Asymptotic Behavior of the Variance of the Best Linear Unbiased Estimator for the Mean of a Discrete-time Singular Stationary Process
- Authors: Babayan N.M.1, Ginovyan M.S.2,3
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Affiliations:
- Russian-Armenian University
- Institute of Mathematics
- Boston University
- Issue: Vol 54, No 6 (2019)
- Pages: 371-380
- Section: Probability Theory and Mathematical Statistics
- URL: https://journals.rcsi.science/1068-3623/article/view/228413
- DOI: https://doi.org/10.3103/S1068362319060074
- ID: 228413
Cite item
Abstract
It is known that for a wide class of discrete-time stationary processes possessing spectral densities f, the variance σn2(f) of the best linear unbiased estimator for the mean depends asymptotically only on the behavior of the spectral density f near the origin, and behaves hyperbolically as n → ∞. In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of σn2(f) as n → ∞. In particular, we show that a necessary condition for σn2(f) to decrease to zero exponentially is that the spectral density f vanishes on a set of positive measure in any vicinity of zero, and if f vanishes only at the origin, then it is impossible to obtain exponential decay of σn2(f), no mater how high the order of the zero of f at the origin.
About the authors
N. M. Babayan
Russian-Armenian University
Author for correspondence.
Email: nmbabayan@gmail.com
Armenia, Yerevan
M. S. Ginovyan
Institute of Mathematics; Boston University
Author for correspondence.
Email: ginovyan@math.bu.edu
Armenia, Yerevan; Boston