1932

Abstract

Over the past few years, interest has increased in models defined on positive and negative integers. Several application areas lead to data that are differences between positive integers. Some important examples are price changes measured discretely in financial applications, pre- and posttreatment measurements of discrete outcomes in clinical trials, the difference in the number of goals in sports events, and differencing of count-valued time series. This review aims at bringing together a wide range of models that have appeared in the literature in recent decades. We provide an extensive review on discrete distributions defined for integer data and then consider univariate and multivariate time-series models, including the class of autoregressive models, stochastic processes, and ARCH-GARCH– (autoregressive conditionally heteroskedastic–generalized autoregressive conditionally heteroskedastic–) type models.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-statistics-032921-022516
2023-03-09
2024-12-08
Loading full text...

Full text loading...

/deliver/fulltext/statistics/10/1/annurev-statistics-032921-022516.html?itemId=/content/journals/10.1146/annurev-statistics-032921-022516&mimeType=html&fmt=ahah

Literature Cited

  1. Abramowitz M, Stegun IA. 1974. Handbook of Mathematical Functions New York: Dover
    [Google Scholar]
  2. Agostini D, Améndola C. 2019. Discrete Gaussian distributions via theta functions. SIAM J. Appl. Algebra Geometry 3:11–30
    [Google Scholar]
  3. Aissaoui SA, Genest C, Mesfioui M. 2017. A second look at inference for bivariate Skellam distributions. Stat 6:179–87
    [Google Scholar]
  4. Akpoue BP, Angers JF. 2017. Some contributions on the multivariate Poisson–Skellam probability distribution. Commun. Stat. Theory Methods 46:149–68
    [Google Scholar]
  5. Al-Osh M, Alzaid A 1987. First order integer valued autoregressive process. J. Time Ser. Anal. 8:3261–75
    [Google Scholar]
  6. Alomani GA, Alzaid AA, Omair MA. 2018. A Skellam GARCH model. Braz. J. Probab. Stat. 32:1200–14
    [Google Scholar]
  7. Alzaid AA, Omair MA. 2010. On the Poisson difference distribution inference and applications. Bull. Malaysian Math. Sci. Soc. 33:117–45
    [Google Scholar]
  8. Alzaid AA, Omair MA. 2012. An extended binomial distribution with applications. Commun. Stat. Theory Methods 41:193511–27
    [Google Scholar]
  9. Alzaid AA, Omair MA. 2014. Poisson difference integer valued autoregressive model of order one. Bull. Malaysian Math. Sci. Soc. 37:2465–85
    [Google Scholar]
  10. Alzaid AA, Omair MA, Alhadlaq WM et al. 2016. Bernoulli difference time series models. Chilean J. Stat. 7:155–65
    [Google Scholar]
  11. Alzaidani AS, Aly EEA. 2022. Regression models for double discrete distributions. Bull. Malaysian Math. Sci. Soc. 45:211–33
    [Google Scholar]
  12. Andersson J, Karlis D. 2014. A parametric time series model with covariates for integers in Z. Stat. Model. 14:2135–56
    [Google Scholar]
  13. Bakouch HS, Kachour M, Nadarajah S. 2016. An extended Poisson distribution. Commun. Stat. Theory Methods 45:226746–64
    [Google Scholar]
  14. Barbiero A. 2014. An alternative discrete skew Laplace distribution. Stat. Methodol. 16:47–67
    [Google Scholar]
  15. Barndorff-Nielsen OE, Lunde A, Shephard N, Veraart AE. 2014. Integer-valued trawl processes: a class of stationary infinitely divisible processes. Scand. J. Stat. 41:3693–724
    [Google Scholar]
  16. Barndorff-Nielsen OE, Pollard DG, Shephard N. 2012. Integer-valued Lévy processes and low latency financial econometrics. Quant. Finance 12:4587–605
    [Google Scholar]
  17. Baroud H. 2011. Analysis of financial data using a difference-Poisson autoregressive model. Master's Thesis Univ. Waterloo Waterloo, Ontario, Can:.
