Optimal Stopping of a Random Sequence with Unknown Distribution

2021 ◽  
Author(s):  
Alexander Goldenshluger ◽  
Assaf Zeevi
Keyword(s):  
Optimal Stopping ◽  
Numerical Experiments ◽  
Random Sequence ◽  
Stopping Rule ◽  
Asymptotically Optimal ◽  
First Order ◽  
Unknown Distribution ◽  
Distributional Information ◽  
The Subject ◽  
General Method

The subject of this paper is the problem of optimal stopping of a sequence of independent and identically distributed random variables with unknown distribution. We propose a stopping rule that is based on relative ranks and study its performance as measured by the maximal relative regret over suitable nonparametric classes of distributions. It is shown that the proposed rule is first-order asymptotically optimal and nearly rate optimal in terms of the rate at which the relative regret converges to zero. We also develop a general method for numerical solution of sequential stopping problems with no distributional information and use it in order to implement the proposed stopping rule. Some numerical experiments illustrating performance of the rule are presented as well.

scholarly journals OPTIMAL SELECTION OF THE k-TH BEST CANDIDATE

2019 ◽  
Vol 33 (3) ◽  
pp. 327-347
Author(s):  
Yi-Shen Lin ◽  
Shoou-Ren Hsiau ◽  
Yi-Ching Yao
Keyword(s):  
Optimal Stopping ◽  
Stopping Rule ◽  
Secretary Problem ◽  
Optimal Selection ◽  
Maximum Probability ◽  
Optimal Stopping Rule ◽  
The Subject ◽  
Selection Of

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of n candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the kth best candidate for k ≥ 2. While the optimal stopping rule for k=1,2 (and all n ≥ 2) is known to be of threshold type (involving one threshold), we solve the case k=3 (and all n ≥ 3) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for p(k, n), the maximum probability of selecting the k-th best of n candidates. It is shown that (i) p(1, n) = p(n, n) > p(k, n) for 1<k<n, (ii) p(k, n) ≥ p(k, n + 1), (iii) p(k, n) ≥ p(k + 1, n + 1) and (iv) p(k, ∞): = lim n→∞p(k, n) is decreasing in k.

Towards the Temporal Approach to Abstract Data Types

1988 ◽  
Vol 11 (1) ◽  
pp. 49-63
Author(s):  
Andrzej Szalas
Keyword(s):  
Temporal Logic ◽  
Abstract Data Types ◽  
Data Types ◽  
First Order ◽  
Abstract Data ◽  
Order Completeness ◽  
General Method

In this paper we deal with a well known problem of specifying abstract data types. Up to now there were many approaches to this problem. We follow the axiomatic style of specifying abstract data types (cf. e.g. [1, 2, 6, 8, 9, 10]). We apply, however, the first-order temporal logic. We introduce a notion of first-order completeness of axiomatic specifications and show a general method for obtaining first-order complete axiomatizations. Some examples illustrate the method.

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

scholarly journals Stabilised explicit Adams-type methods

2021 ◽  
pp. 82-98
Author(s):  
Vasily I. Repnikov ◽  
Boris V. Faleichik ◽  
Andrew V. Moisa
Keyword(s):  
Numerical Experiments ◽  
Step Method ◽  
Type Method ◽  
Constrained Optimisation ◽  
First Order ◽  
Error Constant ◽  
Stability Intervals ◽  
Multi Step Methods ◽  
First Order Methods ◽  
Accuracy And Stability

In this work we present explicit Adams-type multi-step methods with extended stability intervals, which are analogous to the stabilised Chebyshev Runge – Kutta methods. It is proved that for any k ≥ 1 there exists an explicit k-step Adams-type method of order one with stability interval of length 2k. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In the general case, to construct a k-step method of order p it is necessary to solve a constrained optimisation problem in which the objective function and p constraints are second degree polynomials in k variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.

scholarly journals Criterios definidores y conceptuales del enfoque basado en derechos humanos: de la teoría a la práctica

2017 ◽  
pp. 43-86
Author(s):  
Alfonso CHACÓN MATA
Keyword(s):  
Human Rights ◽  
United Nations ◽  
Public Administration ◽  
Public Policies ◽  
The State ◽  
The Public ◽  
First Order ◽  
Derechos Humanos ◽  
The Subject ◽  
Human Rights Based Approach

