Double moving average–EWMA control chart for exponentially distributed quality

A control chart is an ever-popular tool for monitoring the production process. The early detection of a process shift, if any, is the desire of the quality control personnel. In this article, an effective alternative control charting procedure has been developed for the monitoring of exponentially distributed quality characteristic using the double moving average combined with EWMA statistic. The performance of the proposed control chart is examined for different combinations of the shift constant, the EWMA smoothing parameter, the moving average span, and the target in-control average run lengths. It has been observed that the proposed control chart is more efficient in the detection of process shifts as compared to control chart suggested by Khoo and Wang for the same purpose. The proposed control chart is illustrated for practical usage with the help of a synthetic and a real dataset.     

Statistical inference procedure for the mean–variance efficient frontier with estimated parameters

We suggest finite sample tests for the location of the efficient frontier with the estimated parameters in mean–variance space. The exact densities of the test statistics are derived. We implement the introduced testing procedure empirically by considering monthly returns of ten developed stock markets. It is shown that ignoring the uncertainty about the estimated parameters leads to a more frequent reconstruction of the efficient frontier.     

A mixed GWMA–CUSUM control chart for monitoring the process mean

The memory-type control charts are widely used in the process and service industries for monitoring the production processes. The reason is their sensitivity to quickly react against the small process disturbances. Recently, a new cumulative sum (CUSUM) chart has been proposed that uses the exponentially weighted moving average (EWMA) statistic, called the EWMA–CUSUM chart. Similarly, in order to further enhance the sensitivity of the EWMA–CUSUM chart, we propose a new CUSUM chart using the generally weighted moving average (GWMA) statistic, called the GWMA–CUSUM chart, for efficiently monitoring the process mean. The GWMA–CUSUM chart encompasses the existing CUSUM and EWMA–CUSUM charts. Extensive Monte Carlo simulations are used to explore the run length profiles of the GWMA–CUSUM chart. Based on comprehensive run length comparisons, it turns out that the GWMA–CUSUM chart performs substantially better than the CUSUM, EWMA, GWMA, and EWMA–CUSUM charts when detecting small shifts in the process mean. An illustrative example is also presented to explain the implementation and working of the EWMA–CUSUM and GWMA–CUSUM charts.     

Goodness—of—fit for the two—parameter weibull distribution with estimated parameters

Goodness—of—fit statistics based on the empirical distribution function (EDF) are not distribution—free when parameters for the hypothesized distribution are estimated. Tables are percentile values of several EDF statistics are available for the two—parameter Weibull distribution when parameters are estimated by maximum likelihood. To determine how these tabled values change when simpler estimators are employed, percentile scores for EDF goodness—of—fit tests were obtained by Monte—Carlo simulation for maximum likelihood estimators (MLEs), good linear unbiased estimators (GLUEs), and modified Cramer—von Mises, Anderson—Darling, and Watson statistics are presented for GLUEs for both complete and censored samples. Critical values for Kolmogorov—Smirnov statistics were less affected by the method of estimation than were closer for MLEs and MGLUEs than for MGLUEs and GLUEs. On the other hand, MGLUE and GLUE results were much more similar to each other than to the MLE results when censoring was light and sample sizes were large.     

Optimal design for the Bradley–Terry paired comparison model

Paired comparisons are a popular tool for questionnaires in psychological marketing research. The quality of the statistical analysis of the responses heavily depends on the design, i.e. the choice of the alternatives in the comparisons. In this paper we show that the structure of locally optimal designs changes substantially with the parameters in the underlying utility. This fact is illustrated by elementary examples, where the optimal designs can be completely characterized. As an alternative maximin efficient designs are proposed which perform well for all parameter settings. Research supported by grant Ho 1286 of the German Research Council (Deutsche Forschungsgemeinschaft).     

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.     

