Because the normal process capability indices (PCIs) Cp, Cpu, Cpl, and Cpk represent the times that the process standard deviation is within the specification limits; then, based on and by using the direct relations among the parameters of the Weibull, Gumbel (minimum extreme value type I) and lognormal distributions, the Weibull and lognormal PCIs are derived in this paper. On the other hand, because the proposed PCIs Pp, Ppu, Ppl, and Ppk were derived as a function of the mean and standard deviation of the analyzed process, they have the same practical meaning with those of the normal distribution. Results show that the proposed PCIs could be used as the standard Cp, Cpu, Cpl, and Cpk if a short-term variance is analyzed. An application to a set of simulated data is presented. Copyright © 2015 John Wiley & Sons, Ltd.

Non‐normal Capability Indices for the Weibull and Lognormal Distributions

The objective of the paper is to modify the existing process capability index (PCI) for a Weibull distribution and to construct bootstrap confidence intervals (BCIs) for the newly proposed index. Three BCIs that consist of standard, percentile and bias-corrected percentile bootstrap (BCPB) confidence intervals are constructed for the newly proposed index and the existing Pearn and Chen index. The efficiency of the newly proposed index \(C_{\mathrm{GPK}} \) is compared with Pearn and Chen index using their coverage probabilities and average widths. The coverage probabilities and average width of three BCIs were calculated using Monte Carlo simulation studies. The newly proposed index shows better performance than Pearn and Chen index. The results indicate that BCPB confidence interval was more efficient in both cases and outperform other two confidence intervals in all situations. The comparison of average width of BCPB apparently shows that the proposed index performed better in all cases. A real-life example is also provided for a practical application.

Bootstrap Confidence Intervals of the Modified Process Capability Index for Weibull distribution

This paper investigates the efficiency of Gini’s mean difference (GMD) as a measure of variability in two commonly used process capability indices (PCIs), i.e., Cp and Cpk. A comparison has been carried out to evaluate the performance of GMD-based PCIs and Pearn and Chen quantile-based PCIs under low, moderate, and high asymmetry using Weibull distribution. The simulation results, under low and moderate asymmetric condition, indicate that GMD-based PCIs are more close to target values than quantile approach. Beside point estimation, nonparametric bootstrap confidence intervals, such as standard, percentile, and bias corrected percentile with their coverage probabilities also have been calculated. Using quantile approach, bias corrected percentile (BCPB) method is more effective for both Cp and Cpk, where as in case of GMD, both BCPB and percentile bootstrap method can be used to estimate the confidence interval of Cp and Cpk, respectively.

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Capability Indices for Non-Normal Distribution Using Gini’s Mean Difference as Measure of Variability

비대칭 모집단에 대한 공정능력지수의 개발 ( Process Capability Indices for Skewed Populations )

In this study, the volatility of two Asian stock markets, Bursa Malaysia and Singapore Exchange, is estimated. The analysis used data on daily closing prices of the indices of the respective stock markets between July 1, 2019 and August 31, 2020. The sample is split into two subsample periods: Pre-COVID-19 pandemic and during the COVID-19 pandemic. We estimated a standard GARCH, GARCH-M, TGARCH, EGARCH and PGARCH model for each subsample. We chose the best GARCH that yielded the lowest Schwarz information criterion for the normal, skewed normal, Student’s t-distribution, skewed Student’s t-distribution, generalized error distribution (GED) and skewed GED. The results show that both stock market returns are quite persistent, and the persistence decreases for both stock market returns during the pandemic. Furthermore, the normal distribution performed well for Malaysian and Singaporean stock markets before the pandemic and switched to a Student’s t (skewed normal) during the pandemic. The standard GARCH(1,1), GARCH-M(1,1), and EGARCH(1,1) performed well for both stock market returns, and the EGARCH indicates the presence of the leverage effect when stock market returns are negatively correlated to its volatility.

https://archive.aessweb.com/index.php/5002/article/download/2067/3296

The impact of covid-19 pandemic on stock market return volatility: Evidence from Malaysia and Singapore

