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Isotonic estimation in stochastic approximation

Author

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  • Hanson, D. L.
  • Mukerjee, Hari

Abstract

Suppose m(·) is a regression function which has a unique zero [theta]. The Robbins--Monro process Xn+1 = Xn - cnYn is a standard stochastic approximation method used to estimate [theta]. In the literature Xn+1 is used both as the estimator of [theta] after the nth step and as the design/control setting for the process at the (n+1)st step. Apparently the justification for this choice is that Xn --> [theta] a.s. and has various asymptotic optimality properties. Following Frees and Ruppert (1986) we distinguish between design/control settings Xn and estimates [theta]n of [theta]. When m(·) is known to be nondecreasing it is possible to incorporate this prior information into the estimation procedure. We define a new estimator [theta]n using isotonic regression, discuss its strong consistency, and discuss some of its optimality properties; we show that an obvious conjecture about the strong consistency of this estimator is false. One purpose of this note is to generate interest in the possible use of isotonic regression in stochastic approximation.

Suggested Citation

  • Hanson, D. L. & Mukerjee, Hari, 1990. "Isotonic estimation in stochastic approximation," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 279-287, March.
    • Handle: RePEc:eee:stapro:v:9:y:1990:i:3:p:279-287
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