| Count data with excess zeros (zero-inflation) and without zeros (zero-truncation) are fairly common integer-valued time series data in practice. For example, when recording the number of cigarettes each individual smokes a day, there exists many who do not smoke at all. So in this case, there are excess zeros in the data. While in counting the members of the animal communities respectively, each group will have at least an individual, so obviously, this data set does not contain zeros. Many researchers may not consider particularly zero inflation or zero truncation, instead, they simply use a model with Poisson or negative binomial marginal distribution to describe these data. Yet if zero inflation or zero truncation in the data were ignored, the estimation results will be poor with some potentially significant statistic findings probably missing, the misspecifications caused by the ignorance of zero inflation may even lead to erroneous conclusions about the data and bring uncertainty to the research and application. The controlling of integer-valued time series is another important aspect of our research, there are many cases submitted to the integer-valued time series, such as the number of defectives on a production line or the number of the infected with some disease per month. Existing results focus on the Poisson integer-valued autoregressive (INAR) process, but this process cannot deal with overdispersion (variance is greater than mean), which is a common phenomenon in count data. The autocorrelated count data based on a new geometric INAR (NGINAR) process is an alternative to the Poisson one.The main content contains three sections. Fisrt, we propose a first-order mixed integer-valued autoregressive process with zero-inflated generalized power series inno-vations to model zero-inflated time series of counts. Strict stationarity, ergodicity of the process, and some important probabilistic properties are obtained. The condition-al maximum likelihood estimators for the parameters in this process are derived and the performances of the estimators are studied via simulation. Then, we consider the zero truncated Poisson AR(1) model for serially dependent processes of zero truncated Poisson counts. Explicit expressions for the joint higher-order moments and cumulants of this process are constructed after a review of such process. The marginal and serial properties of jumps in this process are also obtained. Based on these results, we use the combined jumps chart to monitor a zero truncated Poisson AR(1) process. Through a bivariate Markov chain approach, we investigate the performance of the combined jumps chart in terms of average run length. Last, we use the combined jumps chart, the CUSUM chart and the combined EWMA chart to detect the shift of parameters in the NGINAR(1) process. We also compare the performance of these charts for the case of an underlying NGINAR(1) proecess in terms of the average run lengths (ARLs).First of all, we introduce two important thinning operators, the binomial thinning operator " o " and the negative binomial thinning operator "* ". Let X be a non-negative discrete random variable. The binomial thinning operator "o" is defined as ?oX=?i=1XYi where ?? [0,1],{Yi} is a sequence of independent random variables with Bernoulli(?) distribution, i.e.,P(Yi=1)=1-P(Yi=0)=?. The negative binomial thinning operator "*" is defined as ?* X=?i=1X Wi, where ? ? [0,1], is a sequence of independent random variables with geometric(?/(1+?)), i.e., P=(Wi=k)=?k/(1+?)k+1,k=0,1,Here we introduce the main results of this thesis.1. First-order mixed integer-valued autoregressive processes with zero-inflated generalized power series innovations (ZIMINAR(1) process)A discrete random variable Y is said to have a zero-inflated generalized power series distribution (ZIGPSD) if Y has the following probability distribution function where 0? ??1, I.