Optimal design of repetitive group sampling plans for Weibull and gamma distributions with applications and comparison to the Birnbaum–Saunders distribution

In this paper, we propose a multiple deferred state repetitive group sampling plan which is a new sampling plan developed by incorporating the features of both multiple deferred state sampling plan and repetitive group sampling plan, for assuring Weibull or gamma distributed mean life of the products. The quality of the product is represented by the ratio of true mean life and specified mean life of the products. Two points on the operating characteristic curve approach is used to determine the optimal parameters of the proposed plan. The plan parameters are determined by formulating an optimization problem for various combinations of producer's risk and consumer's risk for both distributions. The sensitivity analysis of the proposed plan is discussed. The implementation of the proposed plan is explained using real-life data and simulated data. The proposed plan under Weibull distribution is compared with the existing sampling plans. The average sample number (ASN) of the proposed plan and failure probability of the product are obtained under Weibull, gamma and Birnbaum–Saunders distributions for a specified value of shape parameter and compared with each other. In addition, a comparative study is made between the ASN of the proposed plan under Weibull and gamma distributions.     

Economic design of VSI X̄ control chart with correlated non-normal data under multiple assignable causes

It has been recently revealed that the Shewhart control charts with variable sampling interval (VSI) perform better than the traditional Shewhart chart with the fixed sampling interval in detecting shifts in the process. In most of these research works, the normality and independency of the process data or measurements are assumed and that the process is subjected to only one assignable cause. While, in practice, these assumptions usually do not hold, some recent studies are focused on working with only one or two of these violations. In this paper, the situation in which the process data are correlated and follow a non-normal distribution and that there is multiplicity of assignable causes in the process is considered. For this case, a cost model for the economic design of the VSI X? control chart is developed, where the Burr distribution is employed to represent the non-normal distribution of the process data. To obtain the optimal values of the design parameters, a genetic algorithm is employed in which the response surface methodology is applied. A numerical example is presented to show the applicability and effectiveness of the proposed methodology. Sensitivity analysis is also carried out to evaluate the effects of cost and input parameters on the performance of the chart.     

A test for suitability of the preliminary samples for constructing control limits of X¯ chart

Unless the preliminary m subgroups of small samples are drawn from a stable process, the estimated control limits of X¯ chart in phase I can be erroneous, due to which the performance of the chart in phase II can be significantly affected. In this work, a quantitative approach based on extraction of the shape features of control chart patterns in the X¯ chart is proposed for evaluating the stability of the process mean, while the preliminary samples were drawn and thus, the subjectivity associated with the visual analysis of the patterns is eliminated. The effectiveness of the test procedure is evaluated using simulated data. The results show that the proposed approach can be very effective for m≥48. The power of the test can be improved by identifying a new feature that can more efficiently discriminate the cyclic pattern of smaller periodicity from the natural pattern and by redefining the test statistic.     

Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Maximum of the weighted Kaplan–Meier tests for the two-sample censored data

In this paper, some versatile test procedures are considered, which are useful for the case where the association of survival functions is unclear. These procedures are based on a maximum of a class of the weighted Kaplan–Meier statistics. The weight functions used in the procedures account for both the censored and non-censored data points. Numerical simulations with these weight functions show that, under various levels of censoring, the proposed procedures perform well across a wide range of alternative configurations. Implementation of the proposed procedures is illustrated in a real data example.     

z Test for the significance of the mean of a stable probability distribution with 1<α≤2

The analysis of data using a stable probability distribution with tail parameter α<2 (sometimes called a Pareto–Levy distribution) seems to have been avoided in the past in part because of the lack of a significance test for the mean, even though it appears to be the correct distribution to use for describing returns in the financial markets. A z test for the significance of the mean of a stable distribution with tail parameter 1<α≤2 is defined. Tables are calculated and displayed for the 5% and 1% significance levels for a range of tail and skew parameters α and β. Through the use of maximum likelihood estimates, the test becomes a practical tool even when α and β are not that accurately determined. As an example, the z test is applied to the daily closing prices for the Dow Jones Industrial average from 2 January 1940 to 19 March 2010.     

