# ENBIS: European Network for Business and Industrial Statistics

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## ENBIS-13 in Ankara

*15 – 19 September 2013*Abstract submission: 5 February – 5 June 2013

### Phase-I Analysis of Time-Dependent Counts with Missing Observations

*17 September 2013, 16:15 – 16:35*

#### Abstract

- Submitted by
- Christian Weiß
- Authors
- Christian Weiß (Darmstadt University of Technology, Department of Mathematics, Darmstadt), Murat Caner Testik (Hacettepe University, Industrial Engineering Department, Ankara)
- Abstract
- During a Phase I analysis, the available in-control data is used to estimate the required model parameters, which, in turn, are used for control chart design. If we are concerned with serially independent attributes data stemming from a Poisson or binomial distribution, then parameter estimation is essentially based on the sample mean. Assume now that the Phase I data is incomplete, either because some observations were missing or invalid right from the beginning, or because some observations had to be excluded during the Phase I analysis since they were identified as outliers. Then the estimation has to be done from the incomplete data, which is a simple task in the case of independence, because then, the estimates are computed from the reduced data set in the same way as from the full data set (e.g., still based on the mean in the above attributes cases).

If the in-control data are assumed to stem from a time-dependent (though stationary) process, then missing observations severely affect the process of parameter estimation. We consider the case of a Poisson INAR(1) process and a binomial AR(1) process, where the monitored attributes have a first order autoregressive dependence structure. We describe approaches of how to estimate the model parameters in the case of missing observations, and we analyze the performance of these estimators in simulation experiments. Then the ARL performance of some control charts based on such estimated parameters is investigated, also in the case when outliers are not removed from the Phase I data. A real-data example is considered for illustrative purposes.