All articles published by MDPI are made immediately available worldwide under an open access license. No special
permission is required to reuse all or part of the article published by MDPI, including figures and tables. For
articles published under an open access Creative Common CC BY license, any part of the article may be reused without
permission provided that the original article is clearly cited. For more information, please refer to
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature
Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for
future research directions and describes possible research applications.
Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive
positive feedback from the reviewers.
Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world.
Editors select a small number of articles recently published in the journal that they believe will be particularly
interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the
most exciting work published in the various research areas of the journal.
The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive order 2 transition matrix.Stochastic ordering is also used to compare transition matrices referring to pairs of GINAR(1) processes with different values of the marginal mean. We assess and illustrate the implications of these two stochastic ordering results, namely on the properties of the run length of geometric charts for monitoring GINAR(1) counts.
The INAR(1) and GINAR(1) processes were originally proposed by McKenzie [1,2]; the latter model was soon after discussed in more detail by Alzaid and Al-Osh . They rely on the binomial thinning operation due to Steutel and van Harn  which is defined below.
Let X be a non-negative integer-valued r.v. with range and ρ a scalar in . Then the binomial thinning operation on X results in the r.v.
where ∘ represents the binomial thinning operator; is a sequence of i.i.d. Bernoulli r.v. with parameter ρ; is independent of X.
We usually refer to as the r.v. that arises from X by binomial thinning. Furthermore, we define and .
Now that we have defined the binomial thinning operation, a sort of scalar multiplication counterpart in the integer-valued setting, the reader is reminded of the definition of McKenzie’s GINAR(1) process and its main properties.
Let . Then is said to be a GINAR(1) process if is written in the form
where and are independent sequences of i.i.d. Bernoulli r.v. with parameter and of i.i.d. geometric r.v. with parameter p, respectively; the sequence of innovations and are independent; all thinning operations are performed independently of each other and of ; and all the thinning operations at time t are independent of .
According to McKenzie  and Alzaid and Al-Osh , if then is a stationary AR(1) process with marginal distribution.
McKenzie  also adds that is a DTMC with TPM, , where
where represents the indicator function of the set of non-negative integers. These entries can be obtained by taking advantage of a few facts: with probability 1; , for ; the p.f. of the innovations, , is equal to
The autocorrelation function of the GINAR(1) process is equal to
We ought to point out that the GINAR(1) process is a particular case of the generalized geometric INAR(1) or GGINAR(1) process, introduced by (Al-Osh and Aly , Section 3). Moreover, autocorrelated geometric counts can also be modeled by the new geometric INAR(1) or NGINAR(1) process, proposed by Ristić et al.  and relying on the negative binomial thinning operator. Finally, the NGINAR(1) process is a special instance of the ZMGINAR(1) process, the zero-modified geometric first-order integer-valued autoregressive, introduced and thoroughly described by Barreto-Souza .
The remainder of the paper is organized as follows. In Section 2, we shall prove that has two important features stated in the two following theorems.
The TPM of a GINAR(1) process is totally positive of order 2,
i.e., all the minors of the are non-negative.
Let: and be two independent GINAR(1) processes, with parameters and ; and be their corresponding TPM. Then is stochastically smaller than in the usual (or in the Kalmykov order) sense,
if , that is,
in case .
In Section 3, we discuss and illustrate the impact of (6) and (7) on the run length of an upper one-sided geometric chart for monitoring GINAR(1) processes. In Section 4, we sum up our findings and briefly refer to related and future work.
2. Proving the Two Features of the GINAR(1) Process
Demonstrating that the minors of the TPM of a GINAR(1) process are all non-negative is not simple, due to the aspect of the transition probabilities defined in (3). However, by adopting the reasoning of (Morais and Pacheco  Section 2) and resorting to some auxiliary definitions and lemmas in Appendix A.1, we can prove (6).
where: ; , ; a r.v. with p.f. given by (4); and are two independent r.v.
In accordance to Lemmas A1 and A3, stochastically increases with i in the likelihood ratio sense and . Hence, we can invoke the closure of the stochastic order (see Definition A1) under the sum of independent r.v. (see (Shaked and Shanthikumar  p. 46, Theorem 1.C.9) or Karlin and Proschan ) to conclude that
for , i.e., or is a stochastically monotone TPM in the likelihood ratio sense , according to Definition A2. □
The next proof refers to a stochastic ordering between the TPM that govern two DTMC with the same state space, thus associated with what Kulkarni  (pp. 148–149) terms the Kalmykov-dominance or Kalmykov order(see Kalmykov  Theorem 2).