The aim of this paper is to use the General Autoregressive Conditional Heteroscedastic (GARCH) type models for the estimation of volatility of the daily returns of the Kenyan stock market: that is Nairobi Securities Exchange (NSE). The conditional variance is estimated using the data from March 2013 to February 2016. We use both symmetric and asymmetric models to capture the most common features of the stock markets like leverage effect and volatility clustering. The results show that the volatility process is highly persistent, thus, giving evidence of the existence of risk premium for the NSE index return series. This in turn supports the positive correlation hypothesis: that is between volatility and expected stock returns. Another fact revealed by the results is that the asymmetric GARCH models provide better fit for NSE than the symmetric models. This proves the presence of leverage effect in the NSE return series.

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Modeling Stock Market Volatility Using GARCH Models: A Case Study of Nairobi Securities Exchange (NSE)

Process capability analysis is used to determine the process performance as capable or incapable within a specified tolerance. Basic indices Cp, Cpk, Cpm, Cpmk initially developed for normally distributed processes showed inappropriate for processes with non-normal distributions. A number of authors worked on non-normal distributions which were most notably those of Clements, Pearn and Chen, Montgomery and Johnson-Kotz-Pearn (JKP). Obtaining PCIs based on the parameters of non-normal distributions are completely disregarded and ignored. However parameters of some non-normal distributions have significance for knowing the status of process as capable or incapable. In this article we intend to work on the shape parameter of Weibull distribution to calculate PCIs. We work on two data sets for verification and validation purpose. Efficacy of the technique is assessed by bootstrapping the results of estimate and standard error of shape parameter.

/pdf/process-capability-indices-for-shape-parameter-of-weibull-1aik3q8ob2.pdf

Process Capability Indices for Shape Parameter of Weibull Distribution

Process capability indices (PCIs) quantify the ability of a process to produce on target and within specifications performances. Basic indices designed for normal processes gives flawed results for non-normal process. Numerous methods have been proposed for non-normal processes to estimate PCIs in which some of them are based on transformation methods. The Johnson system comprising three types that translate a continuous non-normal distribution to normal. The aim of this paper is to estimate four basic indices for non-normal process using Johnson system with single straightforward procedure. The efficacy of the proposed approach can be assessed for all three Johnson Curves (SB, SU, SL) but result for SU is presented in this paper. PCIs for a data set are estimated and percentiles are obtained by our proposed exact method based on selected Johnson density function which was earlier based on approximate methods without any prior knowledge of density function of non-normal process. We compare our results with other existing methods to estimate PCIs for non-normal process. From statistical analysis we have noted that this modification improve process capability indices.

/pdf/process-capability-indices-under-non-normality-conditions-386g4c53z3.pdf

Process Capability Indices under Non-Normality Conditions using Johnson Systems

Kurtosis is a commonly used descriptive statistics. Kurtosis “Coefficient of excess” is critically reviewed in different aspects and is called as, measuring the fatness of the tails of the density functions, concentration towards the central value, scattering away from the target point or degree of peakedness of probability distribution. Kurtosis is referred to the shape of the distribution but many distributions having same kurtosis value may have different shapes while Kurtosis may exist when peak of a distribution is not in existence. Through extensive study of kurtosis on several distributions, Wu (2002) introduced a new measure called “W-Peakedness” that offers a fine capture of distribution shape to provide an intuitive measure of peakedness of the distribution which is inversely proportional to the standard deviation of the distribution. In this paper the work is extended for different others continuous probability distributions. Empirical results through simulation illustrate the proposed method to evaluate kurtosis by W-peakedness

Suboohi Safdar

Papers

Modeling Stock Market Volatility Using GARCH Models: A Case Study of Nairobi Securities Exchange (NSE)

Process Capability Indices for Shape Parameter of Weibull Distribution

Process Capability Indices under Non-Normality Conditions using Johnson Systems

W-Peakedness: A Simulation based Study