Minimax interpolation of sequences with stationary increments and cointegrated sequences
Volume 3, Issue 1 (2016), pp. 59–78
Pub. online: 1 April 2016
Type: Research Article
Open Access
Open Access
Received
11 March 2016
11 March 2016
Revised
16 March 2016
16 March 2016
Accepted
17 March 2016
17 March 2016
Published
1 April 2016
1 April 2016
Abstract
We consider the problem of optimal estimation of the linear functional $A_{N}\xi ={\sum _{k=0}^{N}}a(k)\xi (k)$ depending on the unknown values of a stochastic sequence $\xi (m)$ with stationary increments from observations of the sequence $\xi (m)+\eta (m)$ at points of the set $\mathbb{Z}\setminus \{0,1,2,\dots ,N\}$, where $\eta (m)$ is a stationary sequence uncorrelated with $\xi (m)$. We propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional in the case of spectral certainty, where spectral densities of the sequences are exactly known. We also consider the problem for a class of cointegrated sequences. We propose relations that determine the least favorable spectral densities and the minimax spectral characteristics in the case of spectral uncertainty, where spectral densities are not exactly known while a set of admissible spectral densities is specified.
References
Bell, W.: Signal extraction for nonstationary time series. Ann. Stat. 12(2), 646–664 (1984). MR0740918. doi:10.1214/aos/1176346512
Box, G.E.P., Jenkins, G.M., Reinsel, G.C.: Time Series Analysis. Forecasting and Control. 3rd edn. Englewood Cliffs, NJ, Prentice Hall (1994). MR1312604
Chigira, H., Yamamoto, T.: Forecasting in large cointegrated processes. J. Forecast. 28(7), 631–650 (2009). MR2744389. doi:10.1002/for.1076
Engle, R.F., Granger, C.W.J.: Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251–276 (1987). MR0882095. doi:10.2307/1913236
Franke, J.: Minimax robust prediction of discrete time series. Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 337–364 (1985). MR0771471. doi:10.1007/BF00532645
Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes. I. Springer, Berlin (2004). MR2058259
Gregoir, S.: Fully modified estimation of seasonally cointegrated processes. Econom. Theory 25(5), 1491–1528 (2010). MR2684793. doi:10.1017/S026646660999065X
Grenander, U.: A prediction problem in game theory. Ark. Mat. 3, 371–379 (1957). MR0090486
Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North–Holland Publishing Company, Amsterdam, New York, Oxford (1979). MR0528295
Johansen, S.: Representation of cointegrated autoregressive processes with application to fractional processes. Econom. Rev. 28, 121–145 (2009). MR2487849. doi:10.1080/07474930802387977
Kolmogorov, A.N.: Selected Works by A.N. Kolmogorov. Vol. II: Probability Theory and Mathematical Statistics. A.N. Shiryayev (ed.) Math Appl. Sov. Ser., vol. 26, Kluwer Academic Publishers, Dordrecht, etc. (1992). MR1153022
Luz, M., Moklyachuk, M.: Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences. Stat. Optim. Inf. Comput. 2(3), 176–199 (2014). MR3351379. doi:10.19139/56
Luz, M., Moklyachuk, M.: Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences. Stat. Optim. Inf. Comput. 3(2), 160–188 (2015). MR3352757. doi:10.19139/132
Luz, M.M., Moklyachuk, M.P.: Interpolation of functionals of stochastic sequences with stationary increments. Theory Probab. Math. Stat. 87, 117–133 (2013). MR3241450. doi:10.1090/S0094-9000-2014-00908-4
Luz, M.M., Moklyachuk, M.P.: Minimax-robust filtering problem for stochastic sequence with stationary increments. Theory Probab. Math. Stat. 89, 127–142 (2014). MR3235180. doi:10.1090/S0094-9000-2015-00940-6
Moklyachuk, M.: Minimax-robust estimation problems for stationary stochastic sequences. Stat. Optim. Inf. Comput. 3(4), 348–419 (2015). MR3435278
Pinsker, M.S.: The theory of curves with nth stationary increments in Hilbert spaces. Izv. Akad. Nauk SSSR, Ser. Mat. 19(5), 319–344 (1955). MR0073957
Pshenichnyi, B.N.: Necessary Conditions of an Extremum. Nauka, Moskva (1982). MR0686452
Rockafellar, R.T.: Convex Analysis. Princeton University Press (1997). MR1451876
Rozanov, Y.A.: Stationary Stochastic Processes. Holden-Day, San Francisco (1967). MR0214134
Salehi, H.: Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes. Ann. Probab. 7(5), 840–846 (1979). MR0542133
Vastola, K.S., Poor, H.V.: An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica 28, 289–293 (1983). MR0740656
Yaglom, A.M.: Correlation theory of stationary and related random processes with stationary nth increments. Mat. Sb. 37(79)(1), 141–196 (1955). MR0071672
Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Vol. 1: Basic Results. Springer, New York etc. (1987). MR0893393
Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Vol. 2: Supplementary Notes and References. Springer, New York, etc. (1987). MR0915557
