Averaged Stochastic Approximation for Sequential Fixed-Width Confidence Intervals under Right Censoring
We construct sequential fixed-width confidence intervals for a parameter of a right-
censored lifetime distribution observed as a stream, where the Fisher information is unknown and
must be learned online. The procedure combines the Polyak–Ruppert averaged Robbins–Monro esti-
mator with a recursively maintained information estimate, both updated in O(1) time and memory.
Under standard regularity and the gain condition cI > 1/2, we prove asymptotic consistency and
Chow–Robbins efficiency (N ε /n ε → 1 a.s. and in mean). The analysis relies on an asymptotically
linear representation of the averaged estimator, whose remainder is controlled almost surely via the
martingale structure of the score. We show that stopping-time efficiency holds for every gain, while
the gain condition governs coverage alone. Simulations on censored exponential and Weibull mod-
els confirm the theory and demonstrate that using the information in place of its reciprocal in the
stopping rule collapses the procedure.
1. Anscombe, F.J.: Large-sample theory of sequential estimation. Proc. Camb. Philos. Soc. 48, 600-607 (1952)
2. Bickel, P.J., Klaassen, C.A.J., Ritov, Y., Wellner, J.A.: Efficient and Adaptive Estimation for Semiparametric
Models. Johns Hopkins University Press, Baltimore (1993)
3. Chen, X., Lee, J.D., Tong, X.T., Zhang, Y.: Statistical inference for model parameters in stochastic gradient
descent. Ann. Statist. 48(1), 251-273 (2020)
4. Chow, Y.S., Robbins, H.: On the asymptotic theory of fixed-width sequential confidence intervals for the mean.
Ann. Math. Statist. 36, 457-462 (1965)
5. Durmus, A., Moulines, E., Naumov, A., Samsonov, S.: Finite-time high-probability bounds for Polyak-Ruppert
averaged iterates of linear stochastic approximation. arXiv:2207.04475 (2022)
6. Fabian, V.: On asymptotic normality in stochastic approximation. Ann. Math. Statist. 39, 1327–1332 (1968)
7. Fang, Y., Xu, J., Yang, L.: Online bootstrap confidence intervals for the stochastic gradient descent estimator. J.
Mach. Learn. Res. 19(78), 1-21 (2018)
8. Gut, A.: Stopped Random Walks: Limit Theorems and Applications, 2nd edn. Springer, New York (2009)
9. Hájek, J.: A characterization of limiting distributions of regular estimates. Z. Wahrscheinlichkeitstheorie Verw.
Gebiete 14, 323-330 (1970)
10. Kushner, H.J., Yin, G.G.: Stochastic Approximation and Recursive Algorithms and Applications, 2nd edn.
Springer, New York (2003)
11. Le Cam, L.: Asymptotic Methods in Statistical Decision Theory. Springer, New York (1986)
12. Le Cam, L., Yang, G.L.: Asymptotics in Statistics: Some Basic Concepts, 2nd edn. Springer, New York (2000)
13. Lee, S., Liao, Y., Seo, M.H., Shin, Y.: Fast and robust online inference with stochastic gradient descent via random
scaling. In: Proc. AAAI Conf. Artif. Intell., vol. 36, pp. 7381-7389 (2022)
14. Louis, T.A.: Finding the observed information matrix when using the EM algorithm. J. R. Stat. Soc. B 44,
226-233 (1982)
15. Luo, L., Song, P.X.-K.: Renewable estimation and incremental inference in generalized linear models with streaming
data sets. J. R. Stat. Soc. B 82(1), 69-97 (2020)
16. Mou, W., Li, C.J., Wainwright, M.J., Bartlett, P.L., Jordan, M.I.: On linear stochastic approximation: fine-grained
Polyak-Ruppert and non-asymptotic concentration. In: Proc. COLT, PMLR 125, 2947–2997 (2020)
17. Moulines, E., Bach, F.: Non-asymptotic analysis of stochastic approximation algorithms for machine learning. In:
Advances in Neural Information Processing Systems, vol. 24 (2011)
18. Nevelson, M.B., Hasminskii, R.Z.: Stochastic Approximation and Recursive Estimation. American Mathematical
Society, Providence (1973)
19. Orchard, T., Woodbury, M.A.: A missing information principle: theory and applications. In: Proc. 6th Berkeley
Symp. Math. Statist. Probab., vol. 1, pp. 697–715 (1972)
20. Pelletier, M.: On the almost sure asymptotic behaviour of stochastic algorithms. Stochastic Process. Appl. 78,
217-244 (1998)
21. Polyak, B.T.: New method of stochastic approximation type. Autom. Remote Control 51, 937-946 (1990)
22. Polyak, B.T., Juditsky, A.B.: Acceleration of stochastic approximation by averaging. SIAM J. Control Optim.
30, 838-855 (1992)
23. Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Statist. 22, 400-407 (1951)
24. Robbins, H., Siegmund, D.: A convergence theorem for non-negative almost supermartingales. In: Optimizing
Methods in Statistics, pp. 233-257. Academic Press, New York (1971)
25. Ruppert, D.: Efficient estimation from a slowly convergent Robbins-Monro process. Tech. Rep. 781, Cornell
University (1988)
26. Toulis, P., Airoldi, E.M.: Asymptotic and finite-sample properties of estimators based on stochastic gradients.
Ann. Statist. 45(4), 1694-1727 (2017)
27. Van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)
28. Woodroofe, M.: Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia (1982)
29. Xu, J., et al.: Scalable estimation and inference with large-scale or online survival data. arXiv:2001.01434 (2020)
Copyright (c) 2026 «ВЕСТНИК НУУз»
Это произведение доступно по лицензии Creative Commons «Attribution-NonCommercial-ShareAlike» («Атрибуция — Некоммерческое использование — На тех же условиях») 4.0 Всемирная.
.jpg)
2.png)
