Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsExtreme value control charts based on Rayleigh distribution
Volume 3(2) · Pages 27–30 · DOI: 10.22271/maths.2018.v3.i2a.208 ↗
Research areasAdvanced Statistical Process MonitoringAdvanced Statistical Methods and ModelsStatistical and Computational Modeling
Abstract
Author-writtenVariable control charts are based on subgroup statistics and variation in the values of the subgroups. In this paper extreme order statistic of the subgroup are considered to develop the control limits to decide upon the in control status of the process. Here, the quality characteristic is assumed to follow Rayleigh (Weibull with shape parameter 2) and gamma with shape parameter 2 distributions are considered. Relevant comparisons are presented with examples.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Variable control charts are widely used tools for monitoring variability in quality control data through graphical procedures based on subgroup statistics. View evidence ↗
- While many control charts assume normal distributions, there is existing research focused on process variates that are non-normally distributed. View evidence ↗
The question
- The study aims to develop and compare alternative control limits for variable control charts based on Rayleigh and gamma (shape parameter 2) distributions. View evidence ↗
What they did
- The authors use extreme order statistics of subgroups to develop the control limits used to determine if a process is in control. View evidence ↗
- The method involves calculating the cumulative distribution functions for both the smallest and largest order statistics within a sample. View evidence ↗
- Percentiles for the Rayleigh distribution are calculated via specific equations, while gamma (2) distribution percentiles are derived using existing CDF tables. View evidence ↗
What they found
- For the Rayleigh distribution, the lower limit (L*) decreases from 0.03675 to 0.01644 and the upper limit (U*) increases from 3.82112 to 4.22129 as the sample size (n) increases from 2 to 10. View evidence ↗
- For the gamma (2) distribution, both constants M*1 and M*2 decrease as sample size (n) increases from 2 to 10, with M*1 moving from 1.27875 to 0.08132 and M*2 from 2.41750 to 0.56950. View evidence ↗
What it means
- The authors suggest that their developed control chart limits are preferable, although they acknowledge this is a hint rather than a definitive proof. View evidence ↗
- The authors state that because the computations were based on specific examples, the conclusions regarding the preferability of one method over another cannot be generalized. View evidence ↗
Visual summary
3 visualsIllustrations explain the paper; charts are replotted from the paper's own table values.
Extreme Order Statistics Concept
AI-generated conceptual illustration (A series of identical metallic spheres on a horizontal axis). It explains the idea and does not show study data.
View source evidence ↗
2: M* 1 vs M* 2
Replotted from Table 2 (page 2) of the original paper. Every bar is the paper's own table value.
View source evidence ↗
3: M* 1 vs M* 2
Replotted from Table 2 (page 2) of the original paper. Every bar is the paper's own table value.
View source evidence ↗Figures & tables
From the original paper
Table 2
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Figure 1
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Figure 2Introduction
In order to monitor the variability in the quality control data by developing graphical procedures for subgroup statistics such as mean, range, standard deviation etc., the variable control charts are popularly used. Depending on the subgroup size and the sampling distribution of the subgroup statistics separate control chart constants and hence control limits would be constructed in practice. However some research works appeared in literature that deals with control charts for non-normally distributed process variates (Edgemen (1989), Kantam and SriRam (2001) [2] developed control charts for process variates which are non- normally distributed and the references therein). Control charts for individual observations in the case of normal process variates are developed and the principle was made use of to propose a procedure for comparison of multiple means known as Analysis of Means (OTT (1967).
On similar lines the well known gamma and exponential distributions are assumed as models of process variate and the corresponding ANOM procedure along with control charts for individual observations are developed by SriRam (2004) [5]. Motivated from these aspects we extend the same principle to two probability models of Rayleigh distribution (Weibull with shape parameter 2) and gamma with shape parameter 2 developed alternative pairs of control limits for variable control charts and compare their appropriateness with that of existing pairs of limits in literature. The percentiles of extreme order statistics in samples from the models and their use in developing limits for variable control charts are presented in the section 2. In section 3, we compare our control limits with those existing in literature with examples.
