Approximations to expected stopping times with applications to sequential estimation

References

• Anscombe , F.J . 1952 . Large sample theory of sequential estimation . Proc. Cambridge Philos. Soc. , 48 : 600 – 607 .
   Web of Science ®Google Scholar
• Aras , G and Woodroofe , M . 1993 . Asymptotic expansions for the moments of a randomly stopped average . Ann. Statist , 21 : 503 – 519 .
   Web of Science ®Google Scholar
• Chang , Y.I and Martinsek , A . 1953–1969 . Fixed size confidence regions for parameters of a logistic regression model . Ann. Statist , 20
   Google Scholar
• Chow , Y.S and Robbins , H . 1965 . On the asymptotic theory of fixedwidth sequential confidence intervals for the mean . Ann. Math. Statist , 36 : 457 – 462 .
   Google Scholar
• Chow , Y.S and Yu , F.K . 1981 . The performance of a sequential procedure for the estimation of the mean . Ann. Statist , 9 : 184 – 189 .
   Web of Science ®Google Scholar
• Chow , Y.S and Martinsek , A.T . 1982 . Bounded regret of a sequential procedure for estimation of the mean . Ann. Statist , 10 : 909 – 914 .
   Web of Science ®Google Scholar
• de la Pena , V and Lai , T.L . 1997 . Moments of randomly stopped U-statistics . Ann. Probab , 25 : 2055 – 2081 .
   Google Scholar
• Dmitrienko , A and Vexler , A . 1996 . Renewal theory results for autoregressive processes . Math. Methods Statist , 5 : 477 – 490 .
   Google Scholar
• Feller , W . 1966 . “ An Introduction to Probability Theory and its Applications II ” . New York : Wiley .
   Google Scholar
• Ghosh , M and Mukhopadhyay , N . 1979 . Sequential point estimation of the mean when the distribution is unspecified . Comm. Statist , 8 : 637 – 652 .
   Google Scholar
• Grambsch , P . 1983 . Sequential sampling based on the observed Fisher information to guarantee the accuracy of the maximum likelihood estimator . Ann. Statist , 11 : 68 – 77 .
   Web of Science ®Google Scholar
• Grambsch , P . 1989 . Sequential maximum likelihood estimation with applications to logistic regression in case-control studies . J. Statist. Plann. Inference , 22 : 355 – 369 .
   Google Scholar
• Hu , I . 1991 . Nonlinear renewal theory for conditional random walks . Ann. Probab , 19 : 401 – 422 .
   Web of Science ®Google Scholar
• Lai , T.L . 1996 . On uniform integrability and asymptotically risk-efficient sequential estimation . Sequential Anal , 15 : 237 – 251 .
   Google Scholar
• Lai , T.L and Siegmund , D . 1977 . A nonlinear renewal theory with applications to sequential analysis, I . Ann. Statist , 5 : 946 – 954 .
   Web of Science ®Google Scholar
• Lai , T.L and Siegmund , D . 1979 . A nonlinear renewal theory with applications to sequential analysis, II . Ann. Statist , 7 : 60 – 76 .
   Web of Science ®Google Scholar
• Lai , T.L and Siegmund , D . 1983 . Fixed accuracy estimation of an autoregressive parameter . Ann.Statist , 11 : 478 – 485 .
   Web of Science ®Google Scholar
• Lalley , S.P . 1986 . Renewal theorem for a class of stationary sequences . Probab. Theory Relat. Fields . 1986 . pp. 196 – 214 .
   Google Scholar
• Martinsek , A . 1983 . Second order approximations to the rist of a sequential procedure . Ann. Statist , 11 : 827 – 836 .
   Web of Science ®Google Scholar
• Robbins , H . 1959 . Sequential estimation of the mean of a normal population . Probability and Statistics , 11 : 235 – 245 . Harold Cramér volume Almquist and Wiksell, Stockholm
   Google Scholar
• Siegmund , D . 1985 . “ Sequential Analysis ” . New York : Springer-Verlag .
   Google Scholar
• Simons , G . 1968 . On the cost of not knowing the variance when making a fixed-width confidence interval for the mean . Ann. Math. Statist , 39 : 1916 – 1952 .
   Google Scholar
• Smith , w.L . 1954 . Asymptotic renewal theorems . Proc. Roy. Soc. Edin. Ser. A , 64 : 9 – 48 .
   Google Scholar
• Smith , W.L . 1958 . Renewal theory and its ramifications . J. Roy. Stat. Soc. Ser. B , 20 : 243 – 302 .
   Google Scholar
• Spitzer , F . 1960 . A Tauberian theorem and its probability interpretation . Trans. Amer. Math. Soc , 94 : 150 – 169 .
   Google Scholar
• Sriram , T.N . 1988 . Sequential estimation of the autoregressive parameter in a first order autoregressive process . Sequential Anal , 7 : 53 – 74 .
   Google Scholar
• Starr , N . 1966 . On the asymptotic efficiency of a sequential procedure for estimating the mean . Ann. Math. Statist , 37 : 1173 – 1185 .
   Google Scholar
• Subkhankulov M.A. Tauberian Theorems With Remainder Terms. Soc. Indust. Appl. Math. Nauka, Moscow 1976 in Russian)
   Google Scholar
• Titchmarsh , E.C . 1939 . “ The Theory of Functions ” . Oxford University Press .
   Google Scholar
• Vexler , A.A and Konev , A.A . 1995 . On the mean number of observations under guaranteed estimation of an autoregression parameter . Automation Remote Control , 56 : 844 – 850 .
   Google Scholar
• Woodroofe , M . 1977 . Second order approximation to sequential poiint and interval estimation . Ann. Statist , 5 : 984 – 995 .
   Web of Science ®Google Scholar
• Woodroofe M. Nonlinear Renewal Theory in Sequential Analysis. Soc. Indust. Appl. Math. Philadelphia 1982
   Google Scholar
• Woodroofe , M . 1990 . On the nonlinear renewal theorem . Ann. Probab , 18 : 1790 – 1805 .
   Web of Science ®Google Scholar
• Yu , K.F . 1989 . On fixed-width confidence intervals associated with maximum likelihood estimation . J. Theoret. Probab , 2 : 193 – 199 .
   Google Scholar
• Zhang , C . 1988 . A nonlinear renewal theory . Ann. Probab , 16 : 793 – 824 .
   Web of Science ®Google Scholar