Approximate confidence intervals for the weighted kappa coefficient of a binary diagnostic test subject to a case–control design

Case–control design to assess the accuracy of a binary diagnostic test (BDT) is very frequent in clinical practice. This design consists of applying the diagnostic test to all of the individuals in a sample of those who have the disease and in another sample of those who do not have the disease. The sensitivity of the diagnostic test is estimated from the case sample and the specificity is estimated from the control sample. Another parameter which is used to assess the performance of a BDT is the weighted kappa coefficient. The weighted kappa coefficient depends on the sensitivity and specificity of the diagnostic test, on the disease prevalence and on the weighting index. In this article, confidence intervals are studied for the weighted kappa coefficient subject to a case–control design and a method is proposed to calculate the sample sizes to estimate this parameter. The results obtained were applied to a real example.     

An improved method of estimation for the parameters of the Birnbaum–Saunders distribution

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution, frequently used for analysing lifetime data. In this paper, we propose a simple method of estimation for the parameters of the two-parameter BS distribution by making use of some key properties of the distribution. Compared with the maximum likelihood estimators and the modified moment estimators, the proposed method has smaller bias, but having the same mean square errors as these two estimators. We also discuss some methods of construction of confidence intervals. The performance of the estimators is then assessed by means of Monte Carlo simulations. Finally, an example is used to illustrate the method of estimation developed here.     

Statistical inference for α-series process with the inverse Gaussian distribution

Statistical inferences for the geometric process (GP) are derived when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). An α-series process, as a possible alternative to the GP, is introduced since the GP is sometimes inappropriate to apply some reliability and scheduling problems. In this study, statistical inference problem for the α-series process is considered where the distribution of first occurrence time is IG. The estimators of the parameters α, μ, and σ² are obtained by using the maximum likelihood (ML) method. Asymptotic distributions and consistency properties of the ML estimators are derived. In order to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators, Monte Carlo simulations are performed. The results showed that the ML estimators are more efficient than the MM estimators. Moreover, two real life datasets are given for application purposes.     

Three-stage and ‘accelerated’ sequential procedures for the mean of a normal population with known coefficient of variation

Three-stage and ‘accelerated’ sequential procedures are developed for estimating the mean of a normal population when the population coefficient of variation (CV) is known. In spite of the usual estimator, i.e. the sample mean, Searls' (1964 Searls, DT. (1964). The utilization of a known coefficient of variation in the estimation procedure. J. Amer. Statist. Assoc, 50: 1225–1226.  ) estimator is utilized for the estimation purpose. It is established that Searls' estimator dominates the sample mean under the two sampling schemes.     

Inference for the common mean of several Birnbaum–Saunders populations

The Birnbaum–Saunders distribution is a widely used distribution in reliability applications to model failure times. For several samples from possible different Birnbaum–Saunders distributions, if their means can be considered as the same, it is of importance to make inference for the common mean. This paper presents procedures for interval estimation and hypothesis testing for the common mean of several Birnbaum–Saunders populations. The proposed approaches are hybrids between the generalized inference method and the large sample theory. Some simulation results are conducted to present the performance of the proposed approaches. The simulation results indicate that our proposed approaches perform well. Finally, the proposed approaches are applied to analyze a real example on the fatigue life of 6061-T6 aluminum coupons for illustration.     

An exact Kolmogorov–Smirnov test for the Poisson distribution with unknown mean

We develop an exact Kolmogorov–Smirnov goodness-of-fit test for the Poisson distribution with an unknown mean. This test is conditional, with the test statistic being the maximum absolute difference between the empirical distribution function and its conditional expectation given the sample total. Exact critical values are obtained using a new algorithm. We explore properties of the test, and we illustrate it with three examples. The new test seems to be the first exact Poisson goodness-of-fit test for which critical values are available without simulation or exhaustive enumeration.