Binomial Thinning Integer-Valued AR (1) with Poisson – α Fold Zero Modified Geometric Innovations

Document Type : Original Research Articles

Authors
Maryam Shalbaf
Gholamali Parham
Rahim Chinipardaz
Department of Statistics, Faculty of Mathematical Sciences and Computer Shahid Chamran University of Ahvaz, Ahvaz, Islamic Republic of Iran
10.22059/jsciences.2021.320996.1007633
Abstract
Real count data time series often show the phenomenon of the overdispersion. In this paper, we introduce the first-order integer-valued autoregressive process. The univariate marginal distribution is derived from the Delaporte distribution and the innovations are convolution of Poisson with -fold zero modified geometric distribution, based on binomial thinning operator, for modelling integer-valued time series with overdispersion. Some properties of the model are derived. The methods of Yule–Walker, conditional lea st squares and conditional maximum likelihood are used for estimating of the parameters, and their asymptotic properties are established. The Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. The model is fitted to time series of the weekly number of syphilis cases that are overdispersed count data.
Keywords
: α-fold zero modified geometric
Binomial thinning
Count time series
Delaporte distribution
INAR (1) models
Subjects
  1. Nadaraya EA. Some new estimates for distribution functions. Theory Probab. Appl. 1964;9:497-500.
  2. Watson G and Leadbetter M. Hazard analysis II. Sankhya Ser. A. 1964;26:101–116.
  3. Singh RS, Gasser T and Prasad B. Nonparametric estimates of distribution functions. Commun Stat Simul Comput. 1983;12:2095-2108.
  4. Falk M. Relative efficiency and deficiency of kernel type estimators of smooth distribution functions. Stat Neerl. 1983;37:73-83.
  5. Reiss R D. Nonparametric estimation of smooth distribution functions. Scand Stat Theory Appl. 1981;8:116-119.
  6. Rice J. Boundary modification for kernel regression. Commun Stat Theory Methods. 1984;13:893–900.
  7. Gasser T and Muller H. Kernels estimation of regression functions. Notes Math. 1979;757:23-68.
  8. Gasser T, Muller H and Mammitzsch V. Kernels for nonparametric curve estimation. J R Stat Soc Series B Stat Methodol. 1985;47:238–252.
  9. Muller H. Smooth optimum kernel estimators near endpoints. Biometrika. 1991;78:521–530.
  • Geenens G. Probit transformation for kernel density estimation on the unit interval, J Am Stat Assoc. 2014;109:346–358.
  • Tenreiro C. Boundary kernels for distribution function estimation. Revstat Stat J. 2013;11:169-190.
  • Tenreiro C. A new class of boundary kernels for distribution function estimation. Commun Stat Theory Methods. 2018;47:5319-5332.
  • Chacón JE and Duong T. Multivariate Kernel Smoothing and Its Applications, Chapman & Hall, 2018.
  • Chen SX. Beta kernel estimators for density functions. Comput Stat Data Anal. 1999;31:131–145.
  • Chen SX. Probability density function estimation using gamma kernels. Ann Inst Stat Math. 2000;52:471-480.
  • Jin X and Kawczak J. Birnbaum–Saunders and lognormal kernel estimators for modelling durations in high frequency financial data. Ann Econ Finance. 2003;4:103-124.
  • Scaillet O. Density estimation using inverse and reciprocal inverse Gaussian kernels. J Nonparametr Stat. 2004;16:217-226.
  • Hirukawa M and Sakudo M. Nonnegative bias reduction methods for density estimation using asymmetric kernels. Comput Stat Data Anal. 2014;75:112–123.
  • Igarashi G. Bias reductions for beta kernel estimation. J Nonparametr Stat. 2016;28:1–
  • Bouezmarni T and Scaillet O. Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data. Econ Theory. 2005;21:390-412.
  • Mombeni HA, Mansouri B and Akhoond MR. Asymmetric kernels for boundary modification in distribution function estimation. Revstat Stat J Forthcoming papers. 2019.
  • Zhang S. A note on the performance of the gamma kernel estimators at the boundary. Stat Probab Lett. 2010;80:548-557.
  •  

 
  • Scott DW. Multivariate Density Estimation, Theory, Practice and Visualization, Second edition. John Wiley & Sons, 2015.
  • Liu R and Yang L. Kernel estimation of multivariate cumulative distribution function. J Nonparametr Stat. 2008;20:661-677.
  • Altman N and Leger C. Bandwidth selection for kernel distribution function estimation. J Stat Plan Inference. 1995;46:195-214.
Journal of Sciences, Islamic Republic of Iran
Volume 33, Issue 1
Winter 2022
Pages 55-63