A Problem in Minimax Variance Polynomial Extrapolation
Levine, A.
Ann. Math. Statist., Tome 37 (1966) no. 6, p. 898-903 / Harvested from Project Euclid
For the problem of optimum prediction by means of $k$th degree polynomial regression, it is shown in [3] how to find the observation points and respective proportions of observations in the interval $\lbrack-1, 1\rbrack$ in order to obtain the minimax variance over the interval $\lbrack -1, t\brack$ of the predicted regression value for all $t \geqq t_1 > 1; t_1$ is the point outside the interval of observations at which the Chebyschev polynomial of degree $k$ is equal to the maximum value of the variance of the least squares estimate in $\lbrack -1, 1\rbrack$. It is shown herein that if the observation points and proportions are chosen as specified in [3], then the maximum of the "least squares" variance in the interval $\lbrack -1, 1\rbrack$ is at -1. As a consequence, an equation is developed which permits the evaluation of $t_1$ as a function of $k$. Moreover, it is shown that $t_1 \rightarrow 1$ as $k \rightarrow \infty$, so that, for large $k$, the solution given in [3] yields an approximation to the minimax variance over the interval $\lbrack -1, t \rbrack$, all $t > 1$.
Publié le : 1966-08-14
Classification: 
@article{1177699371,
     author = {Levine, A.},
     title = {A Problem in Minimax Variance Polynomial Extrapolation},
     journal = {Ann. Math. Statist.},
     volume = {37},
     number = {6},
     year = {1966},
     pages = { 898-903},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177699371}
}
Levine, A. A Problem in Minimax Variance Polynomial Extrapolation. Ann. Math. Statist., Tome 37 (1966) no. 6, pp.  898-903. http://gdmltest.u-ga.fr/item/1177699371/