Nonparametric control charts based on order statistics: Some advances

ABSTRACT

In this article, we introduce new nonparametric Shewhart-type control charts that take into account the location of two order statistics of the test sample as well as the number of observations in that sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution and the average run length (ARL) are all derived. A key advantage of the new charts is that, due to its nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the proposed charts for some typical FAR and ARL values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations, while they seem preferable from a robustness point of view in comparison with the distribution-free control chart of Balakrishnan et al. (2009).     

Empirical Bayes vs. fully Bayes variable selection

For the problem of variable selection for the normal linear model, fixed penalty selection criteria such as AIC, _Cp${\mathrm{C}}_p$, BIC and RIC correspond to the posterior modes of a hierarchical Bayes model for various fixed hyperparameter settings. Adaptive selection criteria obtained by empirical Bayes estimation of the hyperparameters have been shown by George and Foster [2000. Calibration and Empirical Bayes variable selection. Biometrika 87(4), 731–747] to improve on these fixed selection criteria. In this paper, we study the potential of alternative fully Bayes methods, which instead margin out the hyperparameters with respect to prior distributions. Several structured prior formulations are considered for which fully Bayes selection and estimation methods are obtained. Analytical and simulation comparisons with empirical Bayes counterparts are studied.     

Generalized choice models for categorical time series

The NDARMA models of Jacobs and Lewis (1983) allow the modeling of categorical processes with an ARMA-like serial dependence structure. These models can be represented through a backshift mechanism, and we analyze marginal and bivariate properties of the resulting backshift process. Motivated by this backshift mechanism, we define the new class of generalized choice (GC) models, which include the usual NDARMA models as a special case, and we derive results describing the marginal and bivariate distribution of the GC model. We discuss implications concerning DMA(∞) models and the serial dependence structure of NDARMA models. Examples show that the family of GC models allows creating sparsely parametrized models for categorical processes with different types of serial dependence structure.     

A test statistic for weighted runs

A new test statistic based on runs of weighted deviations is introduced. Its use for observations sampled from independent normal distributions is worked out in detail. It supplements the classic χ² test which ignores the ordering of observations and provides additional sensitivity to local deviations from expectations. The exact distribution of the statistic in the non-parametric case is derived and an algorithm to compute p-values is presented. The computational complexity of the algorithm is derived employing a novel identity for integer partitions.     

A note on Läuter-type robust design for polynomial regression models

In this paper, we consider the problem of model robust design for simultaneous parameter estimation among a class of polynomial regression models with degree up to k. A generalized D-optimality criterion, the Ψ_α‐optimality criterion, first introduced by Läuter (1974) is considered for this problem. By applying the theory of canonical moments and the technique of maximin principle, we derive a model robust optimal design in the sense of having highest minimum Ψ_α‐efficiency. Numerical comparison indicates that the proposed design has remarkable performance for parameter estimation in all of the considered rival models.     

Conditional quantile estimation with auxiliary information for left-truncated and dependent data

In this paper, the empirical likelihood method is used to define a new estimator of conditional quantile in the presence of auxiliary information for the left-truncation model. The asymptotic normality of the estimator is established when the data exhibit some kind of dependence. It is assumed that the lifetime observations with multivariate covariates form a stationary α‐mixing sequence. The result shows that the asymptotic variance of the proposed estimator is not larger than that of standard kernel estimator. Finite sample behavior of the estimator is investigated via simulations too.     

The Design of and R Control Charts for Skew Normal Distributed Data

The Shewhart-type control chart is traditionally developed under the normality assumption. In practice, however, this assumption may not hold. Because the skew normal distribution represents a broad distribution class and is more flexible than is the normal distribution, we propose two new control charts to monitor process mean and spread for skew normal distributed data. Moreover, to facilitate practical implementation, tables of charting constants are provided. We conducted simulation studies to compare the false alarm rates, and the results show that new proposed charts perform better than others as skewness increases. Finally, an illustrative example is provided.     

A comparative study on a permutation statistic

We studied asymptotic distribution and finite sample properties of a randomly weighted permutation statistic. The asymptotic normality and the finite sample simulations derived from our studies provided theoretical and numerical justifications for distributional assumption of many useful test statistics used in identifying spatial autocorrelations of mapped data. We compared a new method in computing the mean and the approximated variance of the randomly weighted D statistic, a special permutation statistic, with the Walter’s conditional method. In the numerical illustration of the method, we calculated the standardized values of the D statistic by subtracting the mean from the D statistic and dividing the difference by the standard deviation for the standardized mortality ratios (SMRs) and the life expectancies among the 48 states of the continental USA. Spatial autocorrelations of the SMRs and the life expectancies were found to be statistically significant.     

Statistical analysis for rounded data

When random variables do not take discrete values, observed data are often the rounded values of continuous random variables. Errors caused by rounding of data are often neglected by classical statistical theories. While some pioneers have identified and made suggestions to rectify the problem, few suitable approaches were proposed. In this paper, we propose an approximate MLE (AMLE) procedure to estimate the parameters and discuss the consistency and asymptotic normality of the estimates. For our illustration, we shall consider the estimates of the parameters in AR(p)$\mathit{AR}(p)$ and MA(q)$\mathit{MA}(q)$ models for rounded data.     

