Uniformly best invariant stopping rules

The class of stopping rules for a sequence of i.i.d. random variables with partially known distribution is restricted by requiring invariance with respect to certain transformations. Invariant stopping rules have an intuitive appeal when the optimal stopping problem is invariant with respect to the actual gain function. Uniformly best invariant stopping rules are derived for the gamma distribution with known shape parameter and unknown scale parameter, for the uniform distribution with both endpoints unknown, and for the normal distribution with unknown mean and variance. Some comparisons with previously published results are made.

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Stopping at the maximum of geometric Brownian motion when signals are received

Journal of Applied Probability ◽

10.1017/s0021900200000802 ◽

2005 ◽

Vol 42 (03) ◽

pp. 826-838 ◽

Cited By ~ 2

Author(s):

X. Guo ◽

J. Liu

Keyword(s):

Brownian Motion ◽

Optimal Stopping ◽

Stopping Time ◽

Geometric Brownian Motion ◽

Stopping Rules ◽

Optimal Stopping Problem ◽

Signal Process ◽

Optimal Stopping Time ◽

A Free Boundary Problem ◽

Random Times

Consider a geometric Brownian motion X t (ω) with drift. Suppose that there is an independent source that sends signals at random times τ 1 < τ 2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward S τ , where S t = max(max0≤u≤t X u , s) for some constant s ≥ X 0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−r τ i S τ i | X 0 = x, S 0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ * is governed by a threshold a * such that τ * = inf{τ n : X τ n ≤a * S τ n }. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ * = τ 1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

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Bayesian Estimation for the Centered Parameterization of the Skew-Normal Distribution

Revista Colombiana de Estadística ◽

10.15446/rce.v40n1.55244 ◽

2017 ◽

Vol 40 (1) ◽

pp. 123-140 ◽

Cited By ~ 2

Author(s):

Paulino Pérez-Rodríguez ◽

José A. Villaseñor ◽

Sergio Pérez ◽

Javier Suárez

Keyword(s):

Normal Distribution ◽

Shape Parameter ◽

Mean Squared Error ◽

Scale Parameter ◽

Posterior Distributions ◽

Skew Normal Distribution ◽

Inference Problems ◽

Squared Error ◽

Practical Standpoint ◽

Skew Normal

The skew-normal (SN) distribution is a generalization of the normal distribution, where a shape parameter is added to adopt skewed forms. The SN distribution has some of the properties of a univariate normal distribution, which makes it very attractive from a practical standpoint; however, it presents some inference problems. Specifically, the maximum likelihood estimator for the shape parameter tends to infinity with a positive probability. A new Bayesian approach is proposed in this paper which allows to draw inferences on the parameters of this distribution by using improper prior distributions in the ``centered parametrization'' for the location and scale parameter and a Beta-type for the shape parameter. Samples from posterior distributions are obtained by using the Metropolis-Hastings algorithm. A simulation study shows that the mode of the posterior distribution appears to be a good estimator in terms of bias and mean squared error. A comparative study with similar proposals for the SN estimation problem was undertaken. Simulation results provide evidence that the proposed method is easier to implement than previous ones. Some applications and comparisons are also included.

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Stopping at the maximum of geometric Brownian motion when signals are received

Journal of Applied Probability ◽

10.1239/jap/1127322030 ◽

2005 ◽

Vol 42 (3) ◽

pp. 826-838 ◽

Cited By ~ 9

Author(s):

X. Guo ◽

J. Liu

Keyword(s):