Result (7) can be shown to hold by successively capitalizing on: Lemmas A1 and A2; the closure of under the sum of independent r.v.; ; and implies that the r.v. X is stochastically smaller than the r.v. Y in the usual sense, in short (see Shaked and Shanthikumar  p. 42, Theorem 1.C.1). Then, for , , and :
i.e., if . □
3. Practical Implications in Statistical Process Control
Time series of counts arise naturally in several applications, namely the manufacturing industry, health care, service industry, insurance, and network analysis. Using control charts for monitoring the underlying count processes is essential to swiftly detect changes in such processes and start preventive or corrective actions (see Weiß ). For an overview of control charts for count processes, we refer the reader to Weiß .
As noted by Ristić et al. , counts with geometric marginal distributions play a major role in several areas, for instance reliability, medicine, and precipitation modeling. These counts may refer to the number of machines waiting for maintenance, congenital malformations, or thunderstorms in a day.
In statistical process control, the GINAR(1) process can be used to model, for example, the cumulative counts of conforming items between two nonconforming items when these successive counts are no longer independent, say because the observations are generated by automated high-frequency sampling.
The literature review reveals that no charts have been proposed for monitoring GINAR(1) or GGINAR(1) counts. However, Li et al.  proposed a combined jumps chart, a cumulative sum (CUSUM) chart, and a combined exponentially weighted moving average (EWMA) chart for monitoring the NGINAR(1) counts. Furthermore, Li et al.  described upper and lower one-sided CUSUM charts for monitoring the mean of ZMGINAR(1) counts.
Let us consider that the following quality control chart is being used to detect decreases in the parameter p of the GINAR(1) process.
Let be a GINAR(1) process. The upper one-sided geometric chart makes use of the set of control statistics and triggers a signal at time t if , where U is a fixed upper control limit (UCL) in .
We should bear in mind that the control statistic becomes stochastically smaller in the usual sense as p increases (see Lemma A4). Consequently and as suggested by (Xie et al.  p. 42), it is clear that when an observed value of exceeds the UCL of the chart, this should be taken as a sign that the p has decreased, that is, an indication of a potential increase in the process mean .
The performance of the upper one-sided geometric chart is about to be assessed in terms of the run length (RL), the random number of samples collected before a signal is triggered by this control chart. Consequently, the following first passage time of the stochastic process , under the condition that , is a vital performance measure of this chart for monitoring a GINAR(1) process:
where u is a fixed initial value in the set .
U is chosen in such a way that false alarms are rather infrequent and increases in the process mean (i.e., decreases in p) are detected as quickly as possible. Hence, we should be dealing with a large in-control RL and smaller out-of-control run lengths.
3.1. Significance of
By invoking the first part of Theorem 3.1 of Assaf et al. , we can state that the character of the TPM of the GINAR(1) process leads to the following result.
Let be a GINAR(1) process. Then
i.e., , for .
Corollary 1 implies that has an increasing hazard rate , that is, is a nondecreasing function of (see Kijima  p. 118, Theorem 3.7(ii)). means that signaling, given that no observation has previously exceeded the UCL, becomes more likely as we proceed with the collection of observations provided that .
Note, however, that may not be IHR, for . In fact, the second part of Theorem 3.1 of Assaf et al.  allows us to state that the p.f. is in l and n, i.e., , for . As a consequence, , for , thus we can add that has an decreasing hazard rate .
The next corollary translates the stochastic influence of an increase in the initial value u and can be shown to be valid by capitalizing on (Karlin  pp. 42–43, Theorem 2.1).
Let be a GINAR(1) process. Then, for ,
Let us denote the upper one-sided geometric chart with (resp. ) by Scheme 1 (resp. Scheme 2). Then (10) can be interpreted as follows: the odds of Scheme 1 signaling at sample m against Scheme 2 triggering a signal at the same sample decreases as m increases (see  p. 5).
Result (10) seems quite evident; nevertheless, it would not be valid if the GINAR(1) process was not governed by a TPM.
3.2. Other Comparisons of Run Lengths
The stochastic inequality , for , allows us to stochastically compare two GINAR(1) processes. As a matter of fact, by invoking Lemma A4 and Theorem 6.B.32 of (Shaked and Shanthikumar  p. 282), we can state the next result.
