Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsExtreme value charts and analysis of means based on Dagum distribution
Volume 3(2) · Pages 351–354
Research areasAdvanced Statistical Process Monitoring
Abstract
Author-writtenThe probability model of a quality characteristic is assumed as a Dagum distribution. Variable control charts based on the extreme values of each subgroup are constructed. The control chart constants depend on the probability model of the extreme order statistics of each subgroup and the size of the subgroup. Accordingly, the proposed chart is known as extreme value chart. The technique of analysis of means for a skewed population is applied with respect to Dagum distribution. The results are illustrated by examples on live data.
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
- The Dagum distribution is a unimodal, skewed distribution used for data on the positive real line. View evidence ↗
- Shewhart control charts are widely used by practitioners for statistical quality control. View evidence ↗
- Analysis of Means (ANOM) is a procedure used to categorize subgroup means based on whether they are homogeneous within a category or heterogeneous between categories. View evidence ↗
The question
- The study aims to construct variable control charts based on extreme values and apply the ANOM technique specifically for populations following a Dagum distribution. View evidence ↗
What they did
- The authors calculated control limits for extreme value charts for subgroup sizes (n) ranging from 2 to 10. View evidence ↗
- The goodness of fit for the Dagum model in the provided examples was assessed using the Q-Q plot technique to measure linearity between observed and theoretical quantiles. View evidence ↗
What they found
- The Dagum distribution showed higher correlation coefficients (0.7347, 0.8048, and 0.7741) compared to the Normal distribution (0.2067, 0.4149, and 0.4447) across three examples, indicating it is a better fit. View evidence ↗
- In Example 1 (n=5, k=5, α=0.05), the Normal distribution model found only 1 mean in-control (P=0.2), while the Dagum distribution model found all 5 means in-control (P=1). View evidence ↗
- In Example 2 (n=5, k=3, α=0.05), the Normal model identified 2 means as in-control (P=0.7), whereas the Dagum model identified all 3 means as in-control (P=1). View evidence ↗
- In Example 3 (n=4, k=4, α=0.05), the Normal model found 0 means in-control (P=0), while the Dagum model found all 4 means in-control (P=1). View evidence ↗
What it means
- The authors conclude that the Dagum distribution provides a more reliable decision process for ANOM than the Normal distribution, as it reduces errors associated with incorrectly identifying means as non-homogeneous. View evidence ↗
Visual summary
3 visualsIllustrations explain the paper; charts are replotted from the paper's own table values.
Model Fit Comparison
AI-generated conceptual illustration (mathematical probability distribution curves). It explains the idea and does not show study data.
View source evidence ↗
DD and Normal: Example1 vs Example2 vs Example3
Replotted from Table 7 (page 4) of the original paper. Every bar is the paper's own table value.
View source evidence ↗
P=in/k: Example 1: n=5, k=5, α... vs Example 2: n=5, k=3, α... vs Example 3: n=4, k=4, α...
Replotted from Table 8 (page 4) of the original paper. Every bar is the paper's own table value.
