scholarly journals An asymptotic optimal design

1991 ◽  
Vol 4 (4) ◽  
pp. 357-361 ◽  
Author(s):  
Kamel Rekab
Keyword(s):  
Optimal Design ◽  
Bayesian Approach ◽  
Sequential Design ◽  
Asymptotically Optimal ◽  
Posterior Mean ◽  
Normal Populations

The problem of designing an experiment to estimate the product of the means of two normal populations is considered. A Bayesian approach is adopted in which the product of the means is estimated by its posterior mean. A fully sequential design is proposed and shown to be asymptotically optimal.

scholarly journals A note on the number of irrational odd zeta values

2020 ◽  
Vol 156 (8) ◽  
pp. 1699-1717
Author(s):  
Li Lai ◽  
Pin Yu
Keyword(s):  
Optimal Design ◽  
Lower Bound ◽  
Closed Form ◽  
Rational Functions ◽  
Asymptotically Optimal ◽  
Zeta Values

AbstractWe prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.

scholarly journals On the Estimation of the Credibility Factor: A Bayesian Approach

1995 ◽  
Vol 25 (2) ◽  
pp. 137-151 ◽  
Author(s):  
René Schnieper
Keyword(s):  
Bayesian Approach ◽  
Statistical Models ◽  
Random Variables ◽  
Weighted Average ◽  
Credibility Theory ◽  
Structure Parameters ◽  
A Posteriori ◽  
Unknown Parameters ◽  
Practical Applications ◽  
Posterior Mean

AbstractIn practical applications of Credibility Theory the structure parameters usually have to be estimated from the data. This leads to an estimator of the a posteriori mean which is often biased and where the credibility factor depends on the data. A more coherent approach to the problem would be to also treat the unknown parameters as random variables and to simultaneously estimate the a posteriori mean and the structure parameters. Different statistical models are proposed which allow for such a solution. These models all lead to an estimation of the posterior mean which is a weighted average of the prior mean and of the observed mean, the weights depending on the observations.

scholarly journals Asymptotically Optimal Sequential Design for Rank Aggregation

2022 ◽  
Author(s):  
Xi Chen ◽  
Yunxiao Chen ◽  
Xiaoou Li
Keyword(s):  
Decision Maker ◽  
Sequential Analysis ◽  
Large Deviation ◽  
Asymptotic Optimality ◽  
Rank Aggregation ◽  
Sequential Design ◽  
Final Decision ◽  
Asymptotic Results ◽  
Asymptotically Optimal ◽  
Sequential Procedures

A sequential design problem for rank aggregation is commonly encountered in psychology, politics, marketing, sports, etc. In this problem, a decision maker is responsible for ranking K items by sequentially collecting noisy pairwise comparisons from judges. The decision maker needs to choose a pair of items for comparison in each step, decide when to stop data collection, and make a final decision after stopping based on a sequential flow of information. Because of the complex ranking structure, existing sequential analysis methods are not suitable. In this paper, we formulate the problem under a Bayesian decision framework and propose sequential procedures that are asymptotically optimal. These procedures achieve asymptotic optimality by seeking a balance between exploration (i.e., finding the most indistinguishable pair of items) and exploitation (i.e., comparing the most indistinguishable pair based on the current information). New analytical tools are developed for proving the asymptotic results, combining advanced change of measure techniques for handling the level crossing of likelihood ratios and classic large deviation results for martingales, which are of separate theoretical interest in solving complex sequential design problems. A mirror-descent algorithm is developed for the computation of the proposed sequential procedures.

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