Let and two GINAR(1) processes. If and the initial states are deterministic or random, say , then
The next lemma plays a vital role in the comparison of run lengths and is taken from (Shaked and Shanthikumar  p. 283).
If two stochastic processes and satisfy then
Lemma 1 states what could be considered obvious: if we are dealing with two ordered stochastic processes in the usual sense, the larger stochastic process in the usual sense exceeds the critical level U stochastically sooner also in the usual sense.
By combining Corollary 3 and Lemma 1, we can provide a stochastic flavor to the influence of an increase in p not only on but also on another important RL:
which we coin as overall run length, following (Weiß  Section 20.2.2). refers to a first passage time of the stochastic process under the condition that the initial state coincides with the r.v. . In point of fact, it is reasonable to resort to this performance measure because in practice we do not know , hence it is plausible to rely, for example, on .
The following stochastic ordering results hold for the run lengths of the upper one-sided geometric chart for monitoring GINAR(1) processes:
for and .
Note that we could have also invoked (14) and the closure of the usual stochastic order under mixtures (see Shaked and Shanthikumar  p. 6, Theorem 1.A.3.(d)) to prove (15).
Results (14) and (15) mean that the upper one-sided geometric chart for the GINAR(1) process stochastically increases its detection speed (in the usual sense) as the downward shift in p becomes more extreme. This stochastic ordering result parallels with the notion of a sequentially repeated uniformly powerful test.
3.3. An Illustration
Ristić et al.  found that an NGINAR(1) model with estimated parameters and adequately described the monthly counts of sex offenses reported in the 21st police car beat in Pittsburgh. This data set comprises 144 observations, starting in January 1990 and ending in December 2001.
Note that the GINAR(1) and NGINAR(1) processes share the same geometric marginal distribution; and, as far as the offense data set is concerned, the value of the Akaike information criterion (AIC) for the NGINAR(1) and GINAR(1) models are very close, namely and , respectively, as (Ristić et al.  Table 2) attest. Hence, we are going to consider the upper one-sided geometric chart from Definition 3 with and for monitoring such counts.
An UCL equal to and an initial state (resp. ) yield an in-control ARL of (resp. ). These and other RL-related performance measures used in this subsection are described in Appendix A.2.
The plots of the hazard rate function in Figure 1 give additional insights into the RL performance of the geometric chart as we proceed with the sampling and to the impact of the adoption of a head start. Indeed, it illustrates two results that follow from Corollary 1: and . This last result suggests that the false-alarm rate conveniently decreases in the first samples when we adopt a head start .
According to Brook and Evans , the limiting form of the p.f. of the RL is geometric-like with parameter , where is the maximum real eigenvalue of , regardless of the initial value u of the control statistic . Therefore, it comes as no surprise that the values of the hazard rate functions of and converge to
Furthermore, the hazard rate function of is pointwise below the one of because Corollary 2 establishes that and this result in turn implies , that is, , for (see Definition A4).
We now illustrate the first result of Corollary 4 and also of a consequence of its second result: , for ; is an increasing function of p in the interval, .
In the left panel of Figure 2, we plotted the survival functions of and .
Since , the plot of survival function of is pointwise below the one of , as Figure 2 plainly demonstrates. Hence, the number of samples taken until the detection of a decrease in p by the upper one-sided geometric chart is indeed stochastically smaller than the number of samples we collect until this chart emits a false alarm.
The right panel of Figure 2 refers to the overall ARL function, , for . It increases with p in this particular interval from to . We ought to note that it increases further when we take , therefore the upper one-sided geometric chart cannot detect increases in p in an expedient manner, as we have anticipated.
We wrote a program for Mathematica 10.3 (Wolfram ) to produce all the graphs and results in this subsection.
4. Concluding Remarks
As expertly put by Montgomery and Mastrangelo , the independence assumption is often violated in practice. As a consequence, we often deal with discrete-valued time series, namely when we are dealing with very high sampling rates, as suggested by Weiß and Testik , and Rakitzis et al. .
In this paper, we considered the GINAR(1) count process, resorted to stochastic ordering to prove two features of its TPM, and discussed the implications of these two traits on RL-related performance measures of an upper one-sided geometric control chart that accounts for the autocorrelated character of such process.