LABURPENA: «Giza Eskubideetan Oinarritutako Ikuspegiaren» irismena azaltzeko asmoa dauka artikulu honek, zer-nolako aldagaiak eta kontzeptuak biltzen dituen ulertze aldera. Horren bilakaeraz eta indarraldiaz arituko gara, bai eta horren modalitate aplikatua nola nabarmentzen den azalduko ere. Horretarako, Nazio Batuen Erakundearen esparruan garatu diren ekarpenak eta gaiaren inguruko doktrina aditua erabiliko ditugu. Administrazio Publikoan duen indarra ezin ukatuzkoa da; izan ere, estatuak eta horri atxikitako erakundeek oso kontuan eduki behar dute politika publikoak norbanakoen eta komunitateen beharrei arreta ematen ari zaien jakiteko balio duela ikuspegi horrek, edota politika publiko horiek kontu emanez gardentasun publikoa eratzen ari diren nahiz edozelako diskriminazioa saihesten ari diren jakiteko balio duela. Azkenik, ikuspegi horrek Giza Eskubideen Gorte Interamerikarraren jurisprudentzia-aurrekari batzuetan duen indarraldia eta eragina aztertuko dira. RESUMEN: El presente artículo tiene la intención de exponer los alcances del «Enfoque Basado en Derechos Humanos», con la finalidad de entender que variables y conceptos involucra. Haremos un recuento de su evolución, vigencia y cómo se evidencia su modalidad aplicada, a través de diferentes aportes desarrollados en el marco de la Organización de Naciones Unidas, así como de la doctrina estudiosa del tema. Su vigencia en la Administración Pública es de primer orden, puesto que el Estado y sus entidades adscritas, deben tener muy en cuenta que el enfoque citado, sirve para conocer si las políticas públicas, están atendiendo necesidades de individuos y comunidades concretas; generando transparencia pública a través de rendición de cuentas, así como evitando cualquier tipo de discriminación. Finalmente, se analizará su vigencia e impacto en algunos antecedentes jurisprudenciales de la Corte Interamericana de Derechos Humanos. ABSTRACT: This article intends to expose the scope of the «Human Rights Based Approach», in order to understand what variables and concepts it involves. We will recount its evolution, validity and how its applied modality is evidenced, through different contributions developed within the framework of the United Nations, as well as the doctrine studious of the subject. Its validity in the Public Administration is of the first order, since the State and its affiliated entities must take into account that the aforementioned approach serves to know if public policies are addressing the needs of specific individuals and communities; Generating public transparency through accountability, as well as avoiding any type of discrimination. Finally, its validity and impact will be analyzed in some jurisprudential antecedents of the Inter-American Court of Human Rights.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

MR. PINOCCHIO GOES TO WASHINGTON: LYING IN POLITICS

2004 ◽  
Vol 21 (1) ◽  
pp. 167-201 ◽  
Author(s):  
Robert Weissberg
Keyword(s):  
Hannah Arendt ◽  
Moral Outrage ◽  
Ivory Tower ◽  
Truth Telling ◽  
First Order ◽  
The Subject

A more provocative subject than “lying in politics” is difficult to imagine. Everybody, from the proverbial “Joe Sixpack” to ivory-tower philosophers, can wax eloquently on the subject, if only because easy-to-find, shocking (and occasionally sexually “juicy”) examples abound. If moral outrage were judged an essential vitamin, then condemning dishonesty undoubtedly guarantees a daily megadose. Unfortunately, at least for those who crave self-indulgent outrage, the anti-lying case is less than 100 percent compelling. It is a quagmire of the first order, if only because those who cherish frankness also usually confess to lying. Hannah Arendt once suggested that lies are a necessary and justifiable tool of the statesman's trade. Formulating damnation criteria invites mind-boggling paradoxes, and strident defenders of truth-telling, with scant exception, admit that falsehoods are “sometimes” permitted for “good” reasons. And what might be these “good reasons”? Who can say for sure? Centuries of erudite scholarship on this point might be encapsulated as “Lying to me is bad, but I can consent to deceiving others for noble purposes, as artfully decided by myself.”

scholarly journals Trigonometrically-Fitted Methods: A Review

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1197
Author(s):  
Changbum Chun ◽  
Beny Neta
Keyword(s):  
Initial Value Problems ◽  
Polynomial Interpolation ◽  
Multistep Methods ◽  
Error Reduction ◽  
Trigonometric Polynomials ◽  
Linear Multistep Methods ◽  
Initial Value ◽  
First Order ◽  
The Subject ◽  
Trigonometrically Fitted Methods

Numerical methods for the solution of ordinary differential equations are based on polynomial interpolation. In 1952, Brock and Murray have suggested exponentials for the case that the solution is known to be of exponential type. In 1961, Gautschi came up with the idea of using information on the frequency of a solution to modify linear multistep methods by allowing the coefficients to depend on the frequency. Thus the methods integrate exactly appropriate trigonometric polynomials. This was done for both first order systems and second order initial value problems. Gautschi concluded that “the error reduction is not very substantial unless” the frequency estimate is close enough. As a result, no other work was done in this direction until 1984 when Neta and Ford showed that “Nyström’s and Milne-Simpson’s type methods for systems of first order initial value problems are not sensitive to changes in frequency”. This opened the flood gates and since then there have been many papers on the subject.

A CANADIAN GRID SYSTEM IN 3° TRANSVERSE MERCATOR ZONES

1964 ◽  
Vol 18 (2) ◽  
pp. 147-155
Author(s):  
J. Saastamoinen
Keyword(s):  
Free Choice ◽  
Metropolitan Areas ◽  
Central Meridian ◽  
Conformal Map ◽  
Grid System ◽  
Order Accuracy ◽  
Map Projection ◽  
First Order ◽  
The Subject ◽  
Local Plane

All geodetic networks that rest previously fixed control are best computed in plane coordinates. Under this category fall a great deal of first-order triangulation and, of course, the whole volume of lower-order work—all the way down to the last monuments from which detail surveys originate. A geodetic grid must be designed to meet first-order accuracy and should be established on federal rather than provincial level. Its use requires precomputed tables based on some conformal map projection, preferably the Transverse Mercator (Gauss-Krüger) projection. Congruent projection zones, simple scale factor, free choice of central meridian for local plane coordinates in cities and metropolitan areas—all these features of the Transverse Mercator find no parallel in any other map projection. Following a brief introduction to the subject, a set of tables for a proposed Canadian grid system is presented.

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