Extreme Value Charts & Analysis of Means Based on the Lomax Distribution

Abstract

It is assumed that the probabilistic model of the quality characteristics follows the new weighted exponential distribution. Control charts based on each subgroup\'s extreme values are established. The constants in the control chart are determined by the probability distribution of the extreme value order statistics of the sub-group and the sub-group size. The proposed chart is thus referred to as an extreme values chart. A biased overall mean analysis method (ANOM for truncated population) is used for the Lomax Distribution. Examples based on real time data are used to explain the findings.

Introduction

I. INTRODUCTION

The extreme order statistical percentiles of the LD sample are required to create a control chart that uses extreme observations from a subset of manufacturing processes with LD quality options. Specifically, the first test vector  X=(x1,x2,……. ,xn) from the continuous processing is used as the test statistic on the extreme value control chart. The control chart in extreme value chart displays entire sample observations, but no statistic(s) is/are calculated from it. According to one or both extreme values of the sample,  x1 (test least) and n xn (test most extreme), fall below or above two defined lines (limits), a corrective action is taken. Therefore, this chart is called an extreme value control chart[9].

Many professionals use Shewart control charts as a statistical method [5] for quality control. If the solution is found, the technique shall be adjusted when such charts indicate that an assignable cause exists [1]. In the abstract group statistical for which the control chart is built, the existence of an assignable cause is understood as a signal of heterogeneity [1, 8]. For example, the mean process would be heterogeneous when the figures are sample mean, which would signify differences from the goal mean [4]. Such an analysis is often done by means of means to split a collection of different subset mean into categories [2], so that means are homogenous within a category and heterogeneous between categories and the technique is known as an analysis of means (ANOM) as described by Ott.E.R [7]. The control chart for the mean is read differently using the ANOM technique [6, 10]: grouping of the plotted means within or beyond the control limits. The two means must fall under the control limits in order for all of them to be homogeneous. The probability of any sub-group is equal to the coefficient of confidence, take is as ( 1-α) . This probability statement will be the n th power of the likelihood that the mean of a subgroup fall within the boundaries, provided it is supposed to be independent. I.e. the confidence interval of x for the distribution of samples should be equivalent to ( 1-α)1/n  between two specified bounds. In the rest of this article, the same principle is also applied by LD. We only looked at ANOM control charts [3] in this research since it intends to examine ANOM by employing extreme value statistical control limits.

No new ANOM tables or procedures have been examined by us. However, there is a thorough documentary on ANOM by Rao.C.V [9] and certain similar works are in this direction [11-14] are mentioned in references.

The rest of the paper is described below. Section 2 gives a fundamental exposure to extreme value control diagrams that are supported by average runtime (ARL) and to ANOM. In Section 3, LD is used with an ANOM in conjunction with numerical examples employing extreme value control limits  of LD. The findings and conclusions of Section 4 are provided.

II. MATERIALS AND METHODS

The mathematical and statistical research background of Extreme value charts & ANOM and the methods for the study of LD model are discussed in this part.

Conclusion

According to the decision limits using Normal distribution or the Shewart control limits and ANOM tables of Ott.E.R [7], the number of homogenous mean is 3 and 2 for each data set, and those who are not homogeneous are 2 and 1 respectively. When the ANOM tables of LD are utilized, the number of homogeneous means for the same data sets is 5 and 3, with no deviations from the homogeneity. This indicates that, when the normal distribution model has been applied, certain means have been homogenized, and others have deviated. This decision is valid, even if the data corresponds to the Normal distribution. In comparison, LD is a better model than normal. As a result, we concluded that the decision method of the Normal distribution will be correlated with more error. Henceforth, using proposed LD model is a better option rather than the usual, to achieve homogeneity for ANOM method in some cases.

References

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Copyright

Copyright © 2022 M. Nandini, CH. V. Aruna , O. V. Raja Sekharam, G. V. S. R. Anjaneyulu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.