View source evidence ↗Figures & tables
From the original paper
Table 1
Table 2
Table 3
Table 5
Table 6
Table 4
Table 7
Table 8
Table 9
Figure 1
Figure 2
Figure 3Introduction
The probability density function (pdf) of Dagum distribution (DD) is given by
Dagum distribution (DD) is a skewed, uni model distribution on the positive real line. The mean, median and variance of Dagum distribution are respectively. The Dagum Distribution is positively skewed and mesokurtic. In order to construct à control chart using the extreme observations of a subgroup drawn from the production process with the quality variate following Dagum Distribution. we need the percentiles of extreme order statistics in samples from Dagum Distribution. Specifically, the test statistic on extreme value control chart is the original sample vector X=(x 1 ; x 2 ; x n ) from the ongoing production. In this chart all the individual sample observations are plotted into control chart without calculating any statistic out of them. A corrective action is taken after one or either of the extreme values namely x 1 (sample minimum) and x n (sample maximum) of the sample respectively fall below or above two specified lines (limits). Because of this, the chart is called extreme value control chart. The Shewart control chart is a common tool of statistical quality control for many practitioners. When these charts indicate the presence of an assignable cause, an adjustment of the process is made in the remedy is known. Otherwise the suspected presence of assignable cause is regarded to be an indication of heterogeneity of the subgroup statistic for which the control chart is developed. For instance if the statistic is sample mean, this leads to heterogeneity of process mean indicating departures from target mean. Such an analysis is generally carried out with the help of means to divide a collection of a given number of subgroup means into categories such that means within a category are homogenous and those between categories are heterogeneous and the procedure is called Analysis of Means (ANOM) given by Ott (1967) [7] . For using the ANOM technique the concept of the control chart for means is viewed in a different way-grouping of plotted means to fall within the control limits or some outside the control limits. For the homogeneity of all the means, it is necessary that all the means should fall within the control limits. If (1-α) is taken as the confidence coefficient, we should have the probability of all the subgroup means to fall within the control limits is (1-α). Assuming independence of subgroups the above probability statement becomes n th power of the probability of a subgroup mean to fall within the limits. i. e In the sampling distribution of x the confidence interval for x to lie two specified limits should be equal to (1-α ) 1/n . The same principle is adopted in the rest of this paper through Dagum distribution. Because this paper aims at exploring ANOM using control limits of extreme value statistics we have considered only the control chart aspects. However, a detailed literature about ANOM is available in Rao (2005) [9] and some related works in this direction are Ramig (1983) [8] , Bakir (1994) [1] , Bernard and Wludyka (2001) [2] , Montgomery (2001) [5] , Nelson and Dudewicz (2002) [6] , Rao and Prankumar (2002) [10] , Farnum (2004) [3] , Guirguis and Tobias (2004) [4] . The recently developed ANOM tables or techniques based on various distributions are Srinivasa Rao and Kantam (2012a) [11] , Srinivasa Rao et al. (2012b) [12] , Srinivasa Rao et al. (2012c) [13] and references there in. The rest of the paper is organized as follows. The basic exposure to extreme value control charts is given in Section 2. ANOM applied to Dagum distribution using extreme value control charts of DD is given in Section 3 followed by numerical examples. Summary and conclusions are given in Section 4.
Extreme Value Charts
The given sample observations are assumed to follow Dagum distribution. The control lines are determined by the theory of extreme order statistics based on Dagum distribution model The control lines are to be determined in such a way that an arbitrarily chosen x i of X=(x 1 ; x 2 ; x n )lies with probability
Analysis of Means(ANOM)-Dagum Distribution
When the data variate follows Dagum distribution, suppose
are arithmetic means of k subgroups of size 'n' each drawn from a Dagum model. If these subgroup means are used to develop control charts to assess whether the population from which these subgroups are drawn is operating with admissible quality variations. Depending on the basic population model, we may use the control chart constants developed by us or the popular Shewart constants given in any SQC text book. Generally we say that the process is in control if all the subgroup means fall within the control limits. Otherwise we say the process lacks control. If α is the level of signifi cance of the above decisions we can have the following probability statements.
Using the notation of independent subgroups (3.1) becomes
with equal-tailed probability for each subgroup mean, we can find two constants say L * and U * such that
In 3. We make use of its percentiles in Equation (3.3) for specified 'n' and 'k' to get * L and * U for α = 0:01 and 0.05. These are given in Tables 8 andTable 9. A control chart for averages giving 'In Control' conclusion indicates that all the subgroup means though vary among themselves are homogenous in some sense. This is exactly the null hypothesis in an analysis of variance technique. Hence the constants of Tables 8 and9 can be used as an alternative to analysis of variance technique. For a normal population one can use the tables of Ott (1967) [7] . For a Dagum Distribution our tables can be used. We therefore present below some examples for which the goodness of fit Dagum model is assessed with Q-Q plot technique (strength of linearity between observed and theoretical quantiles of a model) and tested the homogeneity of means involved in each case
Example 1: Wadsworth (1986) [14] : Consider the following data of 25 observations on manufacture of metal products that suspect variations in iron content of raw material supplied by five suppliers. Five ingots were randomly selected from each of the five suppliers. The following table contains the data for the iron determinations on each ingots in percent by weight The goodness of fit of data in these three examples as revealed by Q-Q plot (correlation coefficient) are summarized in the following table, which shows that Dagum distribution is a better model, exhibiting significant linear relation between sample and population quantiles. Treating these observations in the data as a single sample, we have calculated the decision limits for the Normal population, DD population and have given them in the Tables 4 and5 respectively.