    [Google Scholar]
  18. Barreto-Souza W, Bourguignon M. 2015. A skew INAR(1) process on Z. AStA Adv. Stat. Anal. 99:2189–208
    [Google Scholar]
  19. Bhati D, Chakraborty S, Lateef SG. 2020. A discrete probability model suitable for both symmetric and asymmetric count data. Filomat 34:82559–72
    [Google Scholar]
  20. Bourguignon M, Vasconcellos KL. 2016. A new skew integer valued time series process. Stat. Methodol. 31:8–19
    [Google Scholar]
  21. Bulla J, Chesneau C, Kachour M. 2015. On the bivariate Skellam distribution. Commun. Stat. Theory Methods 44:214552–67
    [Google Scholar]
  22. Bulla J, Chesneau C, Kachour M. 2017. A bivariate first-order signed integer-valued autoregressive process. Commun. Stat. Theory Methods 46:136590–604
    [Google Scholar]
  23. Carallo G, Casarin R, Robert CP. 2020. Generalized Poisson difference autoregressive processes. arXiv:2002.04470 [stat.ME]
  24. Catania L, Di Mari R, Santucci de Magistris P. 2020. Dynamic discrete mixtures for high-frequency prices. J. Bus. Econ. Stat. 40:559–77
    [Google Scholar]
  25. Chakraborty S. 2015. Generating discrete analogues of continuous probability distributions—a survey of methods and constructions. J. Stat. Distrib. Appl. 2:16
    [Google Scholar]
  26. Chakraborty S, Chakravarty D. 2016. A new discrete probability distribution with integer support on (–∞,∞). Commun. Stat. Theory Methods 45:2492–505
    [Google Scholar]
  27. Chakraborty S, Chakravarty D, Mazucheli J, Bertoli W. 2021. A discrete analog of Gumbel distribution: properties, parameter estimation and applications. J. Appl. Stat. 48:4712–37
    [Google Scholar]
  28. Chesneau C, Bakouch HS, Tomy L, Veena G. 2022. The Poisson-Lindley difference model with application to discrete stock price change. Int. J. Model. Simul. https://doi.org/10.1080/02286203.2022.2086422
    [Google Scholar]
  29. Chesneau C, Kachour M. 2012. A parametric study for the first-order signed integer-valued autoregressive process. J. Stat. Theory Pract. 6:4760–82
    [Google Scholar]
  30. Chesneau C, Kachour M, Bakouch HS. 2018. A family of bivariate discrete distributions on z2 based on the Rademacher distribution. ProbStat Forum 11:53–66
    [Google Scholar]
  31. Chesneau C, Kachour M, Karlis D. 2015. On some distributions arising from a generalized trivariate reduction scheme. Stat. Methodol. 25:36–50
    [Google Scholar]
  32. Cui Y, Li Q, Zhu F. 2021. Modeling Z-valued time series based on new versions of the Skellam INGARCH model. Braz. J. Probab. Stat. 35:2293–314
    [Google Scholar]
  33. Da Cunha ET, Vasconcellos KL, Bourguignon M 2018. A skew integer-valued time-series process with generalized Poisson difference marginal distribution. J. Stat. Theory Pract. 12:4718–43
    [Google Scholar]
  34. Dasgupta R. 1994. Cauchy equation on discrete domain and some characterization theorems. Theory Probab. Appl. 38:3520–24
    [Google Scholar]
  35. De Castro G. 1952. Note on differences of Bernoulli and Poisson variables. Port. Math. 11:4173–75
    [Google Scholar]
  36. Devroye L. 2002. Simulating Bessel random variables. Stat. Probab. Lett. 57:3249–57
    [Google Scholar]
  37. Djordjević MS. 2017. An extension on INAR models with discrete Laplace marginal distributions. Commun. Stat. Theory Methods 46:125896–913
    [Google Scholar]
  38. Djordjević MS, Ristić MM, Pirković B. 2021. Identifying latent components of the TINAR(1) model. Filomat 35:134469–82
    [Google Scholar]
  39. Freeland R. 2010. True integer value time series. AStA Adv. Stat. Anal. 94:217–29
    [Google Scholar]
  40. Genest C, Mesfioui M. 2014. Bivariate extensions of Skellam's distribution. Probab. Eng. Inform. Sci. 28:3401–17
    [Google Scholar]
  41. Ghosh S, Dutta S, Genton MG. 2017. A note on inconsistent families of discrete multivariate distributions. J. Stat. Distrib. Appl. 4:17
    [Google Scholar]
  42. Gonçalves E, Mendes-Lopes N. 2020. Signed compound Poisson integer-valued GARCH processes. Commun. Stat. Theory Methods 49:225468–92
    [Google Scholar]
  43. Gupta N, Kumar A, Leonenko N. 2020. Skellam type processes of order k and beyond. Entropy 22:111193