{0}(k)=1 for k=0 and 0 else; a(k)>0,g(?) and f(?) are positive, finite, differentiable functions.Definition 1 The ZIMINAR(1) process is a sequence of random variables{Xt} defined by the following recursive equation where (i) p ? [0,1], ? ? [0,1), ?t ? [0,1), w.p. stands for "with probability", the binomial thinning operator "o" and the negative binomial thinning operator "*" have been defined in the former text; (ii){?t} is a sequence of independently and identically distributed (i.i.d.) ZIGPSD with ??= E(?t)<? and ??2= Var(?t)<?;{?t} is independent of both{Yi} and{Wi}; (iii){Yi} and{Wi} are also independent, the thinning operators at time t are performed independently of each other.When parameters are special values, the ZIMINAR(1) process will be reduced. The strict stationarity and ergodicity of the proposed ZIMINAR(1) process are estab-lished in the following theorem.Theorem 1 The exists a unique strict stationary integer-valued random series {Xt} satisfying Eq.(1)and Cov (Xs,?t)=0 for s<t.The unconditional first and second moments of that strict stationary series {Xt} exist.Furthermore,the process is an ergodic process.Now we will consider some probabilistic properties of the ZIMINAR(1)process, such as the k-step ahead conditional mean k-step ahead conditional variance,the mean,variance and the transition probabilities of {Xt},etc.Theorem 2 Suppose that {Xt} follows the ZIMINAR(1)process,then the k-step anead conditional mean and k-step ahead conditional variance are and where c1=p?+(1-p)?,c2=p?2+(1-p)?2,c3=p?(1-?)+(1-p)?(1+?),c4= c3+2c1??,E(Xt)=??/(1-c1)and E(Xt2)=(E?t2)+c4E(Xt))/(1-c2).The mean and variance of the ZTPINAR(1)process are ?X=E(Xt)=??/(1-c1) andFrom the definition of this process,it is obvious that the ZIMINAR(1)model is a first-order Markov process. Therefore.it is sufficient to determine the transition probabilities P(Xt|Xt-1),which can be directly obtained from their definitions.Finally,we study the estimation of the ZIMINAR(1)process.Suppose that we have a realization x1,…,xn from the process.We derive conditional maximum likeli-hood(CML)estimators by maximizing the CML function L=?t=2n P(Xt=xt|Xt-1= xt-1).We can conclude the CML method is reliable and can produce good estimators of parameters,especially for large sample sizes.2.Zero-truncated Poisson first-order integer-valued autoregressive(ZTPINAR(1)) process:moments,jumps and control chartsProposed by Bakouch and Ristic(2010),the ZTPINAR(1)process is defined by the following recursion where(i)?>0,? ?[0,1],the binomial thinning operator "o" has been introduced before;(ii)the marginal distribution of {Xt} is zero-truncated Poisson(?),i.e., P(Xt=k)=??e-?/((1-e-?)?!),?=1,2,(iii)_{?t} is a sequence of i.i.d.random variables,?t is independent of the {Yi} and Xt-l for l?1.According to this definition,?t is a positive random variable with and ? ?[0,1/2] is a sufficient condition for the non-negativity of the probabilities P(?t=j).As the ZTPINAR(1)process is a strictly stationary process,the kth-order joint moment of the random variables Xt,Xt+s1,…,Xt+sk-1(k?2)from the ZTPINAR(1) process exists and is denoted by The closed-form expressions for the joint moments up to order 4 are given in the Theorem 3.2.1.The joint cumulants of the ZTPINAR(1)process can be derived by applying the previous result about the joint moments of the ZTPINAR(1)process.Let be the kth-order joint cumulant of Xt,Xt+s1,…,Xt+sk-1(k?2).The explicit expres-sions for the joint higher-order cumulants are shown in the Corollary 3.2.1.Jump Jt is defined as Xt-Xt-1 and the following theorem presents the results concerning jumps in ZTPINAR(1)processes.Theorem 3 Let {Xt}N0 be a stationary ZTPINAR(1)process,then the moment generating function of Jt is given by Thus,?e obtain E(Jt)=0,Var(Jt)=(2?)/(1-e-?)(1-(?e-?/(1-e-?))(1-?(1-e-?)),and the autocorrelation ?J(k)=Corr(Jt,Jt-k)=(?(k-1(1-e-?)k-1(?(1-e-?)