Imputation-based semiparametric estimation for INAR(1) processes with missing data
Yıl 2020,
Cilt: 49 Sayı: 5, 1843 - 1864, 06.10.2020
Wei Xiong
Dehui Wang
Xinyang Wang
Öz
In applied problems parameter estimation with missing data has risen as a hot topic. Imputation for ignorable incomplete data is one of the most popular methods in integer-valued time series. For data missing not at random (MNAR), estimators directly derived by imputation will lead results that is sensitive to the failure of the effectiveness. In view of the first-order integer-valued autoregressive (INAR(1)) processes with MNAR response mechanism, we consider an imputation based semiparametric method, which recommends the complete auxiliary variable of Yule-Walker equation. Asymptotic properties of relevant estimators are also derived. Some simulation studies are conducted to verify the effectiveness of our estimators, and a real example is also presented as an illustration.
Destekleyen Kurum
National Natural Science Foundation of China, Natural Science Foundation of Jilin Province, Program for Changbaishan Scholars of Jilin Province, Science and Technology Program of Jilin Educational Department during the “13th Five-Year” Plan Period
Proje Numarası
No. 11731015, 11571051, 11501241, 11871028, No. 20150520053JH, 20170101057JC, 20180101216JC, 2015010, No. 2016316
Kaynakça
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process, J. Time Series Anal. 8 (3), 261–275, 1987.
- [2] J. Andersson and D. Karlis, Treating missing values in INAR(1) models: an application
to syndromic surveillance data, J. Time Series Anal. 31 (1), 12-19, 2010.
- [3] I.V. Basawa, P.D. Feigin and C.C. Heyde, Asymptotic properties of maximum likelihood
estimators for stochastic processes, Sankhya A 38 (3), 259-270, 1976.
- [4] X. Chen, A.T.K.Wan and Y. Zhou, Efficient quantile regression analysis with missing
observations, J. Amer. Statist. Assoc. 110, 723-741, 2015.
- [5] X. Cui, J. Guo and G. Yang, On the identifiability and estimation of generalized linear
models with parametric nonignorable missing data mechanism, Comput. Statist. Data
Anal. 107, 64-80, 2017.
- [6] J. Du and Y. Li, The integer-valued autoregressive (INAR(p)) model, J. Time Series
Anal. 12 (2), 129-142, 1991.
- [7] R.K. Freeland and B.P.M. Mccabe, Analysis of low count time series data by poisson
autoregression, J. Time Series Anal. 25 (5), 701-722, 2004.
- [8] P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application, Academic
Press, New York, 1980.
- [9] M.A. Jazi, G. Jones, and C.D. Lai, Integer valued AR(1) with geometric innovations,
J. Iran. Stat. Soc. (JIRSS) 2 (2), 173-190, 2012.
- [10] B. Jia, D. Wang and H. Zhang, A study for missing values in PINAR(1)T processes,
Comm. Statist. Theory Methods 43 (22), 4780-4789, 2014.
- [11] R. Jung, G. Ronning and A. Tremayne, Estimation in conditional first order autoregression
with discrete support, Statist. Papers 46 (2), 195-224, 2005.
- [12] S.A. Khashimov, The central limit theorem for generalized U-statistics for weakly
dependent vectors, Theory Probab. Appl. 38 (3), 563-578, 1993.
- [13] J.K. Kim and C.L. Yu, A semiparametric estimation of mean functionals with nonignorable
missing data, J. Amer. Statist. Assoc. 106 (493), 157-165, 2012.
- [14] J.K. Kim and J. Shao, Statistical Methods for Handling Incomplete Data, CRC Press,
Boco Raton, 2013.
- [15] R.J.A. Little and D.B. Rubin, Statistical Analysis with Missing Data, Second Edition,
John Wiley & Sons, 2002.
- [16] T.A. Louis, Finding the observed information matrix when using the EM algorithm,
J. R. Stat. Soc. Ser. B. Stat. Methodol. 44, 226-233, 1982.
- [17] K. Morikawa, J.K. Kim and Y. Kano, Semiparametric maximum likelihood estimation
with data missing not at random, Canad. J. Statist. 45 (4), 393-409, 2017.
- [18] W.K. Newey and D.L. McFadden, Large sample estimation and hypothesis testing,
in: Handbook of Econometrics, Vol. IV , Engle R.F., McFadden D.L. editors, North
Holland, Amsterdam, 1994.
- [19] M. Pourahmadi, Estimation and interpolation of missing values of a stationary time
series, J. Time Series Anal. 10 (2), 149-169, 1989.
- [20] M.K. Riddles, J.K. Kim and J. Im, Propensity-score-adjustment method for nonignorable
nonresponse, Journal of Survey Statistics and Methodology 4, 215-245, 2016.
- [21] D.B. Rubin, Inference and missing data, Biometrika 63 (3), 581-592, 1976.
- [22] J. Shao and L. Wang, Semiparametric inverse propensity weighting for nonignorable
missing data, Biometrika 103, 175-187, 2016.
- [23] N. Tang, P. Zhao and H. Zhu, Empirical likelihood for estimating equations with
nonignorably missing data, Statist. Sinica 24, 723-747, 2014.
- [24] S. Wang, J. Shao and J.K. Kim, An instrumental variable approach for identification
and estimation with nonignorable nonresponse, Statist. Sinica 24, 1097-1116, 2014.
- [25] K. Yang, D.Wang, B. Jia and H. Li, An integer-valued threshold autoregressive process
based on negative binomial thinning, Statist. Papers 59 (3), 1131-1160, 2018.
