Previous |  Up |  Next


errors-in-variables (EIV); dependent errors; total least squares (TLS); asymptotic normality
Linear relations, containing measurement errors in input and output data, are taken into account in this paper. Parameters of these so-called errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input-output disturbances. Such an estimate is highly non-linear. Moreover in some realistic situations, the errors cannot be considered as independent by nature. Weakly dependent ($\alpha$- and $\varphi$-mixing) disturbances, which are not necessarily stationary nor identically distributed, are considered in the EIV model. Asymptotic normality of the TLS estimate is proved under some reasonable stochastic assumptions on the errors. Derived asymptotic properties provide necessary basis for the validity of block-bootstrap procedures.
[1] Anderson, T. W.: An Introduction to Multivariate Statistical Analysis. John Wiley and Sons, New York 1958. MR 0091588 | Zbl 1039.62044
[2] Billingsley, P.: Convergence of Probability Measures. First edition. John Wiley and Sons, New York 1968. MR 0233396
[3] Bradley, R. C.: Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surveys 2 (2005), 107-144. MR 2178042 | Zbl 1189.60077
[4] Gallo, P. P.: Consistency of regression estimates when some variables are subject to error. Comm. Statist Theory Methods 11 (1982), 973-983. DOI 10.1080/03610928208828287 | MR 0655466 | Zbl 0515.62064
[5] Gallo, P. P.: Properties of Estimators in Errors-in-Variables Models. Ph.D. Thesis, University of North Carolina, Chapel Hill 1982.
[6] Gleser, L. J.: Estimation in a multivariate ``errors in variables'' regression model: Large sample results. Ann. Statist. 9 (1981), 24-44. DOI 10.1214/aos/1176345330 | MR 0600530 | Zbl 0496.62049
[7] Golub, G. H., Loan, C. F. Van: An analysis of the total least squares problem. SIAM J. Numer. Anal. 17 (1980), 6, 883-893. DOI 10.1137/0717073 | MR 0595451
[8] Healy, J. D.: Estimation and Tests for Unknown Linear Restrictions in Multivariate Linear Models. Ph.D. Thesis, Purdue University 1975. MR 2626067
[9] Herrndorf, N.: A functional central limit theorem for strongly mixing sequence of random variables. Probab. Theory Rel. Fields 69 (1985), 4, 541-550. MR 0791910
[10] Ibragimov, I. A., Linnik, Y. V.: Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff 1971. MR 0322926 | Zbl 0219.60027
[11] Lin, Z., Lu, C.: Limit Theory for Mixing Dependent Random Variables. Springer-Verlag, New York 1997. MR 1486580 | Zbl 0889.60001
[12] Pešta, M.: Strongly consistent estimation in dependent errors-in-variables. Acta Univ. Carolin. - Math. Phys. 52 (2011), 1, 69-79. MR 2808295 | Zbl 1228.62085
[13] Pešta, M.: Total least squares and bootstrapping with application in calibration. Statistics: J. Theor. and Appl. Statistics 46 (2013), 5, 966-991. DOI 10.1080/02331888.2012.658806
[14] Rosenblatt, M.: Markov Processes: Structure and Asymptotic Behavior. Springer-Verlag, Berlin 1971. MR 0329037 | Zbl 0236.60002
[15] Utev, S. A.: The central limit theorem for $\varphi$-mixing arrays of random variables. Theory Prob. Appl. 35 (1990), 131-139. DOI 10.1137/1135013 | MR 1050059
Partner of
EuDML logo