×

On Bather’s stochastic approximation algorithm. (English) Zbl 0810.62077

Summary: Stochastic approximation procedures provide a useful technique for detecting the root of an unknown regression function. Based on the idea of averaging J. A. Bather [Asymptotic Statistics, 4th Prague Symp., Prague/Czech. 1988, 13-27 (1989; Zbl 0697.62080)] proposed a new stochastic approximation algorithm. For this algorithm some results will be presented on the rate of convergence as well as on the behaviour for small to moderate sample sizes.

MSC:

62L20 Stochastic approximation

Citations:

Zbl 0697.62080

References:

[1] J. A. Bather: Stochastic approximation: A generalisation of the Robbins-Monro procedure. Proc. Fourth Prague Symp. Asymptotic Statistics, Charles Univ. Prague, August 29-September 2, 1988 (P. Mandl and M. Hušková, Charles Univ., Prague 1989, pp. 13-27.
[2] J. R. Blum: Approximation methods which converge with probability one. Ann. Math. Statist. 25 (1954), 382-386. · Zbl 0055.37806 · doi:10.1214/aoms/1177728794
[3] K. L. Chung: On a stochastic approximation method. Ann. Math. Statist. 25 (1954), 463-483. · Zbl 0059.13203 · doi:10.1214/aoms/1177728716
[4] V. Fabian: On asymptotic normality in stochastic approximation. Ann. Math. Statist. 39 (1968), 1327-1332. · Zbl 0176.48402 · doi:10.1214/aoms/1177698258
[5] G. Kersting: Almost sure approximation of the Robbins-Monro process by sums of independent random variables. Ann. Probab. 5 (1977), 954-965. · Zbl 0374.62082 · doi:10.1214/aop/1176995663
[6] L. Ljung: Strong convergence of a stochastic approximation algorithm. Ann. Statist. 6 (1978), 680-696. · Zbl 0402.62060 · doi:10.1214/aos/1176344212
[7] B. T. Polyak: New method of stochastic approximation type. Automat. Remote Control 51 (1990), 937-946. · Zbl 0737.93080
[8] H. Robbins, S. Monro: A stochastic approximation method. Ann. Math. Statist. 22 (1951), 400-407. · Zbl 0054.05901 · doi:10.1214/aoms/1177729586
[9] D. Ruppert: Almost sure approximations to the Robbins-Monro and Kiefer-Wolfowitz processes with dependent noise. Ann. Probab. 10 (1982), 178-187. · Zbl 0485.62083 · doi:10.1214/aop/1176993921
[10] D. Ruppert: Efficient Estimators from a Slowly Convergent Robbins-Monro Process. Technical Report No. 781, School of Operations Research and Industrial Engineering, Cornell Univ. Ithaca 1988.
[11] D. Ruppert: Stochastic approximation. Handbook of Sequential Analysis. (B. K. Ghosh and P. K. Sen, Marcel Dekker, New York 1991, pp. 503-529.
[12] J. Sacks: Asymptotic distribution of stochastic approximation procedures. Ann. Math. Statist. 29 (1958), 373-405. · Zbl 0229.62010 · doi:10.1214/aoms/1177706619
[13] R. Schwabe: Strong representation of an adaptive stochastic approximation procedure. Stochastic Process. Appl. 23 (1986), 115-130. · Zbl 0614.62107 · doi:10.1016/0304-4149(86)90019-0
[14] R. Schwabe: Stability results for smoothed stochastic approximation procedures. Z. Angew. Math. Mech. 73 (1993), 639-643. · Zbl 0793.65110 · doi:10.1002/zamm.19930730702
[15] J. H. Venter: An extension of the Robbins-Monro procedure. Ann. Math. Statist. 38 (1967), 181-190. · Zbl 0158.36901 · doi:10.1214/aoms/1177699069
[16] H. Walk: Foundations of stochastic approximation. Stochastic Approximation and Optimization of Random Systems, DMV Seminar Blauberen, May 28-June 4, 1989 (L. Jung, G. Pflug and H. Walk, DMV Seminar, Vol. 17, Birkhäuser, Basel 1992, pp. 1-51.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.