For example: the character of the TPM of the GINAR(1) process implies an IHR behaviour of the run length of that same chart; the run length and the overall run length stochastically increase in the usual sense in the interval .
These features of the GINAR(1) process and the associated results are comparable to the ones derived by (Morais  Section 3.2) and Morais and Pacheco [8,28].
It is important to note that the notion of stochastically monotone matrices in the usual sense was introduced by Daley  for real-valued discrete-time Markov chains. Moreover, Karlin  implicitly states that a TPM possesses a monotone likelihood ratio property and, thus, virtually defines stochastically monotone Markov chains in the likelihood ratio sense. Furthermore, the comparison of counting processes and queues in the usual sense can be traced back, for instance, to Whitt  and the multivariate likelihood ratio order of random vectors (or order) is discussed, for example, by (Shaked and Shanthikumar  pp. 298–305).
Coincidentally, the stochastic order in the likelihood sense for stochastic processes or TPM has not been defined up to now, as far as we have investigated. For this reason and the fact that the order is not closed under mixtures (see Shaked and Shanthikumar  p. 33), we did not state or prove the analogue of the two results in Corollary 4.
We also failed to prove that , for , because of two opposing stochastic behaviors of the summands and : the r.v. binomial (resp. ) stochastically increases (resp. decreases) with in the likelihood ratio sense. Had we proven that result, we could have concluded that the larger the upward shifts in the autocorrelation parameter, the longer it takes the upper one-sided geometric chart to detect such a change in .
It would be pertinent to investigate the stochastic properties of the RL and overall RL of lower one-sided geometric charts for detecting increases in the parameter p of a GINAR(1) process.
Another possibility of further work which certainly deserves some consideration is to investigate the extension of Theorems 1 and 2 to the NGINAR(1) process, the novel geometric INAR(1) process proposed by Guerrero et al. , or the new INAR(1) process with Poisson binomial-exponential 2 innovations studied by Zhang et al. , and assess the impact of these two results in the RL performance of upper one-sided charts for monitoring such autocorrelated geometric counts.
We ought to mention that deriving results similar to (6) and (7) seems to be very unlikely for the mixed generalized Poisson INAR process . This follows from the fact that the generalized Poisson distribution has not a p.f.
The author acknowledges the financial support of the Portuguese FCT—Fundação para a Ciência e a Tecnologia, through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID (Center for Computational and Stochastic Mathematics), Instituto Superior Técnico, Universidade de Lisboa.
We are grateful to the three reviewers who selflessly devoted their time to scrutinizing this work and offered pertinent comments that led to an improved version of the original manuscript. The author would also like to thank Christian H. Weiß for the opportunity to celebrate the vital role of “Discrete-valued Time Series” in this special issue of “Entropy”.
Conflicts of Interest
The author declares no conflict of interest.
The following abbreviations are used in this manuscript:
cumulative distribution function
decreasing hazard rate
discrete-time Markov chain
generalized geometric first-order integer-valued autoregressive process
geometric first-order integer-valued autoregressive process
independent and identically distributed
increasing hazard rate
first-order integer-valued autoregressive process
new geometric first-order integer-valued autoregressive process
Pólya frequency of order 2
totally positive of order 2
transition probability matrix
upper control limit
zero-modified geometric first-order integer-valued autoregressive process
Appendix A.1. Auxiliary Definitions and Lemmas
This appendix has the sole purpose of providing a few notions and results that are crucial to prove (6), (7), and some of the implications of these two stochastic ordering results.
The notions of stochastically smaller in the likelihood ratio sense, stochastically monotone in the likelihood ratio sense, and Pólya frequency of order 2 p.f. are taken or follow from (Shaked and Shanthikumar  p. 42), (Kijima  pp. 129–131), and (Kijima  p. 106) (respectively).
Let X and Y be two non-negative integer r.v., with p.f. and . Then X is said to be stochastically smaller than Y in the likelihood ratio sense if
over the union of the supports of the r.v. X and Y. In shorthand notation, .
Let . Then
stochastically increases with i in the likelihood ratio sense because the ratio
Since , is a nonincreasing function of . We still have to verify that : this inequality is valid if has a positive derivative, i.e., if or, equivalently, . Hence stochastically decreases with p in the likelihood ratio sense as long as .
