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.

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 On the asymptotic optimality of greedy index heuristics for multi-action restless bandits

2015 ◽  
Vol 47 (03) ◽  
pp. 652-667
Author(s):  
D. J. Hodge ◽  
K. D. Glazebrook
Keyword(s):  
Differential Equation ◽  
Resource Allocation ◽  
Lagrangian Relaxation ◽  
Asymptotic Optimality ◽  
General Setting ◽  
Resource Constraint ◽  
Asymptotically Optimal ◽  
Restless Bandits ◽  
Multiple Levels

The class of restless bandits as proposed by Whittle (1988) have long been known to be intractable. This paper presents an optimality result which extends that of Weber and Weiss (1990) for restless bandits to a more general setting in which individual bandits have multiple levels of activation but are subject to an overall resource constraint. The contribution is motivated by the recent works of Glazebrook et al. (2011a), (2011b) who discussed the performance of index heuristics for resource allocation in such systems. Hitherto, index heuristics have been shown, under a condition of full indexability, to be optimal for a natural Lagrangian relaxation of such problems in which a resource is purchased rather than constrained. We find that under key assumptions about the nature of solutions to a deterministic differential equation that the index heuristics above are asymptotically optimal in a sense described by Whittle. We then demonstrate that these assumptions always hold for three-state bandits.

scholarly journals Asymptotic Optimality of Estimating Function Estimator for CHARN Model

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Tomoyuki Amano
Keyword(s):  
Time Series ◽  
Least Squares ◽  
Financial Time Series ◽  
Asymptotic Optimality ◽  
Time Series Models ◽  
Estimating Function ◽  
Asymptotically Optimal ◽  
Financial Time ◽  
Conditional Least Squares ◽  
Better Than

CHARN model is a famous and important model in the finance, which includes many financial time series models and can be assumed as the return processes of assets. One of the most fundamental estimators for financial time series models is the conditional least squares (CL) estimator. However, recently, it was shown that the optimal estimating function estimator (G estimator) is better than CL estimator for some time series models in the sense of efficiency. In this paper, we examine efficiencies of CL and G estimators for CHARN model and derive the condition that G estimator is asymptotically optimal.

scholarly journals On the asymptotic optimality of greedy index heuristics for multi-action restless bandits

2015 ◽  
Vol 47 (3) ◽  
pp. 652-667 ◽  
Author(s):  
D. J. Hodge ◽  
K. D. Glazebrook
Keyword(s):  
Differential Equation ◽  
Resource Allocation ◽  
Lagrangian Relaxation ◽  
Asymptotic Optimality ◽  
General Setting ◽  
Resource Constraint ◽  
Asymptotically Optimal ◽  
Restless Bandits ◽  
Multiple Levels

The class of restless bandits as proposed by Whittle (1988) have long been known to be intractable. This paper presents an optimality result which extends that of Weber and Weiss (1990) for restless bandits to a more general setting in which individual bandits have multiple levels of activation but are subject to an overall resource constraint. The contribution is motivated by the recent works of Glazebrook et al. (2011a), (2011b) who discussed the performance of index heuristics for resource allocation in such systems. Hitherto, index heuristics have been shown, under a condition of full indexability, to be optimal for a natural Lagrangian relaxation of such problems in which a resource is purchased rather than constrained. We find that under key assumptions about the nature of solutions to a deterministic differential equation that the index heuristics above are asymptotically optimal in a sense described by Whittle. We then demonstrate that these assumptions always hold for three-state bandits.

OPTIMAL POINTWISE APPROXIMATION OF INFINITE-DIMENSIONAL ORNSTEIN–UHLENBECK PROCESSES

2008 ◽  
Vol 08 (03) ◽  
pp. 519-541 ◽  
Author(s):  
THOMAS MÜLLER-GRONBACH ◽  
KLAUS RITTER ◽  
TIM WAGNER
Keyword(s):  
Spatial Dimension ◽  
Optimal Order ◽  
Optimal Algorithms ◽  
Pointwise Approximation ◽  
Asymptotic Results ◽  
Uhlenbeck Process ◽  
Simulation Experiments ◽  
Asymptotically Optimal ◽  
Infinite Dimensional ◽  
Asymptotic Error

We consider an infinite-dimensional Ornstein–Uhlenbeck process on the spatial domain ]0,1[d driven by an additive nuclear or space–time white noise, and we study the approximation of this process at a fixed point in time. We determine the order of the minimal errors as well as asymptotically optimal algorithms, both of which depend on the spatial dimension d and on the decay of the eigenvalues of the driving Wiener process W in the case of nuclear noise. In particular, the optimal order is achieved by employing drift-implicit Euler schemes with non-uniform time discretizations, while uniform time discretizations turn out to be suboptimal in general. By means of non-asymptotic error bounds and by simulation experiments, we show that the asymptotic results are predictive for the actual errors already for time discretizations with a small number of points.

scholarly journals Efficient Simulation Budget Allocation for Ranking the TopmDesigns

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hui Xiao ◽  
Loo Hay Lee
Keyword(s):  
Convergence Rate ◽  
Optimal Allocation ◽  
Large Deviation ◽  
Budget Allocation ◽  
Rate Function ◽  
Sequential Algorithm ◽  
Allocation Rule ◽  
Asymptotically Optimal ◽  
Efficient Simulation ◽  
Optimal Computing Budget Allocation