    [Google Scholar]
  44. Harandi SS, Alamatsaz M. 2015. Discrete alpha-skew-Laplace distribution. SORT 39:171–84
    [Google Scholar]
  45. Hu X, Andrews B. 2021. Integer-valued asymmetric GARCH modeling. J. Time Ser. Anal. 42:5–6737–51
    [Google Scholar]
  46. Hwang Y, Kim JS, Kweon IS. 2011. Difference-based image noise modeling using Skellam distribution. IEEE Trans. Pattern Anal. Mach. Intell. 34:71329–41
    [Google Scholar]
  47. Inusah S, Kozubowski TJ. 2006. A discrete analogue of the Laplace distribution. J. Stat. Plan. Inference 136:31090–102
    [Google Scholar]
  48. Irwin JO. 1937. The frequency distribution of the difference between two independent variates following the same Poisson distribution. J. R. Stat. Soc. 100:3415–16
    [Google Scholar]
  49. Jayakumar K, Jacob S 2012. Wrapped skew Laplace distribution on integers: a new probability model for circular data. Open J. Stat. 2:1106–14
    [Google Scholar]
  50. Jiang L, Mao K, Wu R. 2014. A Skellam model to identify differential patterns of gene expression induced by environmental signals. BMC Genom. 15:772
    [Google Scholar]
  51. Joe H. 1996. Time series models with univariate margins in the convolution-closed infinitely divisible class. J. Appl. Probab. 33:3664–77
    [Google Scholar]
  52. Kachour M, Truquet L. 2011. A p-order signed integer-valued autoregressive (SINAR(p)) model. J. Time Ser. Anal. 32:3223–36
    [Google Scholar]
  53. Kachour M, Yao J. 2009. First-order rounded integer-valued autoregressive (RINAR(1)) process. J. Time Ser. Anal. 30:4417–48
    [Google Scholar]
  54. Karlis D, Ntzoufras I. 2006. Bayesian analysis of the differences of count data. Stat. Med. 25:111885–905
    [Google Scholar]
  55. Karlis D, Ntzoufras I. 2009. Bayesian modelling of football outcomes: using the Skellam's distribution for the goal difference. IMA J. Manag. Math. 20:2133–45
    [Google Scholar]
  56. Kataria KK, Khandakar M. 2021. Fractional Skellam process of order k. arXiv:2103.09187 [math.PR]
  57. Kemp AW. 1992. Heine-Euler extensions of the Poisson distribution. Commun. Stat. Theory Methods 21:3571–88
    [Google Scholar]
  58. Kemp AW. 1997. Characterizations of a discrete normal distribution. J. Stat. Plan. Inference 63:2223–29
    [Google Scholar]
  59. Kerss A, Leonenko N, Sikorskii A. 2014. Fractional Skellam processes with applications to finance. Fractional Calculus Appl. Anal. 17:2532–51
    [Google Scholar]
  60. Kim HY, Park Y. 2008. A non-stationary integer-valued autoregressive model. Stat. Papers 49:3485–502
    [Google Scholar]
  61. Koopman SJ, Lit R, Lucas A. 2014. The dynamic Skellam model with applications. SSRN https://dx.doi.org/10.2139/ssrn.2406867
    [Google Scholar]
  62. Koopman SJ, Lit R, Lucas A. 2017. Intraday stochastic volatility in discrete price changes: the dynamic Skellam model. J. Am. Stat. Assoc. 112:5201490–503
    [Google Scholar]
  63. Koopman SJ, Lit R, Lucas A, Opschoor A 2018. Dynamic discrete copula models for high-frequency stock price changes. J. Appl. Econom. 33:7966–85
    [Google Scholar]
  64. Kozubowski TJ, Inusah S. 2006. A skew Laplace distribution on integers. Ann. Inst. Stat. Math. 58:3555–71
    [Google Scholar]
  65. Latour A, Truquet L. 2008. An integer-valued bilinear type model Work. Pap. HAL Open Sci. Lyon, France: https://hal.archives-ouvertes.fr/docs/00/37/34/09/PDF/APTBilin-1.pdf
    [Google Scholar]
  66. Lin GD. 1998. On the Mittag–Leffler distributions. J. Stat. Plan. Inference 74:11–9
    [Google Scholar]
  67. Liu T, Yuan X. 2013. Random rounded integer-valued autoregressive conditional heteroskedastic process. Stat. Papers 54:3645–83
    [Google Scholar]
  68. Liu Z, Li Q, Zhu F. 2021. Semiparametric integer-valued autoregressive models on Z. Can. J. Stat. 49:41317–37
    [Google Scholar]
  69. McKenzie E. 1985. Some simple models for discrete variate time series. Water Resourc. Bull. 21:4645–50
    [Google Scholar]
  70. Nastić AS, Ristić MM, Djordjević MS. 2016. An INAR model with discrete Laplace marginal distributions. Braz. J. Probab. Stat. 30:1107–26
    [Google Scholar]
