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Variable Control Charts Based on Percentiles of Exponentiated Inverse Kumaraswamy Distribution

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Variable Control Charts Based on Percentiles of Exponentiated Inverse Kumaraswamy Distribution Journal of Information and Computational Science ISSN: 1548-7741 Variable Control Charts based on Percentiles of Exponentiated Inverse Kumaraswamy Distribution K. Rosaiah1, B. Srinivasa Rao2 and M. Rami Reddy3 1Department of Statistics, Acharya Nagarjuna University, Guntur, INDIA. 2Department of Mathematics & Humanities, R.V.R & J.C College of Engineering, Chowdavaram, Guntur, INDIA. 3Freshman Engineering Department , Lakireddy Balireddy College of Enginerring, Mylavaram, Krishna district, INDIA. 1 [email protected] , 2 [email protected], [email protected] ABSTRACT A variable quality characteristic is assumed to follow the Exponentiated Inverse Kumaraswamy distribution(EIKD). The control charts for the statistics such as mean, median, standard deviation, range and midrange are constructed based on the constants of the percentiles.. The admissibility and ability of the developed percentile limits are assessed by comparing with the existing Shewhart control limits. Keywords and phrases: Most probable, Equi-tailed, Percentiles, Simulation, EIKD. 1. INTRODUCTION The familiar Shewhart control charts are formulated when it is assumed that the quality characteristic follows normal distribution. If 푥1, 푥2, … . 푥푛 is a sample of n observations on a variable quality aspect of a product and if ‘ 푡푛’ is a statistic corresponding to this sample then the Shewhart control limits are E(푡푛) ± 3S.E(푡푛). In statistical quality control theory data is always be considered in small samples to construct the control charts. Hence, it is required to develop an alternate process for the construction of control limits when the population is not normal. In this paper, we assumed that the quality variate follows the Exponentiated inverse Kumaraswamy distribution and developed the control limits for that data in similar to the known Shewhart control limits. If a process quality characteristic is assumed to follow the Exponentiated Inverse Kumaraswamy distribution, the online procedure of such a quality can be monitored through the theory of the Exponentiated Inverse Kumaraswamy distribution. In the non presence of any such specification of the population model, we generally apply the normal distribution and the related constants available. Although, verification and confirmation of the sample data that follows normal distribution is very rare. If the size of the taken sample is not large, the assumption of normality cannot be accepted without any goodness of fit test process. In this case even the central limit theorem cannot be made use of, because it gives only asymptotic normality for any given statistic. Thus, if any non normal distribution is appropriate for a quality variate, an alternate procedure to be established. We introduce the construction of control limits with the assumption of the process variate follows Exponentiated Inverse Kumaraswamy Distribution (EIKD). Volume 9 Issue 10 - 2019 841 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 Let X is a random variable follows Exponentiated Inverse Kumaraswamy distribution(EIKD) with its probability density function as 푓(푥) = 훼훽휆(1 + 푥)−(훼+1)(1 − (1 + 푥)−훼)훽휆−1 ; 0 < 푥 < ∞, 훼, 훽, 휆 > 0. (1.1) The cumulative distribution function of Exponentiated Inverse Kumaraswamy distribution is given by 퐹(푥) = [1 − (1 + 푥)−훼]훽휆 ; 0 < 푥 < ∞, 훼, 훽, 휆 > 0. (1.2) where 훼, 훽 푎푛푑 휆 being shape parameters. Plots of the pdf and cdf of X for selected parameter values of 훼, 훽 푎푛푑 휆 are displayed in figures 1 and 2 respectively. From figure 1, we observe that the curve of EIKD is very nearly normal when the parameters 훼 = 2.5 , 훽 = 2.25 푎푛푑 휆 = 1.5. Reliability function of EIKD is given by 푅(푥) = 1 − {1 − (1 + 푥)−훼}훽휆 (1.3) Then, the Hazard rate function of EIKD is ( )−(훼+1) −훼 훽휆−1 퐻(푥) = 훼훽휆 1+푥 (1−(1+푥) ) ; 0 < 푥 < ∞, 훼, 훽, 휆 > 0. (1.4) 1−{1−(1+푥)−훼}훽휆 Distributional Properties Mean = 휇1 = 휆훽 퐵 (1 − 1 , 훽휆) , 훼 > 1. (1.5) 1 훼 2 Variance = 휇 = 휆훽 퐵 (1 − 2 , 훽휆) − {휆훽 퐵 (1 − 1 , 훽휆)} , 훼 > 2. (1.6) 2 훼 훼 Volume 9 Issue 10 - 2019 842 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 −1 ⁄훼 Mode = 푀 = [( 훼+1 ) − 1] (1.7) 표 훼훽휆+1 Quantile Function −1 1 ⁄훼 푄(푢) = [(1 − 푢훽휆) − 1] where 푢~푈(0,1). (1.8) For 푢 = 1 in 푄(푢) 2 −1 1 ⁄훼 Median =[(1 − (0.5)훽휆) − 1] (1.9) Hence, the pdf of EIKD is unimodal, skewed towards right and becomes more peaked with decreasing the value of λ. Order Statistics 푡ℎ The pdf of the 푘 order statistic 푋(푘) from EIKD is obtained as 푛! 