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The objective of this study is to investigate the long-term impact of the COVID-19 pandemic on the South African retail industry sales using the seasonal autoregressive moving average (SARIMA) from the time series analysis tool pack called Box-Jenkins methodology. The model with the best fit to the total monthly retail sales series is the SARIMA(0,1,1)(0,1,0)12 model as it has the lowest values of the model selection and adequacy measures such as the Akaike’s information criterion, Bayesian information criterion, root mean square error and the mean absolute percentage error. This study concludes that the South African retail industry is remarkably resilient sector because while it was unstable during lockdown, the total retail sales recovered to their pre-intervention levels as soon as less strict lockdown levels were implemented.

This study aims to investigate the lingering impact of the COVID-19 pandemic on the South African total monthly wholesale trade sales using time series Box-Jenkins methodology. The SARIMA(2,1,1)(0,1,1)12 model provides the best fit to the SA’s total monthly wholesale trade sales series as it has the lowest Akaike’s information criterion, Bayesian information criterion, root mean square error and mean absolute percentage error values which serve as model selection and model adequacy metrics. The findings of this study show that the South African wholesale industry trade sales were negatively affected by the COVID-19 pandemic but have fully recovered.

This study assesses the reproducibility of a recent publication on the risk quantification of the South African Financial Index (J580) using skewed distributions. That is, four skewed distributions (Burr, exponential, gamma and Weibull) are fitted to the returns (split into losses and gains) of the J580 dataset. In this paper, we redo the analysis in an effort to highlight some of the quantifiable differences in the values of the descriptives, goodness-of-fit and risk measures for all four distributions. In addition, other goodness-of-fit tests are computed for all four distributions to check consistency, and based on this extension, it is observed that the Weibull is a better model for gains due to a majority of the goodness-of-fit test inferring that and yields better risk measures. Finally, the Burr distribution is recommended for losses as it better captures the heavy tail of the loss returns.

Estimating the threshold for extreme values is essential for anticipating and managing rare and impactful events. This paper discusses four different graphical methods of estimating thresholds using three different stock price datasets. The datasets have different levels of volatility (classified as low, medium, and high). For each of the datasets, thresholds are estimated, and a generalised Pareto distribution is then fitted to the exceedances above each threshold. Subsequently, the mean squared error is calculated for each fitted model, which is then used together with the number of exceedances for each respective threshold as criteria to analyse and make inferences on the most suitable threshold approach when using a dataset that has a specified degree of volatility. It was observed that when dealing with a dataset with low volatility, Pickand plot should be considered for threshold setting. When volatility is very moderate or high, using Hill plot to determine thresholds for extreme values is recommended. The motivation for this paper lies in the need to explore and identify the most effective threshold estimation methods when dealing with different levels of stock price volatility.

The Danish fire insurance data is one of the most recognised and well-known datasets in the empirical insurance claims literature. This dataset is used in many textbooks and articles to illustrate the analysis of fire insurance claims, more specifically in the application of heavy-tailed loss distributions and extreme value theory. In this paper, we provide a short review of publications that used the Danish fire insurance data and conduct an additional analysis. Our additional analysis on the Danish fire insurance data involves investigating the: (i) modality issue using appropriate statistical tests and software, (ii) k-means clustering pattern using different techniques, (iii) effect of using a splicing model on the data, and (iv) differences in results that we obtained as compared to what other cited researchers reported in their earlier publications. In short, the objective of this paper is to highlight the importance of the Danish fire claims dataset by showcasing different models where it has been used to verify certain hypotheses in the empirical actuarial field. Additional analyses are also conducted to illustrate its dense usefulness in the actuarial and extremes field, where real-life datasets are scarce because they are often subjected to a lot of proprietary and privacy laws.

