ON HITTING TIMES FOR MARKOV TIME SERIES OF COUNTS WITH APPLICATIONS TO QUALITY CONTROL

• Examples of time series of counts arise in several areas, for instance in epidemiology, industry, insurance and network analysis. Several time series models for these counts have been proposed and some are based on the binomial thinning operation, namely the integer-valued autoregressive (INAR) model, which mimics the structure and the autocorrelation function of the autoregressive (AR) model. The detection of shifts in the mean of an INAR process is a recent research subject and it can be done by using quality control charts. Underlying the performance analysis of these charts, there is an indisputable popular measure: the run length (RL), the number of samples until a signal is triggered by the chart. Since a signal is given as soon as the control statistic falls outside the control limits, the RL is nothing but a hitting time. In this paper, we use stochastic ordering to assess: – the ageing properties of the RL of charts for the process mean of Poisson INAR(1) output; – the impact of shifts in model parameters on this RL. We also explore the implications of all these properties, thus casting interesting light on this hitting time for a Markov time series of counts. Key-Words: • statistical process control; run length; phase-type distributions; stochastic ordering. AMS Subject Classification: • 60E15, 62P30. 456 M.C. Morais and A. Pacheco On Hitting Times for Markov Time Series of Counts in Quality Control 457 1. THE INAR(1) PROCESS Time series of counts become apparent in areas such as: • epidemiology — the number of new cases of some infectious and notifiable diseases is monitored monthly to assess and surveil the incidence of acute viral infections such as poliomyelitis, as reported by Zeger (1988) and Silva (2005, pp. 145–147); • industry — the monthly number of accidents in a manufacturing plant (Silva et al., 2009), the number of defects per sample (Weiss, 2009a) and the number non-conforming units within a sample of finite size counts (Weiss, 2009b,c) have to be controlled; • insurance — modelling the number of claim counts is an extremely important part of insurance pricing (Boucher et al., 2008); • network analysis — the number of intrusions on computers and network systems (Weiss, 2009d, p. 11) also requires surveillance. In some cases, the integer values of the time series are large and continuousvalued models can be (and are frequently) used. However, when the time series consists only of small integer numbers, ARMA processes are of limited use for modelling purposes, namely because the multiplication of an integer-valued random variable (r.v.) by a real constant may lead to a non-integer r.v. (Silva, 2005, p. 22). A possible way out is to replace the scalar multiplication by a random operation, such as the binomial thinning operation. This operation can be thought as the scalar multiplication counterpart in the integer-valued setting which preserves the integer structure of the process, it is due to Steutel and Van Harn (1979) and may be stated as follows. Definition 1.1. Let X be a discrete r.v. with range N0 = {0, 1, . } and α a scalar in [0, 1]. Then the binomial thinning operation on X results in the following r.v. :