Summary and Conclusion
The usage of normal model resulted in homogeneity for some means and deviation for some other means, indicating a possible rejection of those means. This decision is valid if Normal distribution is a good fit to the data. As a comparision, we have alredy established by Q-Q plot that Dagum distribution is a better model than Normal as supported by the Q-Q plot correlation coefficient of each data set with Normal as well as Dagum distribution separately. We have observed that more error is likely to be associated with decision process of Normal distribution. Therefore, more number of the means to be homogenous with the help of Dagum distribution(Table-5) is a better decision than some means to be away from homogeneity using Normal, ANOM procedure.
Figures and tables from the paper
| N | Z(1) 0.00135 | Z(n) 0.99865 |
|---|---|---|
| 2 | 1.37E-06 | 57.67655 |
| 3 | 6.08E-07 | 70.65498 |
| 4 | 3.42E-07 | 81.59453 |
| 5 | 2.19E-07 | 99.94373 |
| 6 | 1.52E-07 | 91.23162 |
| 7 | 1.12E-07 | 107.955 |
| 8 | 8.55E-08 | 115.4116 |
| 9 | 6.76E-08 | 122.4148 |
| 10 | 5.47E-08 | 129.0384 |
Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.
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| N | D3* | D4* |
|---|---|---|
| 2 | 7.609E-07 | 32.08687 |
| 3 | 3.382E-07 | 39.30708 |
| 4 | 1.902E-07 | 45.39302 |
| 5 | 1.218E-07 | 50.75436 |
| 6 | 8.458E-08 | 55.60112 |
| 7 | 6.214E-08 | 60.05801 |
| 8 | 4.758E-08 | 64.20626 |
| 9 | 3.759E-08 | 68.1023 |
| 10 | 3.045E-08 | 71.78721 |
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| p | 0.99865 | 0.995 | 0.99 | 0.975 | 0.95 | 0.05 | 0.025 | 0.01 | 0.005 | 0.00135 |
|---|---|---|---|---|---|---|---|---|---|---|
| n=2 | 11.468 | 5.1834 | 3.9115 | 2.5239 | 1.8169 | 0.0321 | 0.0147 | 0.0061 | 0.0035 | 0.0009 |
| 3 | 9.1636 | 4.7826 | 3.5575 | 2.3065 | 1.6691 | 0.0714 | 0.0451 | 0.0263 | 0.0162 | 0.0085 |
| 4 | 7.7031 | 3.9709 | 2.9331 | 2.0182 | 1.5083 | 0.1009 | 0.0706 | 0.0469 | 0.0292 | 0.0155 |
| 5 | 5.8986 | 3.6035 | 2.7199 | 1.8759 | 1.4330 | 0.1266 | 0.0941 | 0.0676 | 0.0513 | 0.0338 |
| 6 | 5.4368 | 3.4367 | 2.4518 | 1.7638 | 1.4001 | 0.1513 | 0.1168 | 0.0887 | 0.0653 | 0.0320 |
| 7 | 5.1056 | 3.2389 | 2.5704 | 1.7609 | 1.3269 | 0.1687 | 0.1317 | 0.1005 | 0.0798 | 0.0542 |
| 8 | 5.3257 | 3.0336 | 2.2657 | 1.6277 | 1.2457 | 0.1875 | 0.1487 | 0.1212 | 0.1040 | 0.0761 |
| 9 | 4.8355 | 2.6043 | 2.1420 | 1.5916 | 1.2802 | 0.1992 | 0.1651 | 0.1307 | 0.1091 | 0.0811 |
| 10 | 4.2427 | 2.5570 | 2.0747 | 1.5372 | 1.2430 | 0.2133 | 0.1787 | 0.1366 | 0.1185 | 0.0982 |
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| Brand1 | Brand2 | Brand3 |
|---|---|---|
| 100 | 76 | 108 |
| 96 | 80 | 100 |
| 92 | 75 | 96 |
| 96 | 84 | 98 |
| 92 | 82 | 100 |