- [26] H. Zhang, D. Wang and F. Zhu, Inference for INAR(p) processes with signed generalized
power series thinning operator, J. Statist. Plann. Inference 140 (3), 667-683,
2010.
- [27] H. Zheng, I.V. Basawa and S. Datta, Inference for pth-order random coefficient
integer-valued autoregressive processes, J. Time Series Anal. 27 (3), 411-440, 2006.
Yıl 2020,
Cilt: 49 Sayı: 5, 1843 - 1864, 06.10.2020
Wei Xiong
Dehui Wang
Xinyang Wang
Proje Numarası
No. 11731015, 11571051, 11501241, 11871028, No. 20150520053JH, 20170101057JC, 20180101216JC, 2015010, No. 2016316
Kaynakça
- [1] M.A. Al-Osh and A.A. Alzaid, First-order integer-valued autoregressive (INAR(1))
process, J. Time Series Anal. 8 (3), 261–275, 1987.
- [2] J. Andersson and D. Karlis, Treating missing values in INAR(1) models: an application
to syndromic surveillance data, J. Time Series Anal. 31 (1), 12-19, 2010.
- [3] I.V. Basawa, P.D. Feigin and C.C. Heyde, Asymptotic properties of maximum likelihood
estimators for stochastic processes, Sankhya A 38 (3), 259-270, 1976.
- [4] X. Chen, A.T.K.Wan and Y. Zhou, Efficient quantile regression analysis with missing
observations, J. Amer. Statist. Assoc. 110, 723-741, 2015.
- [5] X. Cui, J. Guo and G. Yang, On the identifiability and estimation of generalized linear
models with parametric nonignorable missing data mechanism, Comput. Statist. Data
Anal. 107, 64-80, 2017.
- [6] J. Du and Y. Li, The integer-valued autoregressive (INAR(p)) model, J. Time Series
Anal. 12 (2), 129-142, 1991.
- [7] R.K. Freeland and B.P.M. Mccabe, Analysis of low count time series data by poisson
autoregression, J. Time Series Anal. 25 (5), 701-722, 2004.
- [8] P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application, Academic
Press, New York, 1980.
- [9] M.A. Jazi, G. Jones, and C.D. Lai, Integer valued AR(1) with geometric innovations,
J. Iran. Stat. Soc. (JIRSS) 2 (2), 173-190, 2012.
- [10] B. Jia, D. Wang and H. Zhang, A study for missing values in PINAR(1)T processes,
Comm. Statist. Theory Methods 43 (22), 4780-4789, 2014.
- [11] R. Jung, G. Ronning and A. Tremayne, Estimation in conditional first order autoregression
with discrete support, Statist. Papers 46 (2), 195-224, 2005.
- [12] S.A. Khashimov, The central limit theorem for generalized U-statistics for weakly
dependent vectors, Theory Probab. Appl. 38 (3), 563-578, 1993.
- [13] J.K. Kim and C.L. Yu, A semiparametric estimation of mean functionals with nonignorable
missing data, J. Amer. Statist. Assoc. 106 (493), 157-165, 2012.
- [14] J.K. Kim and J. Shao, Statistical Methods for Handling Incomplete Data, CRC Press,
Boco Raton, 2013.
- [15] R.J.A. Little and D.B. Rubin, Statistical Analysis with Missing Data, Second Edition,
John Wiley & Sons, 2002.
- [16] T.A. Louis, Finding the observed information matrix when using the EM algorithm,
J. R. Stat. Soc. Ser. B. Stat. Methodol. 44, 226-233, 1982.
- [17] K. Morikawa, J.K. Kim and Y. Kano, Semiparametric maximum likelihood estimation
with data missing not at random, Canad. J. Statist. 45 (4), 393-409, 2017.
- [18] W.K. Newey and D.L. McFadden, Large sample estimation and hypothesis testing,
in: Handbook of Econometrics, Vol. IV , Engle R.F., McFadden D.L. editors, North
Holland, Amsterdam, 1994.
- [19] M. Pourahmadi, Estimation and interpolation of missing values of a stationary time
series, J. Time Series Anal. 10 (2), 149-169, 1989.
- [20] M.K. Riddles, J.K. Kim and J. Im, Propensity-score-adjustment method for nonignorable
nonresponse, Journal of Survey Statistics and Methodology 4, 215-245, 2016.
- [21] D.B. Rubin, Inference and missing data, Biometrika 63 (3), 581-592, 1976.
- [22] J. Shao and L. Wang, Semiparametric inverse propensity weighting for nonignorable
missing data, Biometrika 103, 175-187, 2016.
- [23] N. Tang, P. Zhao and H. Zhu, Empirical likelihood for estimating equations with
nonignorably missing data, Statist. Sinica 24, 723-747, 2014.
- [24] S. Wang, J. Shao and J.K. Kim, An instrumental variable approach for identification
and estimation with nonignorable nonresponse, Statist. Sinica 24, 1097-1116, 2014.
- [25] K. Yang, D.Wang, B. Jia and H. Li, An integer-valued threshold autoregressive process
based on negative binomial thinning, Statist. Papers 59 (3), 1131-1160, 2018.
- [26] H. Zhang, D. Wang and F. Zhu, Inference for INAR(p) processes with signed generalized
power series thinning operator, J. Statist. Plann. Inference 140 (3), 667-683,
2010.
- [27] H. Zheng, I.V. Basawa and S. Datta, Inference for pth-order random coefficient
integer-valued autoregressive processes, J. Time Series Anal. 27 (3), 411-440, 2006.