Extreme Order Statistics Based Control Limits If X is a process variate having a Rayleigh distribution (Weibull distribution with shape parameter 2) has the probability density function and distribution function given as −𝑥2 𝑥 𝑓(𝑥) = 2𝜎2 ⁄ ; 𝑥≥0 (2.1) 𝜎2 𝑒 −𝑥2 𝐹(𝑥) = 1 −𝑒 2𝜎2 ⁄ ; 𝑓𝑜𝑟 𝑥 Є [0, ∞) (2.2) In the standard form 2Department of Science and Humanities, Vijaya Institute of Pharmaceutical Sciences for Women, Vijayawada, Andhra Pradesh, India R.R.L. Kantam 3Departments of Statistics, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India Correspondence B Sri Ram Department of Science and Humanities, Acharya Nagarjuna University College of Engineering and Technology, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India In case of Rayleigh Distribution σ is estimated by MLE i.e 2 n f (z) = z. 2 z e 2 ; z≥0 (2.3) x i i 1 2 2 z e n F(z) = 1- say s under repeated sub-grouping ˆ = s For Rayleigh distribution the control limits are R D R D for z€[0, ∞) (2.4) Similarly in case of gamma distribution (2) the probability density function and cumulative distribution function * 3 * 2 , x L x e 2 f (x) = (2.5) Where * 2 D = 2 d and * 3 D = 2 d U , * 2 D and x x e F(x) = 1- (1+ * 3 D are given in the following table For Rayleigh distribution the control limits are s U s L * * , , ) (2.6) In the standard form z e f (z) = z. ; z≥0 (2.7) z e F(z) = 1- *L and * U are given in the following table Control chart constants for R-Chart Rayleigh distribution Rayleigh distribution Table 2.1 * 3 D n * 2 D 2 0.825 1.128 3 0.748 1.693 4 0.709 2.059 5 0.670 2.326 6 0.648 2.534 7 0.627 2.704 8 0.614 2.847 9 0.600 2.970 10 0.588 3.076 Table 2.2 n *L (1+z) for z€[0, ∞) (2.8) Here we made an attempt to get the pair of control limits using extreme order statistics in samples from Rayleigh distribution and gamma distribution with shape parameter 2. The percentiles of extreme order statistics for standard Rayleigh distribution can be obtained from the following steps. The cumulative distribution function of smallest order statistic X(1) is given by F1(x) = 1-(1-F(x))n (2.9) The cumulative distribution function of the largest order statistic X(n) given by Fn(x) = (F(x))n (2.10) In case of Rayleigh distribution CDF of the smallest order statistics X(1) and largest order statistics X(n) are given by F1(x) = P(X(i)<L) = 0.00135 (2.11) F1(x) = P(X(i)<U) = 0.99865 (2.12) With as the conjecture, we made an attempt to get the pair of control limits using extreme order statistics in samples from Rayleigh and gamma distribution.
| n | M* 1 | M* 2 |
|---|---|---|
| 2 | 1.27875 | 2.41750 |
| 3 | 0.62727 | 1.68667 |
| 4 | 0.38135 | 1.30250 |
| 5 | 0.26034 | 1.06800 |
| 6 | 0.19109 | 0.90417 |
| 7 | 0.14739 | 0.7900 |
| 8 | 0.11786 | 0.69875 |
| 9 | 0.09685 | 0.62889 |
| 10 | 0.08132 | 0.56950 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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| n | L* | U* |
|---|---|---|
| 2 | 0.03675 | 3.82112 |
| 3 | 0.03001 | 3.92576 |
| 4 | 0.02599 | 3.99836 |
| 5 | 0.02325 | 4.05378 |
| 6 | 0.02122 | 4.09850 |
| 7 | 0.01965 | 4.13594 |
| 8 | 0.01838 | 4.16809 |
| 9 | 0.01733 | 4.19625 |
| 10 | 0.01644 | 4.22129 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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| n | D * 2 | D* 3 |
|---|---|---|
| 2 | 0.825 | 1.128 |
| 3 | 0.748 | 1.693 |
| 4 | 0.709 | 2.059 |
| 5 | 0.670 | 2.326 |
| 6 | 0.648 | 2.534 |
| 7 | 0.627 | 2.704 |
| 8 | 0.614 | 2.847 |
| 9 | 0.600 | 2.970 |
| 10 | 0.588 | 3.076 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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The extreme value control charts based on Rayleigh distribution L= ) 99865 .0 log( 2 n (2.13) * U 2 0.03675 3.82112 3 0.03001 3.92576 4 0.02599 3.99836 5 0.02325 4.05378 6 0.02122 4.09850 7 0.01965 4.13594 8 0.01838 4.16809 9 0.01733 4.19625 10 0.01644 4.22129 Constants of R-chart gamma (2) distribution gamma (2) distribution Table 2.3 U = ) ) 99865 .0 ( 1 log( 2 / 1 n (2.14) The extreme value control charts based on gamma (2) distribution 1-e-L(1+L) =1-(0.00135)1/n (2.15) 1-e-U(1+U) = (0.99865)1/n (2.16) By using equations (2.13), (2.14) we can calculate percentiles of Rayleigh distribution