We consider the problem of ranking the topmdesigns out ofkalternatives. Using the optimal computing budget allocation framework, we formulate this problem as that of maximizing the probability of correctly ranking the topmdesigns subject to the constraint of a fixed limited simulation budget. We derive the convergence rate of the false ranking probability based on the large deviation theory. The asymptotically optimal allocation rule is obtained by maximizing this convergence rate function. To implement the simulation budget allocation rule, we suggest a heuristic sequential algorithm. Numerical experiments are conducted to compare the effectiveness of the proposed simulation budget allocation rule. The numerical results indicate that the proposed asymptotically optimal allocation rule performs the best comparing with other allocation rules.

scholarly journals Tools to Estimate the First Passage Time to a Convex Barrier

2005 ◽  
Vol 42 (1) ◽  
pp. 61-81
Author(s):  
Ola Hammarlid
Keyword(s):  
Limit Distribution ◽  
Sequential Analysis ◽  
Large Deviation ◽  
First Passage Time ◽  
Stopping Time ◽  
Random Variable ◽  
Passage Time ◽  
First Passage ◽  
Linear Barrier ◽  
Asymptotic Normal

The first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear barrier is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties.

scholarly journals Group Decision Support System Using The Analytic Network Process and Borda Methods for Selecting

2019 ◽  
Vol 13 (4) ◽  
pp. 333
Author(s):  
Beta Yudha Mahindarta ◽  
Retantyo Wardoyo
Keyword(s):  
Decision Support ◽  
Decision Support System ◽  
Decision Maker ◽  
Support System ◽  
Group Decision ◽  
Housing Development ◽  
Final Decision ◽  
Group Decision Support System ◽  
Individual Decision ◽  
Selection Of

The amount of land for the current location of housing development has resulted in developers choosing the location of housing development regardless of the condition of the land, infrastructure, socio-economic. To overcome this problem a computer system is needed in the form of a GDSS that can assist in the selection of Housing Development Locations.This study aims to implement a GDSS with ANP and Borda methods to determine the selection of the right and fast housing development location. GDSS is needed because there are 3 Individual Decision Makers, DM-1  assessing based on Land Conditions, DM-2 assessing Infrastructure-based, DM-3 assess the Socio-Economic and Decision Maker based groups to make the final decision. The ANP method is used to weight the criteria from each alternative location, to the alternative ranking of housing construction locations for each individual Decision Maker. The Borda method is used to combine the results of ranking carried out by the Group Decision Maker so that it gets the final ranking as a determinant of the Location of Housing Development.The final result of this research is a decision support system that can help developers to get a priority recommendation according to the needs of the developer.

scholarly journals A Restless Bandit Model for Resource Allocation, Competition, and Reservation

2021 ◽  
Author(s):  
Jing Fu ◽  
Bill Moran ◽  
Peter G. Taylor
Keyword(s):  
Resource Allocation ◽  
Numerical Experiments ◽  
Asymptotic Optimality ◽  
Sufficient Condition ◽  
Limited Capacity ◽  
Index Policy ◽  
Asymptotically Optimal ◽  
Restless Bandit ◽  
Simple Sufficient Condition ◽  
Resource Capacity

In “A Restless Bandit Model for Resource Allocation, Competition and Reservation,” J. Fu, B. Moran, and P. G. Taylor study a resource allocation problem with varying requests and with resources of limited capacity shared by multiple requests. This problem is modeled as a set of heterogeneous restless multi-armed bandit problems (RMABPs) connected by constraints imposed by resource capacity. Following Whittle’s idea of relaxing the constraints and Weber and Weiss’s proof of asymptotic optimality, the authors propose an index policy and establish conditions for it to be asymptotically optimal in a regime where both arrival rates and capacities increase. In particular, they provide a simple sufficient condition for asymptotic optimality of the policy and, in complete generality, propose a method that generates a set of candidate policies for which asymptotic optimality can be checked. Via numerical experiments, they demonstrate the effectiveness of these results even in the pre-limit case.

scholarly journals FORMAL AND DAILY CARE DECISION-MAKING INVOLVEMENT IN AFRICAN-AMERICAN DEMENTIA DYADS

2019 ◽  
Vol 3 (Supplement_1) ◽  
pp. S915-S915
Author(s):  
Kalisha Bonds ◽  
MinKyoung Song ◽  
Karen Lyons ◽  
Martha Driessnack
Keyword(s):  
Decision Making ◽  
African American ◽  
Decision Maker ◽  
Sense Of Self ◽  
Final Decision ◽  
The Novel ◽  
Structured Interviews ◽  
Daily Care ◽  
Care Decision

Abstract Decision-making involvement (e.g., verbal and/or nonverbal communication) of persons with dementia (PWD) has been associated with quality of life of PWDs and their caregivers, underscores personhood, and reduces ethical dilemmas for caregivers regarding the PWD’s care. Yet, no study has explored the decision-making involvement in formal and daily care of both members of African-American dementia dyads (i.e., African-American PWDs and their African-American caregivers), limiting our understanding of how these dyads navigate decision-making during the dementia trajectory. This study took a closer look through in-depth, semi-structured interviews with African-American dementia dyads as they reflected on their decision-making surrounding formal and daily care. A pilot study of five dyadic interviews, each averaging 45 minutes, was completed. We used a combination of quantitative content analysis, decision-making matrices and I-poems created from I-statements of the dyad regarding their decision-making involvement. Decision-making matrices (i.e., diagrams of the degree of sharing, the balance of power within the dyad, and the final decision maker in formal and daily care) were constructed across interviews. The pairing of traditional analyses with the novel use of I-poems traces participants’ sense of self, ensuring their voice is retained. There was agreement within all five dyads regarding the final decision maker(s) in formal and daily care. Between dyads, daily decision-making involvement was led by African American PWDs; whereas, formal care decision-making involvement of African American PWDs varied. Findings highlight the importance of a deeper understanding of formal and daily care decision-making involvement within and between African-American dementia dyads and potential clinical implications.

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