Chebyshev polynomials and least squares estimation based on one-dependent random variables

AbstractOne-dependent random variables appear in several fields of statistical work, e.g. in time series analysis and sampling theory. We consider such random variables Y1,…,Yk,k ϵ N, with E(>Yi)=μ, i ϵ {1, …,k}, and regular tridiagonal covaria nce matrix Σ. The real parameter μ can be estimated by the method of least squares, which leads to the best linear unbiased estimator úopt. We give representations of úopt and V(úopt) in terms of Chebyshev polynomials, which turn out to be a helpful tool in the analysis of structure and properties of úopt. Using well-known relations, we give some more sum formulas and formulas concerning products of Chebyshev polynomials, which may also be of interest in other contexts