In many situations, the quality of a process or product can be characterized by a functional relationship or profile. It is well-known that the independence assumptions of the error terms within or between profiles are not always valid and could be violated due to within or between profile autocorrelation. Since most of the process capability indices (PCIs) have been developed for simple linear profiles (SLPs) without considering autocorrelation, this paper provides some novel methods to analyze the capability of SLP under each of the two different autocorrelation effects separately, as well as the case where both autocorrelation effects are present. We assume that the first-order autoregressive AR(1) model explains the within- and between-profile autocorrelation in error terms. To evaluate the process capability, a new functional index called C p ‴ (P r o f i l e) is introduced for SLP with independent errors, and then it is modified to include the three possible cases of within, between, and simultaneous autocorrelation. The simulation results demonstrate that the proposed schemes outperform existing schemes regarding bias and mean square error (MSE) criteria. Moreover, bootstrap confidence intervals for the proposed index are obtained. Finally, an illustrative example in the chemical industry is used to demonstrate the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Control charts, which are one of the major tools in the Statistical Process Control (SPC) domain, are used to monitor a process over time and improve the final quality of a product through variation reduction and defect prevention. As a novel development of control charts, referred to as profile monitoring, the study variable is not defined as a quality characteristic; it is a functional relationship between some explanatory and response variables which are monitored in such a way that the major aim is to check the stability of this model (profile) over time. Most of the previous works in the area of profile monitoring have focused on the development of different theories and assumptions, but very little attention has been paid to the practical application in real-life scenarios in this field of study. To address this knowledge gap, this paper proposes a monitoring framework based on the idea of profile monitoring as a data-driven method to monitor the internet usage of a telecom company. By definition of a polynomial model between the hours of each day and the internet usage within each hour, we propose a framework with three monitoring goals: (i) detection of unnatural patterns, (ii) identifying the impact of policies such as providing discounts and, (iii) investigation of general social behaviour variations in the internet usage. The results shows that shifts of different magnitudes can occur in each goal. With the aim of different charting statistics such as Hoteling T2 and MEWMA, the proposed framework can be properly implemented as a monitoring scheme under different shift magnitudes. The results indicate that the MEWMA scheme can perform well in small shifts and has faster detection ability as compared to the Hoteling T2 scheme. [ABSTRACT FROM AUTHOR]

The homogeneously weighted moving average (HWMA) control chart is a new memory-type chart that allocates a specific weight to the current sample and the remaining weight is distributed equally to the previous samples. In this paper, the HWMA control chart is proposed for monitoring count data. This chart is based on the Conway–Maxwell (COM) distribution, which can be used to model under-spread and over-spread count data. The performance of the proposed chart is evaluated in terms of the average run-length (ARL), standard deviation of the run-length (SDRL), median run-length (MRL) as well as the expected ARL, SDRL and MRL values for both location and dispersion shifts in the process. The sensitivity of the new control chart is compared with those of some existing well-known COM-Poisson memory-type control charts in terms of the out-of-control ARL. The results of this study reveal that the proposed control chart performs competitively well with the existing charts in detecting shifts in the location and dispersion parameters in many situations. Numerical examples are given to demonstrate the application of the proposed chart. [ABSTRACT FROM AUTHOR]

The use of the 2-of-(H + 1) runs-rules and synthetic schemes to improve the performance of the currently available X ¯ schemes in monitoring the process mean under the combined effect of measurement errors and autocorrelation is proposed. To maximize the detection ability of the 2-of-(H + 1) runs-rules and synthetic schemes, we implement the modified side-sensitive (MSS) design approach for the charting regions as we show it yields the best possible performance out of all the available designs. These new monitoring schemes incorporate the additive model with a constant standard deviation and a first-order autoregressive model to the computation of the control limits in order to account for measurement errors and autocorrelation, respectively. Moreover, to construct a dedicated Markov chain matrix, the abovementioned models and some sampling methods are incorporated into the values of probability elements which are then used to derive closed-form expressions for the zero- and steady-state run-length distribution. A real life example is used to illustrate the practical implementation of the proposed schemes. [ABSTRACT FROM AUTHOR]

In order to reduce the effect of autocorrelation on the X ¯ monitoring scheme, a new sampling strategy is proposed to form rational subgroup samples of size n. It requires sampling to be done such that: (i) observations from two consecutive samples are merged, and (ii) some consecutive observations are skipped before sampling. This technique which is a generalized version of the mixed samples strategy is shown to yield a better reduction of the negative effect of autocorrelation when monitoring the mean of processes with and without measurement errors. For processes subjected to a combined effect of autocorrelation and measurement errors, the proposed sampling technique, together with multiple measurement strategy, yields an uniformly better zero-state run-length performance than its two main existing competitors for any autocorrelation level. However, in steady-state mode, it yields the best performance only when the monitoring process is subject to a high level of autocorrelation, for any given level of measurement errors. A real life example is used to illustrate the implementation of the proposed sampling strategy. [ABSTRACT FROM AUTHOR]