  71. Navarro J, Ruiz J. 2005. A note on the discrete normal distribution. Adv. Appl. Stat. 5:2229–45
    [Google Scholar]
  72. Ntzoufras I, Palaskas V, Drikos S. 2021. Bayesian models for prediction of the set-difference in volleyball. IMA J. Manag. Math. 32:4491–518
    [Google Scholar]
  73. Omair A, Alzaid A, Odhah O. 2016. A trinomial difference distribution. Rev. Colomb. Estad. 39:11–15
    [Google Scholar]
  74. Omair MA, Alomani GA, Alzaid AA. 2022. Bivariate distributions on Z2. Bull. Malaysian Math. Sci. Soc. 45:425–44
    [Google Scholar]
  75. Ong S, Shimizu K, Min Ng C. 2008. A class of discrete distributions arising from difference of two random variables. Comput. Stat. Data Anal. 52:31490–99
    [Google Scholar]
  76. Ord J. 1968. The discrete Student's t distribution. Ann. Math. Stat. 39:51513–16
    [Google Scholar]
  77. Pedeli X, Karlis D. 2011. A bivariate INAR(1) process with application. Stat. Model. 11:4325–49
    [Google Scholar]
  78. Pirković BA, Ristić MM, Nastić AS. 2022. Random environment integer-valued autoregressive process with discrete Laplace marginal distributions. REVSTAT Stat. J. In press
    [Google Scholar]
  79. Ristić MM, Bakouch HS, Nastić AS. 2009. A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. J. Stat. Plan. Inference 139:72218–26
    [Google Scholar]
  80. Roy D. 2003. The discrete normal distribution. Commun. Stat. Theory Methods 32:101871–83
    [Google Scholar]
  81. Sangpoom S, Bodhisuwan W. 2016. The discrete asymmetric Laplace distribution. J. Stat. Theory Pract. 10:173–86
    [Google Scholar]
  82. Santos C, Pereira I, Scotto M 2019. Periodic INAR(1) models with Skellam-distributed innovations. Computational Science and Its Applications – ICCSA 2019 S Misra, O Gervasi, B Murgante, E Stankova, V Korkhov et al.64–78. New York: Springer
    [Google Scholar]
  83. Scotto MG, Weiß CH, Gouveia S. 2015. Thinning-based models in the analysis of integer-valued time series: a review. Stat. Model. 15:6590–618
    [Google Scholar]
  84. Seetha Lekshmi V, Sebastian S 2017. An integer valued process with generalized discrete Laplace marginals. J. Probab. Stat. Sci. 15:2219–32
    [Google Scholar]
  85. Shahtahmassebi G, Moyeed R. 2014. Bayesian modelling of integer data using the generalised Poisson difference distribution. Int. J. Stat. Probab. 3:135
    [Google Scholar]
  86. Shahtahmassebi G, Moyeed R. 2016. An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores. Stat. Neerl. 70:3260–73
    [Google Scholar]
  87. Shephard N, Yang JJ. 2017. Continuous time analysis of fleeting discrete price moves. J. Am. Stat. Assoc. 112:5191090–106
    [Google Scholar]
  88. Skellam J. 1946. The frequency distribution of the difference between two Poisson variates belonging to different populations. J. R. Stat. Soc. Ser. A 109:3296
    [Google Scholar]
  89. Steutel F, Harn KV. 1979. Discrete analogues of self–decomposability and stability. Ann. Probab. 7:5893–99
    [Google Scholar]
  90. Szablowski P. 2001. Discrete normal distribution and its relationship with Jacobi theta functions. Stat. Probab. Lett. 52:3289–99
    [Google Scholar]
  91. Tomy L, Veena G. 2022. A retrospective study on Skellam and related distributions. Austrian J. Stat. 51:1102–11
    [Google Scholar]
  92. Wang D, Zhang H. 2010. Generalized RCINAR(p) process with signed thinning operator. Commun. Stat. Simul. Comput. 40:113–44
    [Google Scholar]
  93. Weiß CH. 2018. An Introduction to Discrete-Valued Time Series New York: Wiley
    [Google Scholar]
  94. Xu Y, Zhu F. 2022. A new GJR-GARCH model for Z-valued time series. J. Time Ser. Anal. 43:3490–500
    [Google Scholar]
  95. Zhang H, Wang D, Zhu F. 2010. Inference for INAR(p) processes with signed generalized power series thinning operator. J. Stat. Plan. Inference 140:3667–83
    [Google Scholar]
  96. Zhang H, Wang D, Zhu F. 2012. Generalized RCINAR(1) process with signed thinning operator. Commun. Stat. Theory Methods 41:101750–70
    [Google Scholar]
/content/journals/10.1146/annurev-statistics-032921-022516
Loading
/content/journals/10.1146/annurev-statistics-032921-022516
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error