-1))/2 for k=1,2,The combined jumps chart is used to control the ZTPINAR(1)process. Let l,u,k ?N0,N0={0,1,…}with 0?l<u and k?u-l. The process is con-sidered as being in control unless Xt (?) [l,u]or Jt (?)[-k,k].The average run length(ARL)is widely used to measure and compare the perfor-mance of the control charts.The ARL of the combined jumps chart can be computed by using the Markov chain method when the parameters(?,?)and control limits (l,u,k)are given. From computation results we can demonstrate the capability of the zTPINAR(1)combined jumps chart in terms of ARLs.First,the ZTPINAR(1) combined jumps chart performs very well in detecting the increasing shifts of ?(which corresponds to larger process mean and jumps).Then,it becomes clear that the ZT-PINAR(1)combined jumps chart is sensitive to a decrease in ?(which corresponds to larger jumps).3.Effective control charts for monitoring the NGINAR(1)processA new geometric first-order integer-valued autoregressive(NGINAR(1))process was proposed by Ristic et al.(2009).The process{Xt}is given by where(i)? ?[0,1],the negative binomial thinning operator"*" has been introduced before?(ii)?>0,the marginal distribution of{Xt} is geometrie(?/(1+?)),i.e., P(Xt=x)=?x/(1+?)x+1,x=0,1,…;(iii){?t}is a sequence of i.i.d.random variables independent of {Wi}N,Xt-l and ?t are independent for l?1.According to this definition,?t is a non-negative random variable with which is a mixture of two random variables with geometric(?/(1+?))distribution and geometric(?/(1+?))distribution.P(?t=l)is well defined for ? ?[0,?/(1+?)],while it is undefined for the values outside of this range.Now we present some properties of the statistic Jt in a stationary NGINAR(1) process.Theorem 4 Le {Xt}N0 be a stationary NGINAR(1)process,then the moment generating function of Jt is given by Thus,we obtain E(Jt)=0,Var(Jt)=2(1-?)?(1+?)and the autocorrelation ?J(k)= ?k-1(?-1)/2,k=1,2,.We use three kinds of charts to monitoring the NGINAR(1)process,i.e.,the combined jumps chart,the cumulative sum(CUSUM)chart and the combined expo-nentially weighted moving average(EWMA)chart.Let lc,uc,k ?N0 with 0?lc<uc and k?uc-lc.The observed combined jumps (Xt,Jt)are plotted in the combined jumps chart with control region CR(lc,uc,k)= {lc,…,uc}×{-k,…,k).The process is considered as being in control unless Xt (?)[lc,uc] or Jt (?)[-k,k].Let c0,k1,h ?N0 with u<k1<h.Starting with C0=c0,a CUSUM statistic with reference value k1 is obtained by Ct=max(0,Xt-k1+Ct-1),t?1.The observed {Ct} are plotted in the CUSUM chart with control region CR(h,k1)=No×{0,…,h). The process is considered as being in control unless Ct(?)[0,h].Let lc,uc.le,ue,z.?N0,? ?(0,1],0?lc<uc and 0?le<ue.Starting with Z0=z0,an EWMA statistic is given by zt=round(?Xt+(1-?)Zt-1)),t=1,2,…, where the function round(.)is given by round(x)=z iff the integer z ?(x-1/2,x+ 1/2];the initial value,z0,is usually set at round of the in-control process mean. The observed(Xt,Zt)are plotted in the combined EWMA chart with control region CR(?,lc,uc,le,uc)={lc,…,uc)×{le,…,ue].The process is considered as being in control unless Xt(?)[lc,uc]or Zt(?)[le,ue].The Markov chain approach is also used in computing the ARL of the combined EWMA chart,the CUSUM chart and the combined EWMA chart of the NGINAR(1) process.From the computation,we have the folloing conclusions.First, the combined jumps chart,the CUSUM chart and the combined EWMA chart perform very well in detecting the increasing shifts of ?,with ARL values much lower than the corresponding in-control ARL.All charts perform similar in given situations so we can choose any one to detect the p increasing shifts.Second, we find that the combined jumps chart can detect the ?decreasing shifts. Last,the properties of the other two charts are different from that of the combined jumps chart.These two charts have the potential to detect ? increasing shifts.The ability of the combined EWMA chart to detect the ? increasing shif is weaker than the CUSUM chart. |
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