Classical monitoring schemes are typically designed under the assumption of known process parameters, perfect measurements and normality. In real-life applications, these assumptions are often violated. Thus, their Phase II performances are negatively affected by both measurement errors and parameter estimation. In this paper, the performance of the homogenously weighted moving average (HWMA) scheme is investigated under the assumption of unknown process parameters with and without measurement errors using the characteristics of the run-length distribution through intensive simulations. The negative effect of measurement errors is reduced using multiple measurements sampling strategy. It is found that the negative effect of the measurement errors is higher as the smoothing parameter increases and the larger the Phase I sample size, the smaller the negative effect of measurement errors. An illustrative example is given to demonstrate the implementation in real-life applications. [ABSTRACT FROM AUTHOR]

For independent and identically distributed observations, and those with measurement errors only, the adaptive designs (i.e. variable sampling sizes (VSS), variable sampling intervals (VSI) and the latter two combined to form VSSI) have been thoroughly discussed. However, no research exists for processes under the combined effect of autocorrelation and measurement errors; thus, such adaptive Shewhart X ¯ schemes are proposed. The Markov chain approach for adaptive designs are used to evaluate the run-length distribution properties with two special sampling strategies (i.e. s -skip and multiple measurements) incorporated to reduce the combined negative effect of autocorrelation and measurement inaccuracy. Using numerous run-length metrics, it is shown that the combination of the two sampling strategies with the VSSI design reduces this negative effect considerably and improves the detection ability of the X ¯ scheme by a significant margin as compared with using the fixed sample size and sampling interval (FSSI), VSS and VSI designs. Autocorrelation level has a higher negative effect as compared with the measurement inaccuracy level. For high levels of autocorrelation (ϕ > 0.8), the s -skip strategy has little influence in reducing the negative effect; but the VSSI design maintains better performance than the other designs. Finally, a real-life example is used to illustrate its implementation. [ABSTRACT FROM AUTHOR]

Monitoring schemes are typically designed under the assumption of perfect measurements. However, in real-life applications, data tend to be subjected to measurement errors, that is, a difference between the real quantities and the measured ones mostly exist even with highly sophisticated advanced measuring instruments. Thus, in this paper, the negative effect of measurement errors on the performance of the homogenously weighted moving average (HWMA) scheme is studied using the linear covariate error model for constant and linearly increasing variance. Monte Carlo simulations are used to evaluate the performance of the proposed HWMA scheme in terms of the run-length characteristics. It is observed that as the smoothing parameter increases, measurement errors have a higher negative effect on the performance of the HWMA X ¯ scheme. More importantly, it is shown that the negative effect of measurement errors is reduced by using multiple measurements and/or by increasing the slope coefficient of the covariate error model. Moreover, the performance of the HWMA X ¯ scheme is compared with the corresponding exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) X ¯ schemes. An illustrative example is provided to help in implementing this monitoring scheme in a real-life situation. [ABSTRACT FROM AUTHOR]

Side-sensitive Shewhart-type designs have been shown in a variety of process monitoring contexts to increase the performance of traditional charts when the process observations are independent and identically distributed. In this paper, we investigate whether this is also the case when the observations are from an autocorrelated process. Thus, we propose a synthetic chart and a 2-of-(H + 1) runs-rules chart using a side-sensitive design approach and a skipping sampling strategy for monitoring correlated observations from a stationary first-order autoregressive process. The resulting schemes have an improved zero-state and steady-state OOC performance as compared to the currently available non-side-sensitive Shewhart-type synthetic, runs-rules and the basic X ¯ monitoring schemes. A Markov chain technique is used to derive closed-form average run-length expressions for the proposed schemes and overall performance measures are used to evaluate the efficiency of these monitoring schemes. [ABSTRACT FROM AUTHOR]