Optimal design of the double sampling X¯ chart with estimated parameters based on median run length

The double sampling (DS) $\overline{X}$ chart when the process parameters are unknown and have to be estimated from a reference Phase-I dataset is studied. An expression for the run length distribution of the DS $\overline{X}$ chart is derived, by conditioning and taking parameter estimation into account. Since the shape and the skewness of the run length distribution change with the magnitude of the mean shift, the number of Phase-I samples and sample sizes, it is shown that the traditional chart’s performance measure, i.e. the average run length, is confusing and not a good representation of a typical chart’s performance. To this end, because the run length distribution is highly right-skewed, especially when the shift is small, it is argued that the median run length (MRL) provides a more intuitive and credible interpretation. From this point of view, a new optimal design procedure for the DS $\overline{X}$ chart with known and estimated parameters is developed to compute the chart’s optimal parameters for minimizing the out-of-control MRL, given that the values of the in-control MRL and average sample size are fixed. The optimal chart which provides the quickest out-of-control detection speed for a specified shift of interest is designed according to the number of Phase-I samples commonly used in practice. Tables are provided for the optimal chart parameters along with some empirical guidelines for practitioners to construct the optimal DS $\overline{X}$ charts with estimated parameters. The optimal charts with estimated parameters are illustrated with a real application from a manufacturing company.     

The variable sampling interval run sum X‾ control chart

Traditional control charts for process monitoring are based on taking samples from the process at fixed length sampling intervals. More recently, research works focused on the use of variable sampling intervals (VSIs), where the lengths of the sampling intervals are varied according to the process quality. A short sampling interval is considered when the process quality indicates a possible out-of-control situation while a long sampling interval is considered, otherwise. In this paper, the VSI run sum (RS) $\bar{X}$ chart is proposed with its optimal scores and parameters determined using an optimization technique to minimize the out-of-control average time to signal (ATS) or the adjusted average time to signal (AATS). A Markov-chain method is used to evaluate both the ATS and AATS of the proposed chart, for the zero and steady state cases, respectively. Results show that the VSI RS $\bar{X}$ chart is considerably more efficient than the basic RS $\bar{X}$ chart. The VSI RS $\bar{X}$ chart performs generally well compared with other competing charts, such as the standard $\bar{X}$, synthetic $\bar{X}$, exponentially weighted moving average (EWMA) $\bar{X}$, VSI $\bar{X}$ and VSI EWMA $\bar{X}$ charts. The sensitivity of the VSI RS $\bar{X}$ chart can be enhanced further by adding more scoring regions or a head-start feature. An illustrative example is presented to explain the implementation of the proposed VSI RS $\bar{X}$ chart.     

Radial Basis Function generated Finite Differences for option pricing problems

In this paper we present a numerical method to price options based on Radial Basis Function generated Finite Differences (RBF-FD) in space and the Backward Differentiation Formula of order 2 (BDF-2) in time. We use Gaussian RBFs that depend on a shape parameter $\epsilon$. The choice of this parameter is crucial for the performance of the method. We chose $\epsilon$ as $\mathrm{const}?h^{?1}$ and we derive suitable values of the constant for different stencil sizes in 1D and 2D. This constant is independent of the problem parameters such as the volatilities of the underlying assets and the interest rate in the market. In the literature on option pricing with RBF-FD, a constant value of the shape parameter is used. We show that this always leads to ill-conditioning for decreasing $h$, whereas our proposed method avoids such ill-conditioning. We present numerical results for problems in 1D, 2D, and 3D demonstrating the useful features of our method such as discretization sparsity, flexibility in node placement, and easy dimensional extendability, which provide high computational efficiency and accuracy.     

OWave control chart for monitoring the process mean

In this paper a control chart for monitoring the process mean, called OWave (Orthogonal Wavelets), is proposed. The statistic that is plotted in the proposed control chart is based on weighted wavelets coefficients, which are provided through the Discrete Wavelets Transform using Daubechies $\mathit{db}2$ wavelets family. The statistical behavior of the wavelets coefficients when the mean shifts are occurring is presented, and the distribution of wavelets coefficients in the case of normality and independence assumptions is provided. The on-line algorithm of implementing the proposed method is also provided. The detection performance is based on simulation studies, and the comparison result shows that OWave control chart performs slightly better than Fixed Sample Size and Sampling Intervals control charts ($\overline{X}$, EWMA, CUSUM) in terms of Average Run Length. In addition, illustrative examples of the new control chart are presented, and an application to Tennessee Eastman Process is also proposed.