by maintaining some desired false alarm rate
.
Similar to the probability limit approach, we also observe two circumstances: One when
is known and the other when
is unknown. When
is known, the limits can be expressed as
When we don’t have any previous value of
, we will estimate
beforehand (typically in a phase I study) and use it as follows:
Table 3 illustrates the different values of factors
and
. These values are calculated by using
L coefficients, which are expressed in
Table 2. By using these values, we can easily develop a control chart for the inverse Maxwell parameter.
The main purpose of constructing a control chart is to identify whether or not shift is available in a process. Therefore, we will test the following hypothesis.
Null hypothesisor(i.e., no shift is available in the process)
versus
Alternative hypothesisor(i.e., shift is available in the process.)
Here, indicates that the shift is available in the process.