Publication Date: 2020
Journal Of The Iranian Statistical Society (17264057)19(2)pp. 1-13
This paper focuses on the empirical autocovariance operator of H-valued periodically correlated processes. It will be demonstrated that the empirical estimator converges to a limit with the same periodicity as the main process. Moreover, the rate of convergence of the empirical autocovariance operator in Hilbert-Schmidt norm is derived. © 2020. All Rights Reserved
Publication Date: 2020
Theory of Probability and Mathematical Statistics (00949000)101pp. 119-127
The Wold decomposition of stationary processes is widely applied in time series prediction and provides interesting insights into the structure of stationary stochastic processes. In 1971, Kallianpur and Mandrekar, using the notion of resolution of identity and unitary operators, presented the Wold decomposition for weakly stationary stochastic processes with values in infinite dimensional separable Hilbert spaces. This paper aims to expand the idea of Wold decomposition to Hilbertian periodically correlated processes, applying the concept of L-closed subspaces. © 2020 American Mathematical Society.
Publication Date: 2019
Journal of Statistical Computation and Simulation (15635163)89(8)pp. 1423-1436
Functional time series is a popular method of forecasting in functional data analysis. The Box-Jenkins methodology for model building, with the aim of forecasting, includes three iterative steps of model identification, parameter estimation and diagnostic checking. Portmanteau tests are one of the most popular diagnostic checking tools. In particular, they are applied to find if the residuals of the fitted model are white noise. Gabrys and Kokoszka [Portmanteau test of independence for functional observations. J Am Stat Assoc. 2007;102(480):1338–1348.] proposed a portmanteau test of independence for functional observation based on Box and Pierce's statistic. Their statistic is too sensitive to the lag value, specially when the sample size is small. Here, two modifications of Gabrys and Kokoszka statistic are presented, which have superior properties in small samples. The efficiency of the modified statistics is demonstrated through a simulation study. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.
Publication Date: 2019
Statistics (02331888)53(2)pp. 283-300
Autoregressive Hilbertian (ARH) processes are of great importance in the analysis of functional time series data and estimation of the autocorrelation operators attracts the attention of various researchers. In this paper, we study estimators of the autocorrelation operators of periodically correlated autoregressive Hilbertian processes of order one (PCARH(1)), which is an extension of ARH(1) processes. The estimation method is based on the spectral decomposition of the covariance operator and considers two main cases: known and unknown eigenvectors. We show the consistency in the mean integrated quadratic sense of the estimators of the autocorrelation operators and present upper bounds for the corresponding rates. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.
Publication Date: 2017
Far East Journal of Mathematical Sciences (09720871)102(1)pp. 111-119
Although in classical theory of time series analysis, it is customary to consider white noise processes as the error term, in functional time series analysis, this assumption can be put in abeyance. An approach to weaken this assumption is to consider the notion of weakly dependent functional processes. In this paper, we study the periodograms and their asymptotic properties in L2 -m -approximable processes that constitute a special class of weakly dependent functional processes. © 2017 Pushpa Publishing House, Allahabad, India.
Publication Date: 2016
Indian Journal of Science and Technology (discontinued) (09746846)9(30)
Following recent research work on the autocorrelation of autoregressive Hilbertian discrete time processes, we likewise give an estimator for the autocorrelation of periodically correlated autoregressive Hilbertian processes and then prove the strong consistency of the estimator.
Publication Date: 2011
Statistical Inference for Stochastic Processes (15729311)14(2)pp. 177-188
We consider periodically correlated autoregressive processes in Hilbert spaces. Our studies on these processes involve existence, covariance structure, estimation of the covariance operators, strong law of large numbers and central limit theorem. © 2011 Springer Science+Business Media B.V.