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| 1 | 2 | 3 | 4 |
|---|---|---|---|
| 58.2 | 56.3 | 50.1 | 52.9 |
| 57.2 | 54.5 | 54.2 | 49.9 |
| 58.4 | 57 | 55.4 | 50 |
| 55.8 | 55.3 | 54.9 | 51.7 |
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| 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| 3.46 | 3.59 | 3.51 | 3.38 | 3.29 |
| 3.48 | 3.46 | 3.64 | 3.40 | 3.46 |
| 3.56 | 3.42 | 3.46 | 3.37 | 3.37 |
| 3.39 | 3.49 | 3.52 | 3.46 | 3.32 |
| 3.40 | 3.50 | 3.49 | 3.39 | 3.38 |
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| — | DD | Normal |
|---|---|---|
| Example1 | 0.7347 | 0.2067 |
| Example2 | 0.8048 | 0.4149 |
| Example3 | 0.7741 | 0.4447 |
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| — | — | No. of Counts | — | — | — |
|---|---|---|---|---|---|
| [LDL, UDL](Ott 1967) | In | P=in/k | Out | Out/k | |
| Example 1: n=5, k=5, α = 0.05 | [3.517, 3.879] | 1 | 0.2 | 4 | 0.8 |
| Example 2: n=5, k=3, α = 0.05 | [87.82, 95.52] | 2 | 0.7 | 1 | 0.3 |
| Example 3: n=4, k=4, α = 0.05 | [26.14, 82.84] | 0 | 0 | 4 | 1 |
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| — | — | No. of Counts | — | — | — |
|---|---|---|---|---|---|
| [LDL, UDL](Table-7) | In | P=in/k | Out | Out/k | |
| Example 1: n=5, k=5, α = 0.05 | [1.06517, 7.4039] | 5 | 1 | 0 | 0 |
| Example 2: n=5, k=3, α = 0.05 | [29.3617, 185.2045] | 3 | 1 | 0 | 0 |
| Example 3: n=4, k=4, α = 0.05 | [15.8977, 124.8574] | 4 | 1 | 0 | 0 |
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References
10 in the original paper- Bakir ST. Means using ranks for randomised complete block design Communications in Statistics-Simulation and Computation. 1994; 23:547-568.
- Bernard AJ, Wludyka PS. Robust I-sample Analysis of means type randomization tests for variances Journal of Statistical Computation and Simulation. 2001; 69:57-88.
- Farnum NR. Analysis of Means Tables using mathematical processors Quality Engineering. 2004; 16:399-405.
- Guirguis GH, Tobias RD. On the computation of the distribution for the analysis of means Communications in Statistics-Simulation and Computation. 2004; 16:861887.
- Montgomery DC. Design and Analysis of Experiments Fifth edition, John Wiley and Sons, New York, 2004.
- Nelson PR, Dudewicz EJ. Exact Analysis of Means with Unequal Variances Technometrics. 2002; 44:152-160.
- Ott ER. Analysis of Means-A graphical procedure Industrial Quality Control. 1967; 24:101-109.
- Ramig PF. Applications of Analysis of Means Journal of Quality Technology. 1983; 15:19-25.
- Rao CV. Analysis of Means-A review Journal of Quality Technology. 2005; 37:308-315.
- Rao CV, Prankumar M. ANOM-type Graphical Methods for testing the Equality of Several Correlation Coeffcients Gujarat Statistical Review. 2002; 29:47-56.
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