A Modified Robbins-Monro Procedure Approximating the Zero of a Regression Function from Below

Setup adjustment for discrete-part manufacturing processes with asymmetric cost functions

“…Anbar (1977) proposed a biased stochastic approximation procedure, further studied by Krasulina (1998), for the problem of one-side convergence. Both authors treated an abstract estimation problem from the point of view of stochastic approximation and did not consider process adjustment scenarios.…”

“…Using the law of the repeated logarithm, Anbar (1977) demonstrated that the convergence of Y n as n → ∞ when b n converges to zero is in n 1 2 (log(log n)) − 1 2 . However, practical insights on the selection of the sequence {b n } N n=1 had not been provided in previous literature.…”

“…For a given selection of the k n 's, the bias terms {b n } N n=1 can be determined by equating the right hand side of expression (8) to m n . Although the approaches in Anbar (1977) and Krasulina (1998) utilize the harmonic sequence…”

“…The idea of having the quality characteristic converge to the optimal setting from the lower cost side is related to certain stochastic approximation techniques in which a bias term is added to allow for one-side convergence, as discussed by Anbar (1977) and Krasulina (1998). These efforts were oriented to finding asymptotic or long-term one-sided convergence conditions of stochastic approximation methods instead of process adjustment applications.…”

One-sided convergence of continuous processes of stochastic approximation

“…By analogy with the process of Anbax (see [1]), we propose to consider the stochastic differential equation…”