푓 (푘) = 훼훽휆(1 + 푥)−(훼+1){1 − (1 + 푥)−훼}훽휆푘−1 [1 − {1 − (1 + 푥)−훼}훽휆] 푛−푘 푥 (푘−1)!(푛−푘)! (1.10) For 푘 = n, the pdf 푋(푛) is −(훼+1) −훼 푛훽휆−1 푓푥(푛) = 푛훼훽휆(1 + 푥) {1 − (1 + 푥) } (1.11) For 푘 = 1, the pdf of 푋(푛) is given by −(훼+1) −훼 훽휆−1 −훼 훽휆 푛−1 푓푥(1) = 푛훼훽휆(1 + 푥) {1 − (1 + 푥) } [1 − {1 − (1 + 푥) } ] (1.12) The parameter estimations and other distributional properties are thoroughly discussed by Kawsar Fatima, et al. (2018) [2]. Skewed distributions developed for statistical quality control methods by many authors. Some of them are Edgeman (1989) [3]- Inverse Gaussian Distribution, Gonzalez and Viles (2000) [4]-Gamma Distribution, Kantam and Sriram (2001) [5]-Gamma Distribution, Kantam et al (2006) [7]-Log logistic Distribution, Subba Rao and Kantam (2008) [8]- double exponential distribution, Srinivasa Rao and Sarth Babu (2012) [10]-Linear Failure Rate Model, Srinivasa Rao and Kantam (2012) [11]-Half Logistic Distribution, Srinivasa Rao et al (2015) [12]-New Weibull Pareto Distribution, Srinivasa Rao et al (2016) [13]-Half Normal Distribution, and Rosaiah et al(2018) [14]- Gumbel Distribution and references there in. Since EIKD is a skewed distribution, this paper makes an effort to study in a comparative way. EIKD is also a model of skewed distribution which was not considered to develop the control charts. EIKD is also an adequate probability model for life testing and reliability studies. Therefore, if a lifetime data is regarded as a quality Volume 9 Issue 10 - 2019 843 www.joics.org Journal of Information and Computational Science ISSN: 1548-7741 data, construction of control charts is desirable for the use by specialists. In this paper we made an attempt to deal with this situation and solve it to the maximum extent. The remaining paper is designed as follows. The fundamental concept and the construction of control charts for the statistics- mean, median, standard deviation, range and midrange are imparted in Section 2. The relative comparison of the prepared control limits with the Shewhart limits is presented in Section 3. Summary and conclusions are provided in Section 4. 2. PERCENTILE CONSTANTS OF THE CONTROL CHARTS 2.1. Mean chart (풙̅ – chart) Let 푥1, 푥2, 푥3, … … 푥푛 be a random sample with n observations is assumed to have been taken from EIKD with α= 2.5, β=2.25 and 휆=1.5. If it is considered that the sample is a sub group of an industrial process data to estimate the true average, the sample mean ‘ 푥̅ ’ provides whether the process mean is around the specified average or not, through repetitive sampling. Statistically saying, we have to identify the ‘most probable’ limits in which the sample means 푥̅ lies. Here the term ‘most probable’ is a relevant concept that is used in the population sense. Since the current procedure in the construction of control charts are for normal distribution only, the theory of 3σ limits is considered as the ‘most probable’ limits. It is usually aware that these 3σ limits of normal distribution gives the probability of 0.9973 that any sample value falls within that limits. Hence, we must develop two limits of the sampling distribution of sample mean in EIKD in such a way that the probability that any sample mean falls within these limits is 0.9973. Symbolically, it is required to obtain L and U so that P(L ≤ 푥̅ ≤ U ) = 0.9973 (2.1.1) where 푥̅ is the sample mean of size n. With the assumption of equi-tailed principle, L and U are considered 0.00135 and 0.99865 percentile factors respectively of the sampling distribution of the sample mean 푥̅ . We resorted to the empirical sampling distribution of 푥̅ over simulation there by calculating its percentile values. These are given in the Table 1. Table 1: Percentile values of Mean in EIKD n 0.99865 0.99 0.975 0.95 0.05 0.025 0.01 0.00135 2 14.5772 7.1579 5.0452 3.7924 0.4034 0.3317 0.2620 0.1688 3 11.4804 6.1088 4.4578 3.4571 0.5055 0.4298 0.3543 0.2529 4 10.2557 5.5858 4.0784 3.2698 0.5731 0.4971 0.4202 0.3069 5 9.2143 4.9766 3.8234 3.0775 0.6356 0.5614 0.4847 0.3796 6 8.2078 4.7438 3.6386 2.9766 0.6753 0.6015 0.5280 0.4001 7 7.9036 4.5151 3.4702 2.8626 0.7172 0.6438 0.5704 0.4618 8 7.2808 4.2443 3.3617 2.8082 0.7510 0.6790 0.6110 0.5006 9 6.6885 4.0334 3.2601 2.7501 0.7741 0.6998 0.6327 0.5158 10 6.5647 3.8697 3.1929 2.6878 0.7980 0.7288 0.6520 0.5462 To calculate the control limits for sample mean, the percentile values obtained in the above table are utilized in the following way.
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