New Control Charts for a Multivariate Gamma Distribution

In this study, we introduce a multivariate gamma distribution, then, by defining a new statistic, three control charts called the MG charts, are proposed for this distribution. The first control chart is based on the exact distribution of this statistic, the second control chart is based on the Satterthwaite approximation, and the last is based on the normal approximation. The efficiency of the proposed control charts is evaluated by the average run length (ARL) criterion. The results show that whenever the magnitude of the parameter shifts c<1, the control chart based on the exact distribution has smaller ARL1s, while for c>1, the control chart based on Satterthwaite approximation show smaller values of ARL1s. Besides, the results show that for large changes (c>2.5), all proposed charts almost have equal ARL1s; i.e. they have the same performance.


Structure of a multivariate gamma control chart based on the exact distribution
Let , p j ,..., 2 , 1 = be random variables following a gamma distribution with the shape parameter − 0 and the scale parameter . Suppose 0 is a random variable that follows a gamma distribution with the shape parameter 0 and the scale parameter . Assume that The multivariate gamma distribution structure can be described with bellow example: Suppose that a system has p independent components so that , is longevity i-th item, and has a gamma distribution with the shape parameter − 0 and the scale parameter . When i-th item fails, it is repaired or replaced by another item. Assume that 0 is the time of repaired or replaced of i-th item, and has a gamma distribution with the shape parameter 0 and the scale parameter . In this case = + 0 denotes the restart time of i-th item. Define the statistic as follows: (2) The probability density function of is given by: where, * = (∑ ) − 0 =1 and ) , , ( The probability density function of given in Equation (3) is derived based on the method of variable transformation in the Appendix. Fixing Type-I error at , the lower and the upper control limits (LCL and UCL, respectively) are determined by solving the following two equations: Under this control chart, the out-of-control probability ( 0 ) based on a single sample when the process is in-control is given by: The in-control ARL denoted by ARL 0 is the expected number of subgroups to be examined until the process is declared to be out-of-control when the process is truly in control. It is given by: Now, suppose those shape parameters are changed from 0 to 1 = 0 , p i ,..., 2 , 1 = for a constant . Then, the probability of the process being declared out-of-control based on the single sample when the process is changed is given by: 1 = ( ≤ | = 1 ) + ( ≥ | = 1 ). (9) Note that, the control limits are calculated when the process is in-control. The out-of-control ARL ( 1 ) for the changing process is given as: (10)

3-The efficiency evaluation of the chart
In this section, the efficiency of the proposed control chart is evaluated using the ARL criterion. There are two types of ARL namely the in control (ARL 0 ) and the out-of-control (ARL 1 ) ARLs. The larger the value of the in-control and the smaller the value of the out-of-control ARL the better the performances of the chart are. Here, the ARL 1 s of the chart are examined for a 3-variable and a 2-variable process when ARL 0 s are fixed at 200 and 370 for each case. The results based on various parameter changes are shown in Tables 1 and 2, respectively. These results are obtained using MATLAB software.    The conclusions made based on the results in Tables 1 and 2 are summarized as follow: 1-It is observed that for > 1, when increases, the ARL 1 decreases for all cases. 2-ARL 1 decreases rapidly when > 2.
3-Most of the results show that if > 1, small changes of may not be detected (ARL 1 > ARL 0 ).

Structure of the multivariate gamma control chart based on approximation
In this section, two approximations are employed to obtain the control limits; the Satterthwaite approximation (Casella and Berger (2002)) and the Wilson-Hilferty (WH) approximation (Wilson and Hilferty (1931)).

