Variable control charts based on percentiles of the new Rayleigh-Pareto distribution

Introduction

The well-known Shewhart control charts are developed under the assumption that the quality characteristic follows a normal distribution. If In quality control studies data is always in small samples only. Therefore if the population is not normal there is a need to develop separate procedure for the construction of control limits.

In this paper we assume that the quality variate follows the new Rayleigh-Pareto model and develop control limits for such data on par with the presently available control limits. If a process quality characteristic is assumed to follow the new Rayleigh-Pareto distribution the online process of such a quality can be controlled through the theory of the new Rayleigh-Pareto distribution. In the absence of any such specification of the population model we generally use the normal distribution and the associated constants available in all standard text books of statistical quality control. However, normality is only an assumption that is rarely verified and found to be true. Unless the sample is very large in size this assumption may not be taken for granted without proper goodness of fit test procedure. At the same time central limit theorem cannot be made use of, because central limit theorem gives only asymptotic normality for any statistic. Therefore, if a distribution other than normal is a suitable model for a quality variate, separate procedures are to be developed. We present the construction of quality control charts when the process variate is assumed to follow the new Rayleigh-Pareto distribution. Let X be a random variable from a Pareto distribution with its cumulative distribution function (cdf) for The NRPD has a cdf of the form

x R the survival is function of the Pareto distribution and is given by

is the pdf of a Rayleigh distribution and is given by ( )

The cdf of the NRPD is given by ( )

The probability density function (pdf) is given by

The hazard function is given by

From the hazard function the following can be observed:

 the failure rate is constant, which makes the NRPD suitable for modeling systems or components with failure rate 2. If , 1   the hazard is an increasing function, which makes the NRPD suitable for modeling components that wears faster with time.

If

, 1   the hazard is a decreasing function, which makes the NRPD suitable for modeling components that wears slower with time. The distributional properties are:

The pdf of largest order statistics ( ) n x is given by

The pdf of the smallest order statistic ( ) x1 is given by ( )

The other distributional properties are thoroughly discussed by Nasiru and Luguterah (2015) [8] . Skewed distributions to develop statistical quality control methods are attempted by many authors. Some of them are Edge-man (1989) [3] -Inverse Gaussian Distribution, Gonzalez and Viles (2000) [4] -Gamma Distribution, Kantam and Sriram (2011) [5] -Gamma Distribution, Chan and Cui (2003) [2] have developed control chart constants for skewed distributions where the constants are dependent on the coefficient of skewness of the distributions, Kantam et al (2006) [6] -Log logistic Distribution, Betul and Yaziki (2006) [1] -Burr Distribution, Subba Rao and Kantam (2008) [11] -Double exponential distribution, Kantam and Rao (2010) [7] control charts for process variate, Rao and Sarath Babu (2012) [9] -Linear failure rate distribution, Rao and Kantam (2012) [10] -Half logistic distribution, K. Rosaiah, R.R.L. Kantam, B. Srinivas Rao (2012) [12] Variable control charts for Half logistic distribution, Srinivas Rao Boyapati, Suleman Nasiru, K.N.V.R. Lakshmi (2015) [13] -Variable Control Charts Based on Percentiles of the new Weibull-Pareto Distribution and references there in.

NRPD is another situation of skewed distribution which is paid much attention with respect to development of control charts in the present study. If 0.5, 2.5, 1.5

Then the hazard function indicates a decreasing failure rate function (shown in the graph), which makes the NRPD suitable for modeling components that wears slower with time. At the same time it is one of the probability models applicable for life testing and reliability studies also. Accordingly, if a lifetime data is considered as a quality data, development of control charts for the same is desirable for the use by practitioners. Since NRPD is a skewed distribution, this paper makes an attempt to study in a comparative manner. An attempt is made in this paper to address this problem and solve it to the extent possible. The rest of the paper is organized follows. The basic theory and the development of control charts for the statisticsaverage, median, midrange, range and standard deviation are presented in Section 2. The comparative study to the developed control limits in relation to the Shewarts limits is given in Section 3. Summary and conclusions are given in Section 4.

this is considered as a subgroup of an industrial process data with a targeted population average, under repeated sampling the statistic x gives whether the process average is around the targeted mean or not. Statistically speaking, we have to find the 'most probable' limits with in which x fails. Here the phrase 'most probable' is a relative concept which is to be considered in the population sense. As the existing procedures are for normal distribution only, the concept of  3 control limits is taken as the 'most probable' limits. It is well known that  3 limits of normal distribution include % 73 . 99 of probability. Hence, we have to search two limits of the sampling distribution of sample mean in NRPD such that the probability content of those limits is .

. 0

Symbolically we have to find Lower control limit and Upper control limit (L and U) such that 9973 . 0

Where x is the mean of the sample size n. taking the equi-tailed concept L, U are respectively 00135 . 0 and 99865 . 0 percentiles of the sampling distribution of x . We resorted to the empirical sampling distribution of x through simulation there by computing its percentiles. These are given in Table 1.

Median -chart

We have to search two limits of the sampling distribution of sample median in NRPD such that the probability content of these limits is 9973 . 0

. Symbolically, we have to find U L, such that

(

Where m is the median of sample size n. Through simulation, the percentiles observed are given Table 3.

are the percentile constants of median chart and are given in Table 4.

Midrange-chart

We have to search two limits of the sampling distribution of sample midrange in NRPD such that the probability content of these limits is 9973 . 0

. Symbolically, we have to find U L, such that

(

Where M is the midrange of sample size n. Through simulation, the percentiles observed are given Table 5.

are the percentile constants of midrange chart for NRPD process data given in Table 6.

R-chart

We have to search two limits of the sampling distribution of sample

Range in NRPD such that the probability content of these limits is 9973 . 0

13)

Where R is the range of sample of size n. Through simulation, the percentiles observed are given in Table 7.

(

Are the percentile constants of R chart for NRPD process data and are given in Table 8.

 -chart

We have to search two limits of the sampling distribution of sample standard deviation in NRPD such that the probability content of these limits is 9973 . 0

. Symbolically, we have to find U L, such that

(

Where s is the standard deviation of sample of size n. Through simulation the percentiles observed are given in Table 9.

Comparative study

The control chart constants for the statistics mean, median, midrange, range and standard deviation developed in section 2 are based on the population described by NRPD. In order to use this for a data, the data is confirmed to follow NRPD. Therefore the power of the control limits can be accessed through their application for a true NRPD data in relation to the application for Shewhart limits. With this back drop we have made this comparative study simulating random samples of size n= 2,3,…,10 from NRPD and calculated the control limits using the constants of section 2 for mean, median, midrange, range and standard deviation in succession. The number of statistic values that have fallen within the respective control limits is evaluated and is named as NRPD coverage probability. Similar count for control limits using Shewhart constants available in quality control manuals are also calculated. These are named as Shewhart coverage probability. The coverage probabilities under the two schemes namely true NRPD, Shewhart limits are presented in the following Tables 11,12,13,14 and 15.

Summary and Conclusions

The Tables 11, 12, 13, 14 and 15 show that for a true NRPD if the Shewhart limits are used in a mechanical way it would result in less confidence coefficient about the decision of process variation for mean, median, midrange, range and standard deviation charts. Hence if a data is confirmed to follow NRPD, the usage of Shewhart constants in all the above charts is not advisable and exclusive evaluation and application of NRPD constants is preferable in statistical quality control.

References

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