## The Annals of Statistics

### The Convergence of General Step-Length Algorithms for Regular Optimum Design Criteria

#### Abstract

For a regular optimality criterion function $\Phi$, a sequence of design measures $\{\xi_n\}$ is generated using the iteration $\xi_{n+1} = (1 - \alpha_n)\xi_n + \alpha_n\xi_n$, where $\xi_n$ is chosen to minimize $\nabla \Phi(M(\xi_n), M(\xi))$ over all $\xi$ and $\{\alpha_n\}$ is a prescribed sequence of numbers from (0, 1). This is called a general step-length algorithm for $\Phi$. Typical conditions on $\{\alpha_n\}$ are $\alpha_n \rightarrow 0$ and $\Sigma_n\alpha_n = \infty$. In this paper, a dichotomous behavior of $\{\xi_n\}$ is proved under the above conditions on $\{\alpha_n\}$ for $\Phi$ satisfying some mild regularity conditions. Sufficient conditions for convergence to optimal designs are also established. This can be applied to show that the $\{\xi_n\}$ as constructed above do converge to an optimal design for most of the trace-related and determinant-related design criteria.

#### Article information

Source
Ann. Statist., Volume 6, Number 6 (1978), 1273-1285.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176344373

Digital Object Identifier
doi:10.1214/aos/1176344373

Mathematical Reviews number (MathSciNet)
MR523762

Zentralblatt MATH identifier
0396.62059

JSTOR