Let be an irreducible DTMC with TPM . Then is said to be stochastically monotone in the likelihood ratio sense if
for all i. In this case, we write or .
Let X be a non-negative r.v. with probability function (p.f.) . If
i.e., , then X is said to have a Pólya frequency of order 2 p.f. and we write .
Since , we can state that these two ratios are certainly nondecreasing functions of x over the support of the corresponding p.f.
The concepts of stochastically smaller in the usual sense in the univariate and multivariate cases and stochastically smaller in the hazard rate sense in the univariate case can be found in (Shaked and Shanthikumar  pp. 3, 17, 266), whereas on p. 281 of this same reference the stochastic ordering of stochastic processes in the usual sense is defined.
Let X and Y be two non-negative integer r.v., with p.f. and and c.d.f. and . Then: X is said to be stochastically smaller than Y in the usual sense if
X is said to be stochastically smaller than Y in the hazard rate sense in case
The stochastic orders , , and can be related: (see Shaked and Shanthikumar  pp. 18, 43, Theorems 1.B.1, 1.C.1). Moreover, provided that these expectations exist.
Let be an r.v. with geometric distribution with parameter p. Then
Equation (A8) follows in a straightforward manner: , for ; thus, , for , when .
Let and be two vectors in ; then we write if , for . Additionally, recall that is called an upper set if whenever and (see Shaked and Shanthikumar  p. 266).
Let and be two dimensional random vectors. Then is said to be smaller than in the usual sense if
for every upper set in . We write .
Let and be two discrete-time stochastic processes with a common state space . Then is said to be stochastically smaller than in the usual sense if
for every and . In this case, we write .
As a consequence of Definition A6, implies that , for all .
Appendix A.2. Run Length Related Performance Measures
The run length of the upper one-sided geometric chart, , is the first passage time to the set of states , where . Thus, we can use the Markov chain approach proposed by Brook and Evans  and provide the expected value of ,
where: is the th vector of the orthogonal basis for ; represents an identity matrix with rank ; is the sub-stochastic matrix that governs the transitions between the states in , with entries given by (3); is a column-vector with ones.
We can also add the survival and hazard rate functions of are equal to
The overall ARL of the upper one-sided geometric chart is given by
(see (Weiß  Section 20.2.2) or Weiß and Testik ).
McKenzie, E. Some simple models for discrete variate time series. Water Resour. Bull.1985, 21, 645–650. [Google Scholar] [CrossRef]
McKenzie, E. Autoregressive moving-average processes with negative-binomial and geometric marginal distributions. Adv. Appl. Probab.1986, 18, 679–705. [Google Scholar] [CrossRef]
Steutel, F.W.; van Harn, K. Discrete analogues of self-decomposability and stability. Ann. Probab.1979, 7, 893–899. [Google Scholar] [CrossRef]
Al-Osh, M.A.; Aly, E.-E.A.A. First order autoregressive time series with negative binomial and geometric marginals. Commun. Stat.–Theory Methods1992, 21, 2483–2492. [Google Scholar] [CrossRef]
Ristić, M.M.; Bakouch, H.S.; Nastić, A.S. A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. J. Stat. Plan. Inference2009, 139, 2218–2226. [Google Scholar] [CrossRef]
Barreto-Souza, W. Zero-Modified geometric INAR(1) process for modelling count time series with deflation or inflation of zeros. J. Time Ser. Anal.2015, 36, 839–852. [Google Scholar] [CrossRef]
Morais, M.C.; Pacheco, A. On hitting times for Markov time series of counts with applications to quality control. REVSTAT–Stat. J.2016, 4, 455–479. [Google Scholar]
Shaked, M.; Shanthikumar, J.G. Stochastic Orders; Springer: New York, NY, USA, 2007. [Google Scholar]
Karlin, S.; Proschan, F. Pólya type distributions of convolutions. Ann. Math. Stat.1960, 31, 721–736. [Google Scholar] [CrossRef]
Kulkarni, V.G. Modeling and Analysis of Stochastic Systems; Chapman and Hall: London, UK, 1995. [Google Scholar]
