Information inequalities for the minimax risk of sequential estimators are derived in the case where the loss is measured by the squared error of estimation plus a linear functional of the number of observations. The results are applied to construct minimax sequential estimators of: the failure rate in an exponential model with censored data, the expected proportion of uncensored observations in the proportional hazards model, the odds ratio in a binomial distribution and the expectation of exponential type random variables.
@article{bwmeta1.element.bwnjournal-article-zmv25i1p85bwm, author = {Les\l aw Gajek and B. Mizera-Florczak}, title = {Information inequalities for the minimax risk of sequential estimators (with applications)}, journal = {Applicationes Mathematicae}, volume = {25}, year = {1998}, pages = {85-100}, zbl = {0895.62083}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv25i1p85bwm} }
Gajek, Lesław; Mizera-Florczak, B. Information inequalities for the minimax risk of sequential estimators (with applications). Applicationes Mathematicae, Tome 25 (1998) pp. 85-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv25i1p85bwm/
[000] M. Alvo (1977), Bayesian sequential estimation, Ann. Statist. 5, 955-968. | Zbl 0368.62061
[001] Y. S. Chow, H. Robbins and D. Siegmund (1971), Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston. | Zbl 0233.60044
[002] S. Csörgő (1988), Estimation in the proportional hazards model of random censorship, Statistics 19, 437-463.
[003] S. Csörgő and J. Mielniczuk (1988), Density estimation in the simple proportional hazards model, Statist. Probab. Letters 6, 419-426. | Zbl 0691.62039
[004] L. Gajek (1987), An improper Cramér-Rao lower bound, Zastos. Mat. 19, 241-256. | Zbl 0644.62027
[005] L. Gajek (1988), On minimax value in the scale model with truncated data, Ann. Statist. 16, 669-677. | Zbl 0645.62011
[006] L. Gajek and U. Gather (1991), Estimating a scale parameter under censorship, Statistics 22, 529-549. | Zbl 0742.62028
[007] J. C. Gardiner and V. Susarla (1984), Risk-efficient estimation of the mean exponential survival time under random censoring, Proc. Nat. Acad. Sci. U.S.A. 81, 5906-5909. | Zbl 0557.62074
[008] J. C. Gardiner and V. Susarla (1991), Some asymptotic distribution results in time-sequential estimation of the mean exponential survival time, Canad. J. Statist. 19, 425-436.
[009] J. C. Gardiner, V. Susarla and J. van Ryzin (1986), Time sequential estimation of the exponential mean under random withdrawals, Ann. Statist. 14, 607-618. | Zbl 0603.62088
[010] J. A. Koziol and S. B. Green (1976), A Cramér-von Mises statistic for randomly censored data, Biometrika 63, 465-474. | Zbl 0344.62018
[011] E. L. Lehmann (1983), Theory of Point Estimation, Wiley, New York. | Zbl 0522.62020
[012] R. Magiera (1977), On sequential minimax estimation for the exponential class of processes, Zastos. Mat. 15, 445-454. | Zbl 0371.62115
[013] B. Mizera (1996), Lower bounds on the minimax risk of sequential estimators, Statistics 28, 123-129. | Zbl 0864.62055
[014] W. Rudin (1976), Principles of Mathematical Analysis, McGraw-Hill, New York. | Zbl 0346.26002
[015] M. Tahir (1988), Asymptotically optimal Bayesian sequential point estimation with censored data, Sequential Anal. 7, 227-237. | Zbl 0689.62066
[016] J. Wolfowitz (1947), The efficiency of sequential estimates and Wald's equation for sequential processes, Ann. Math. Statist. 19, 215-230. | Zbl 0032.04203
[017] M. Woodroofe (1982), Nonlinear Renewal Theory in Sequential Analysis, CBMS-NSF Regional Conf. Ser. Appl. Math. 39, SIAM, Philadelphia. | Zbl 0487.62062