A class of distribution-free control charts

Summary.  A class of Shewhart-type distribution-free control charts is considered. A key advantage of these charts is that the in-control run length distribution is the same for all continuous process distributions. Exact expressions for the run length distribution and the average run length (ARL) are derived and properties of the charts are studied via evaluations of the run length distribution probabilities and the ARL. Tables are provided for implementation for some typical ARL values and false alarm rates. The charts proposed are preferable from a robustness point of view, have attractive ARL properties and would be particularly useful in situations where one uses a classical Shewhart   X -chart. A numerical illustration is given.     

Mixed multivariate EWMA-CUSUM control charts for an improved process monitoring

Multivariate exponential weighted moving average and cumulative sum charts are the most common memory type multivariate control charts. They make use of the present and past information to detect small shifts in the process parameter(s). In this article, we propose two new multivariate control charts using a mixed version of their design setups. The plotting statistics of the proposed charts are based on the cumulative sum of the multivariate exponentially weighted moving averages. The performances of these schemes are evaluated in terms of average run length. The proposals are compared with their existing counterparts, including HotellingT², MCUSUM, MEWMA, and MC1 charts. An application example is also presented for practical considerations using a real dataset.     

A note on the queue with a random set-up time under a restricted admissibility policy with a Bernoulli vacation schedule

This paper deals with the queue size distribution of an M^x/G/1 queue with a random set-up time and with a Bernoulli vacation schedule under a restricted admissibility policy. This generalizes the model studied by Madan and Choudhury [Sankhyá 66 (2004) 175–193].     

Robust multivariate Bayesian forecasting under functional distortions in the -metric

This paper investigates robustness of multivariate forecasting in the Bayesian framework. The minimax approach is used to construct robust statistical procedures under deviations from hypothetical assumptions. The deviations are defined as functional distortions using the χ²-pseudo-metric. Two cases of deviations are considered: distortions of parameter distribution and distortions of joint distribution of observations and parameters. Explicit forms for the guaranteed upper risk functional are obtained and integral equations for robust prediction statistics are given for both cases.     

New 3-level response surface designs constructed from incomplete block designs

Box and Behnken [1958. Some new three level second-order designs for surface fitting. Statistical Technical Research Group Technical Report No. 26. Princeton University, Princeton, NJ; 1960. Some new three level designs for the study of quantitative variables. Technometrics 2, 455–475.] introduced a class of 3-level second-order designs for fitting the second-order response surface model. These 17 Box–Behnken designs (BB designs) are available for 3–12 and 16 factors. Although BB designs were developed nearly 50 years ago, they and the central-composite designs of Box and Wilson [1951. On the experimental attainment of optimum conditions. J. Royal Statist. Soc., Ser. B 13, 1–45.] are still the most often recommended response surface designs. Of the 17 aforementioned BB designs, 10 were constructed from balanced incomplete block designs (BIBDs) and seven were constructed from partially BIBDs (PBIBDs). In this paper we show that these seven BB designs constructed from PBIBDs can be improved in terms of rotatability as well as average prediction variance, D$D$- and G$G$-efficiency. In addition, we also report new orthogonally blocked solutions for 5, 8, 9, 11 and 13 factors. Note that an 11-factor BB design is available but cannot be orthogonally blocked. All new designs can be found at http://www.math.montana.edu/∼$\sim$jobo/bbd/.     

CUSUM chart for detecting range shifts when monotonicity of likelihood ratio is invalid

It is often encountered in the literature that the log-likelihood ratios (LLR) of some distributions (e.g. the student t distribution) are not monotonic. Existing charts for monitoring such processes may suffer from the fact that the average run length (ARL) curve is a discontinuous function of control limit. It implies that some pre-specified in-control (IC) ARLs of these charts may not be reached. To guarantee the false alarm rate of a control chart lower than the nominal level, a larger IC ARL is usually suggested in the literature. However, the large IC ARL may weaken the performance of a control chart when the process is out-of-control (OC), compared with a just right IC ARL. To overcome it, we adjust the LLR to be a monotonic one in this paper. Based on it, a multiple CUSUM chart is developed to detect range shifts in IC distribution. Theoretical result in this paper ensures the continuity of its ARL curve. Numerical results show our proposed chart performs well under the range shifts, especially under the large shifts. In the end, a real data example is utilized to illustrate our proposed chart.     

A two phases queueing system with Bernoulli vacation schedule under multiple vacation policy

This paper deals with a single server Poisson arrival queue with two phases of heterogeneous service along with a Bernoulli schedule vacation model, where after two successive phases service the server either goes for a vacation with probability p (0≤p≤1) or may continue to serve the next unit, if any, with probability q(=1−p). Further the concept of multiple vacation policy is also introduced here. We obtained the queue size distributions at a departure epoch and at a random epoch, Laplace Stieltjes Transform of the waiting time distribution and busy period distribution along with some mean performance measures. Finally we discuss some statistical inference related issues.     

Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics

In this article we propose three distribution-free (or nonparametric) statistical quality control charts for monitoring a process center when an in-control target center is not specified. These charts are of the Shewhart-type, the exponentially moving average-type, and the cumulative sum-type. The constructions of the proposed charts require the availability of an initial reference sample taken when the process was operating in-control to calculate an estimator for the unknown in-control target process center. This estimated center is then used in the calculation of signed-rank-like statistics based on grouped observations taken periodically from the process output. As long as the in-control process underlying distribution is continuous and symmetric, the proposed charts have a constant in-control average run length and a constant false alarm rate irrespective of the process underlying distribution. Other advantages of the proposed distribution-free charts include their robustness against outliers and their superior efficiency over the traditional normal-based control charts when applied to processes with moderate- or heavy-tailed underlying distributions, such as the double exponential or the Cauchy distributions.