Kalmykov, G.I. On the partial ordering of one-dimensional Markov processes. Theory Probab. Its Appl.1962, 7, 456–459. [Google Scholar] [CrossRef]
Weiß, C.H. SPC methods for time-dependent processes of counts—A literature review. Cogent Math.2015, 2, 1–11. [Google Scholar] [CrossRef]
Weiß, C.H. An Introduction to Discrete-Valued Time Series; Wiley: Hoboken, NJ, USA, 2018. [Google Scholar]
Li, C.; Wang, D.; Zhu, F. Effective control charts for monitoring the NGINAR(1) process. Qual. Reliab. Eng. Int.2016, 32, 877–888. [Google Scholar] [CrossRef]
Li, C.; Wang, D.; Sun, J. Control charts based on dependent count data with deflation or inflation of zeros. J. Stat. Comput. Simul.2019, 89, 3273–3289. [Google Scholar] [CrossRef]
Xie, M.; Goh, T.N.; Kuralmani, V. Statistical Models and Control Charts for High-Quality Processes; Springer Science+Business Media, LLC: New York, NY, USA, 2002. [Google Scholar]
Assaf, D.; Shaked, M.; Shanthikumar, J.G. First-passage times with PFr densities. J. Appl. Probab.1985, 22, 185–196. [Google Scholar] [CrossRef]
Kijima, M. Markov Processes for Stochastic Modeling; Chapman and Hall: London, UK, 1997. [Google Scholar]
Karlin, S. Total positivity, absorption probabilities and applications. Trans. Am. Math. Soc.1964, 11, 33–107. [Google Scholar] [CrossRef]
Morais, M.J.C. Stochastic Ordering in the Performance Analysis of Quality Control Schemes. Ph.D. Thesis, Universidade Técnica de Lisboa, Lisbon, Portugal, 2002. [Google Scholar]
Weiß, C.H. Categorical Time Series Analysis and Applications in Statistical Quality Control. Ph.D. Thesis, Fakultät für Mathematik und Informatik der Universität Würzburg, Würzburg, Germany, 2009. [Google Scholar]
Brook, D.; Evans, D.A. An approach to the probability distribution of CUSUM run length. Biometrika1972, 59, 539–549. [Google Scholar] [CrossRef]
Wolfram Research, Inc. Mathematica; Version 10.3; Wolfram Research, Inc.: Champaign, IL, USA, 2015. [Google Scholar]
Montgomery, D.C.; Mastrangelo, C.M. Some statistical process control methods for autocorrelated data. J. Qual. Technol.1991, 23, 179–193. [Google Scholar] [CrossRef]
Weiß, C.H.; Testik, M.C. The Poisson INAR(1) CUSUM chart under overdispersion and estimation error. IIE Trans.2011, 43, 805–818. [Google Scholar] [CrossRef]
Rakitzis, A.C.; Weiß, C.H.; Castagliola, P. Control charts for monitoring correlated counts with a finite range. Appl. Stoch. Model. Bus. Ind.2017, 33, 733–749. [Google Scholar] [CrossRef]
Morais, M.C.; Pacheco, A. On stochastic ordering and control charts for traffic intensity. Seq. Anal.2016, 35, 536–559. [Google Scholar] [CrossRef]
Whitt, W. Comparing counting processes and queues. Adv. Appl. Probab.1981, 13, 207–220. [Google Scholar] [CrossRef]
Shaked, M.; Shanthikumar, J.G. Stochastic Orders and Their Applications; Academic Press: San Diego, CA, USA, 1994. [Google Scholar]
Guerrero, M.B.; Barreto-Souza, W.; Ombao, H. Integer-valued autoregressive processes with prespecified marginal and innovation distributions: A novel perspective. Stoch. Models2022, 38, 70–90. [Google Scholar] [CrossRef]
Zhang, J.; Zhu, F.; Khan, N.M. A new INAR model based on Poisson-BE2 innovations. Commun. Stat. Theory Methods2022. [Google Scholar] [CrossRef]
Huang, J.; Zhu, F.; Deng, D. A mixed generalized Poisson INAR model with applications. J. Stat. Comput. Simul.2022. [Google Scholar] [CrossRef]
Weiß, C.H.; Testik, M.C. CUSUM monitoring of first-order integer-valued autoregressive processes of Poisson counts. J. Qual. Technol.2009, 41, 389–400. [Google Scholar] [CrossRef]
Hazard rate functions of and .
Hazard rate functions of and .
Survival function of , for and (black and gray solid lines); overall ARL function, , for .
Survival function of , for and (black and gray solid lines); overall ARL function, , for .
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely
those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or
the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas,
methods, instructions or products referred to in the content.