Brownian Motion ◽

Optimal Stopping ◽

Stopping Time ◽

Geometric Brownian Motion ◽

Stopping Rules ◽

Optimal Stopping Problem ◽

Signal Process ◽

Optimal Stopping Time ◽

A Free Boundary Problem ◽

Random Times

Consider a geometric Brownian motion Xt(ω) with drift. Suppose that there is an independent source that sends signals at random times τ1 < τ2 < ⋯. Upon receiving each signal, a decision has to be made as to whether to stop or to continue. Stopping at time τ will bring a reward Sτ, where St = max(max0≤u≤tXu, s) for some constant s ≥ X0. The objective is to choose an optimal stopping time to maximize the discounted expected reward E[e−rτiSτi | X0 = x, S0 = s], where r is a discount factor. This problem can be viewed as a randomized version of the Bermudan look-back option pricing problem. In this paper, we derive explicit solutions to this optimal stopping problem, assuming that signal arrival is a Poisson process with parameter λ. Optimal stopping rules are differentiated by the frequency of the signal process. Specifically, there exists a threshold λ* such that if λ>λ*, the optimal stopping problem is solved via the standard formulation of a ‘free boundary’ problem and the optimal stopping time τ* is governed by a threshold a* such that τ* = inf{τn: Xτn≤a*Sτn}. If λ≤λ* then it is optimal to stop immediately a signal is received, i.e. at τ* = τ1. Mathematically, it is intriguing that a smooth fit is critical in the former case while irrelevant in the latter.

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CHARACTERIZATIONS OF OPTIMAL POLICIES IN A GENERAL STOPPING PROBLEM AND STABILITY ESTIMATING

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964814000035 ◽

2014 ◽

Vol 28 (3) ◽

pp. 335-352 ◽

Cited By ~ 2

Author(s):

Evgueni Gordienko ◽

Andrey Novikov

Keyword(s):

Optimal Stopping ◽

Transition Probabilities ◽

Measurable Space ◽

Stopping Rules ◽

Stopping Times ◽

Optimal Stopping Problem ◽

The Stability ◽

Image Position ◽

Optimal Stopping Times ◽

Natural Classes

We consider an optimal stopping problem for a general discrete-time process X1, X2, …, Xn, … on a common measurable space. Stopping at time n (n = 1, 2, …) yields a reward Rn(X1, …, Xn) ≥ 0, while if we do not stop, we pay cn(X1, …, Xn) ≥ 0 and keep observing the process. The problem is to characterize all the optimal stopping times τ, i.e., such that maximize the mean net gain: $$E(R_\tau(X_1,\dots,X_\tau)-\sum_{n=1}^{\tau-1}c_n(X_1,\dots,X_n)).$$ We propose a new simple approach to stopping problems which allows to obtain not only sufficient, but also necessary conditions of optimality in some natural classes of (randomized) stopping rules.In the particular case of Markov sequence X1, X2, … we estimate the stability of the optimal stopping problem under perturbations of transition probabilities.

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Methodological Remarks Regarding Optimal Stopping Tasks and the Implications for Sampling Biases

10.31234/osf.io/5j74d ◽

2021 ◽

Author(s):

Didrika Sahira van de Wouw ◽

Ryan McKay ◽

Nicholas Furl

Keyword(s):

Optimal Stopping ◽

Sampling Rate ◽

Optimal Stopping Problem ◽

Main Study ◽

Optimal Model ◽

Pilot Studies ◽

Mean And Variance ◽

The Mean ◽

Task Features ◽

Sampling Biases

This paper investigates a type of optimal stopping problem where options are presented in sequence and, once an option has been rejected, it is impossible to go back to it. With previous research finding mixed results of undersampling and oversampling biases on these kinds of optimal stopping tasks, the question remaining is what causes people to sample too much or too little compared to models of optimality? In two pilot studies and a main study, we explored task features that could lead to over- versus undersampling on number-based tasks. We found that, regardless of task features, there were no significant differences in human sampling rate across conditions. Nevertheless, we observed differences in sampling biases across conditions due to varying sampling rates of the optimal model. Our optimal model, like most models used for this type of optimal stopping problem, requires that researchers specify the mean and variance of a theoretical distribution, from which the options are generated. We show that different ways of specifying this generating distribution can lead to different model sampling rates, and consequently, differences in sampling biases. This highlights that a correct specification of the generating distribution is critical when investigating sampling biases on optimal stopping tasks.

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A Minimax Procedure for Selecting the Population with the Largest (Smallest) Scale Parameter

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319720304 ◽

1972 ◽

Vol 21 (3-4) ◽

pp. 143-154 ◽

Cited By ~ 3

Author(s):

J. B. Ofosu

Keyword(s):

Weibull Distribution ◽

Normal Distribution ◽

Shape Parameter ◽

Scale Parameter ◽

Scale Parameters ◽

Parameter Problem

Summary A MINIMAX procedure is given for selecting the population with the largest scale parameter from k gamma populations with unknown scale parameters and a common known shape parameter. The main results are applied to the scale parameter problem for the Weibull distribution as well as the normal distribution with known and unknown mean.

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