Case1: Satterthwaite approximation
The Satterthwaite approximation is a suitable approach widely used today (Casella and Berger (2002)). Assume that the statistic in Equation (2) approximately follows a gamma distribution with the shape parameter and the scale parameter . In the other words: and ( ) = 2 . (13) Using the method of moment estimation (MME) and the Satterthwaite approximation, (17) Hence, from (16) and (17) it can be concluded that and In the following, we assume that is known. From (11) it can be concluded that 2 has a chi-squared distribution with 2 degrees of freedom. In other words 2 ≈ 2 2 .
(20) Thus, the control limits of the chart with the probability of Type-I error can be obtained as: Now, suppose that the shape parameter of the gamma distribution is changed from 0 to 1 = 0 (i.e one or all of the s are changed), for a constant . By (19), it can be concluded that scale parameter is shifted to 1 . Thus, the probability of the process being declared out-of-control based on the single sample when the process is changed is given by: As a result, the out-of-control ARL ( 1 ) for the changing process is obtained as: Having 0 the shape parameter of the gamma distribution when the process is in control, the traditional Shewharttype control limits using the above normal approximation can be written as follows: and In Phase-I monitoring, the unknown parameters and are estimated based on the historical data set. Then, under normal approximation, the out-of-control probability when the process is in-control 0 is obtained as 0 = ( * > | = 0 )+ ( * < | = 0 ) = 2Φ(−3).
(31) Thus, the ARL for the in-control process is given by In what follows, we assume that the shape parameter of the gamma distribution is changed from 0 to 1 = 0 (i.e one or all of the s are changed) for a constant . By (19), it can be concluded that the scale parameter is shifted to 1 . Then, the probability of the process being declared out-of-control based on the single sample is given by 1 = ( * > | = 1 )+ ( * < | = 1 ) Note that, the control limits are calculated when the process is in-control. Besides, the out-of-control ARL for the shifted process is given by:

Efficiency comparison
Comparison among the efficiency of the three control charts proposed in Section 4 is demonstrated in this section.
Here, the 1 of the charts are compared for a 3-variable and a 2-variable process when ARL 0 s is fixed at 370 for each case. Moreover, ARL 0 = 200 is considered when = 3. The results based on various parameter changes are shown in Tables 3-5.  Table 3. The ARL 1 s corresponding to the three proposed control charts when = 3 , = 4 and ARL 0 = 370. Table 4. The ARL 1 s corresponding to the three proposed control charts when = 2 , = 4 and ARL 0 = 370. = (9,7,9) and 0 = 2 = (5,1,3) and 0 = 0.5 = (4,4,4) and 0 = 2  The results in Tables 3-5 show that when < 1, the control chart based on the exact distribution has smaller ARL1 values. For example, in the case ARL 0 = 370, = (9,7,9) and = 0.7 (the first column of Table 3), the value of ARL1 for the exact distribution is 10.34, while it is 11.98 for the Satterthwaite approximation and 12.3 for the normal approximation. In addition, when > 1, the control chart based on the Satterthwaite approximation has smaller ARL1 values; thus it can detect the changes earlier. Besides, based on the results in the above tables one can conclude that for large changes ( > 2.5), the three charts have almost an equal performance in terms of the out-of-control ARL. So, can conclude that the Satterthwaite approximation and the normal approximation are good and appropriate approximations.

Conclusions and remarks
In the present article, a multivariate gamma distribution with positive correlations was introduced. Then, by defining a statistic, three control charts were proposed to monitor processes modeled by this distribution. The first control chart was based on the exact distribution of the defined statistic, the second control chart was based on the Satterthwaite approximation, and the third was based on the Wilson-Hilferty approximation. As the Wilson-Hilferty approximation is used to approximate a gamma distribution by a normal distribution, a Shewhart control chart was constructed based on this approximation. The efficiency of these charts was compared in terms of out-of-control ARL, when ARL 0 remains constant. It was observed that whenever the magnitude of the parameter shifts < 1, the control chart based on the exact distribution had smaller ARL1s, while for > 1, the control chart based on Satterthwaite approximation showed smaller values of ARL1s. Besides, it was observed that for large changes ( > 2.5), all proposed charts had almost equal ARL1s; i.e. they had the same performance.