and for percentiles of gamma (2) distribution we use cumulative distribution function tables for gamma (2) distribution from SriRam (2004) [5] and solve the equations (2.15), (2.16) and respective percentiles are tabulated in Tables (2.1) and (2.2). In this case gamma distribution (2), if σ is not known, it is estimated by R where 2 d =E[Zn-Z1] taken from Gupta 2 d (1960). n M*1 M*2 2 1.27875 2.41750 3 0.62727 1.68667 4 0.38135 1.30250 5 0.26034 1.06800 6 0.19109 0.90417 7 0.14739 0.7900 8 0.11786 0.69875 9 0.09685 0.62889 10 0.08132 0.56950 Example 1 Consider the following data Sample Sample No Sample Observations Total Sample Range 42 65 75 78 87 42 45 68 72 90 19 24 80 81 81 36 54 69 77 84 42 51 57 59 78 51 74 75 78 132 60 60 72 95 138 Fig 1.1 18 20 27 42 60 15 30 39 62 84 69 109 113 118 153 71.09 65.87 63.97 66.27 58.62 86.27 89.95 39.92 52.05 115.54 S.D (s) 1 2 3 4 5 6 7 8 9 10 11 12 64 90 93 109 112 61 78 94 109 136 347 317 285 320 287 410 425 167 230 562 468 478 45 48 62 48 36 81 78 42 69 84 48 75 95.15 99.01 Total 716 903.71 For Sub-group size 5 the control chart constants from the above table are The control limits for Rayleigh distribution R D R D * 3 * 2 , , * 2 D = 0.670, * 3 D = 2.326 R D R D Fig 1.2 Range control Chart based on gamma (2) distribution * 3 * 2 , = [35.2889, 122.5104] The Control limits for Rayleigh distribution are s U s L * * , n 2 x i i 1 s s = n , s = k , k is the Sample number *L = 0.02325, * U = 4.05378 From above Rayleigh distribution distribution table at n=5, s = 75.31 Fig 1.3 Example 2 Consider the following data Sample Sample No Sample Observations Total Sample Range s U s L * * , = [1.2379, 215.8725] From the above considered example all the sample points are within the control limits. Control limits for gamma distribution with shape parameter 2 is given by R M R M * 2 * 1 , for n=5, R = 52.67, 14 8 12 12 11 10 13 8 11 12 16 13 15 12 14 11 11.70 10.65 13.13 13.09 S.D (s) 1 2 3 4 5 10 10 8 8 46 42 52 52 36 6 5 5 4 2 * 1 M = 0.26034, * 2 M = 1.06800 R M R M 9.05 Total 22 57.62 For Sub-group size 4 the control chart constants from the above table are From above Rayleigh distribution table at n=4, R = 4.4 R D R D * 3 * 2 , , * 2 D = 0.709, * 2 * 1 , = [13.71211, 56.25156] From above data six points are outside the limits and the respective control chart based on examples are given below. Range control Chart based on Rayleigh distribution Range control Chart based Range values Range control Chart based S.D * 3 D = 2.059 R D R D * 3 * 2 , = [3.1196, 9.0596] The Control limits for Rayleigh distribution are s U s L * * , n 2 x i i 1 s s = n , s = k , k is the Sample number *L = 0.02599, * U = 3.99836 From above Rayleigh distribution table at n=4, s = 11.524 s U s L * * , = [0.29951, 46.0771] All the sample points within the control limits.
Control limits for gamma distribution with shape parameter 2 is given by R M R M * 2 * 1 , for n=4, R = 4.4, * 1 M = 0.38135, * 2 M = 1.3025 R M R M
Conclusions
Since the whole computation is carried out with examples. The conclusions from the above calculations about preferability of one over the other cannot generalize. However, it gives a hint that our control chart limits are preferable.
Figures and tables from the paper
References
5 in the original paper- Edgeman. Inverse Gaussian Control Charts, Australian Journal of Statistics, 1989; 31(1):435-446.
- Kantam RRL, Sriram B. Variable Control Charts based on Gamma Distribution. IAPQR, 2001.
- OTT.E.R Analysis of mean-A graphical procedure”. Industrial Quality Control. 1967.
- Gupta SS. Order statistics for the gamma distribution”, Technimetrics, 1960; 2:243-262.
- Sriram B. Some problems of Quality and Reliability in gamma Type Models, Ph.D Thesis submitted to Acharya Nagarjuna University. 2004. * 2 * 1 , = [1.67794, 5.734] From the above considered example one sample point is outside the control limits and the respective control charts based examples are given below. Range control Chart based on Rayleigh distribution Range control Chart based Range values Range control Chart based S.D Fig 2.1 Fig 2.2 Range control Chart based on gamma (2) distribution Fig 2.3
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