Christian H. Weiß

Controlling correlated processes of Poisson counts

The class of INARMA models is well suited to model the autocorrelation structure of processes with Poisson marginals arising in context of statistical quality control. After reviewing briefly the basic principles and important members of this broad family of models, we concentrate on the INAR(1) model, which is of particular relevance for quality control. We suggest four approaches to control such count processes, and compare their run length performance in a simulation study. Results show that only some of the out-of-control situations considered can be controlled effectively with the discussed control schemes. Copyright

Compound Poisson INAR(1) processes: Stochastic properties and testing for overdispersion

The compound Poisson INAR(1) model for time series of overdispersed counts is considered. For such CPINAR(1) processes, explicit results are derived for joint moments, for the k-step-ahead distribution as well as for the stationary distribution. It is shown that a CPINAR(1) process is strongly mixing with exponentially decreasing weights. This result is utilized to design a test for overdispersion in INAR(1) processes and to derive its asymptotic power function. An application of our results to a real-data example and a study of the finite-sample performance of the test are presented.

Properties of the exponential EWMA chart with parameter estimation

Count rates may reach very low levels in production processes with low defect levels. In such settings, conventional control charts for counts may become ineffective since the occurrence of many samples with zero defects would cause control statistic to be consistently zero. Consequently, the exponentially weighted moving average (EWMA) control chart to monitor the time between successive events (TBE) or counts has been introduced as an effective approach for monitoring processes with low defect levels. When the counts occur according to a Poisson distribution, the TBE observations are distributed as exponential. Although the assumption of exponential distribution is a reasonable choice as a model of TBE observations, its parameter, i.e. the mean (also the standard deviation), is rarely known in practice and its estimate is used in place of the unknown parameter when constructing the exponential EWMA chart. In this article, we investigate the effects of parameter estimation on the performance measures (average run length, standard deviation, and percentiles of the run length distribution) of the exponential EWMA control chart. A comprehensive analysis of the conditional performance measures of the chart shows that the effect of estimation can be serious, especially if small samples are used. An investigation of the marginal performance measures, which are calculated by averaging the conditional performance measures over the distribution of the parameter estimator, allows us to provide explicit sample size recommendations in constructing these charts to reach a satisfactory performance in both the in-control and the out-of-control situation. Copyright

A New Class of Autoregressive Models for Time Series of Binomial Counts

The binomial AR(1) model of McKenzie (1985) for time series of binomial counts has a well-interpretable structure and applies well to several real-world problems. After a brief review of important properties of this model, we propose and investigate a new class of pth order autoregressive models, which coincide with the binomial AR(1) model for p = 1. Special cases of this new model family are discussed, each having a different autocorrelation structure. A real-data example demonstrates that these higher-order models have a great potential to be applied in practice.

The INARCH(1) Model for Overdispersed Time Series of Counts

The INARCH(1) model for overdispersed time series of counts has a simple structure, a parsimonious parametrization, and a great potential for applications in practice. We analyze two approaches to approximate the marginal process distribution: a Markov chain approach and the Poisson–Charlier expansion. Then approaches for estimating the two model parameters are discussed. We derive explicit expressions for the asymptotic distribution of the maximum likelihood and conditional least squares estimators. They are used for constructing simultaneous confidence regions, the finite-sample performance of which is analyzed in a simulation study. A real-data example from economics illustrates the application of the INARCH(1) model.

Thinning-based models in the analysis of integer-valued time series: a review:

This article aims at providing a comprehensive survey of recent developments in the field of integer-valued time series modelling, paying particular attention to models obtained as discrete counterparts of conventional autoregressive moving average and bilinear models, and based on the concept of thinning. Such models have proven to be useful in the analysis of many real-world applications ranging from economy and finance to medicine. We review the literature of the most relevant thinning operators proposed in the analysis of univariate and multivariate integer-valued time series with either finite or infinite support. Finally, we also outline and discuss possible directions of future research.

The Poisson INAR(1) CUSUM chart under overdispersion and estimation error

The Poisson INAR(1) CUSUM chart has been proposed to monitor integer-valued autoregressive processes of order 1 with Poisson marginals. The effectiveness of this chart has been shown under the assumptions of Poisson marginals and known in-control process parameters, but these assumptions may not be very well satisfied in practical applications. This article investigates the practical issues concerning applications of the Poisson INAR(1) CUSUM chart, considering average run lengths obtained through a bivariate Markov chain approach. First, the effects of deviations from the assumed Poisson model are investigated when there is overdispersion. Design recommendations for achieving robustness are provided along with an extension, the Winsorized Poisson INAR(1) CUSUM chart. Next, analyzing the conditional average run length performance under some hypothetical cases of parameter estimation, it is shown that estimation errors may severely affect the chart’s performance. The marginal average run length performance is used to derive sample size recommendations. An example for monitoring the number of beds occupied at a hospital emergency department is used to illustrate the proposed approach.

Integer-valued autoregressive models for counts showing underdispersion

The Poisson distribution is a simple and popular model for count-data random variables, but it suffers from the equidispersion requirement, which is often not met in practice. While models for overdispersed counts have been discussed intensively in the literature, the opposite phenomenon, underdispersion, has received only little attention, especially in a time series context. We start with a detailed survey of distribution models allowing for underdispersion, discuss their properties and highlight possible disadvantages. After having identified two model families with attractive properties as well as only two model parameters, we combine these models with the INAR(1) model (integer-valued autoregressive), which is particularly well suited to obtain auotocorrelated counts with underdispersion. Properties of the resulting stationary INAR(1) models and approaches for parameter estimation are considered, as well as possible extensions to higher order autoregressions. Three real-data examples illustrate the application of the models in practice.

EWMA Monitoring of Correlated Processes of Poisson Counts

Abstract Statistical quality control is often concerned with processes of Poisson counts. If these counts exhibit serial dependence, a popular approach is to use a Poisson INAR(1) model to describe the autocorrelation structure of the process. In this article, we develop a strategy to monitor a Poisson INAR(1) process, which is based on a combination of the c — and an EWMA chart. Since the resulting bivariate process is a Markov chain, ARL s can be computed exactly with the well-known Markov chain approach. We provide explicit design recommendations and investigate the performance of the combined EWMA chart towards a shift in one of the two model parameters.

A Two‐Sided Cumulative Sum Chart for First‐Order Integer‐Valued Autoregressive Processes of Poisson Counts

Count data processes are often encountered in manufacturing and service industries. To describe the autocorrelation structure of such processes, a Poisson integer-valued autoregressive model of order 1, namely, Poisson INAR(1) model, might be used. In this study, we propose a two-sided cumulative sum control chart for monitoring Poisson INAR(1) processes with the aim of detecting changes in the process mean in both positive and negative directions. A trivariate Markov chain approach is developed for exact evaluation of the ARL performance of the chart in addition to a computationally efficient approximation based on bivariate Markov chains. The design of the chart for an ARL-unbiased performance and the analyses of the out-of-control performances are discussed. Copyright