Testing behavior of the reversed hazard rate

Kayid, Mohamed; Al-nahawati, H.; Ahmad, Ibrahim A.

doi:10.1016/j.apm.2010.11.054

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…. Also referred to by some communities as the failure rate, it is a fundamental quantity within several fields of economics and mathematics and has found a lot of applications in survival analysis [KP02], reliability theory [RH04], pricing [HR09,GPZ21] and even forensic analysis [KAnA11]. For our results, we utilize the integral of the hazard rate, H(x) = x 0 h(x) dx, which we call the cumulative hazard rate of D. In particular, we study distributions in which H is an arbitrary polynomial, subject to being a valid cumulative hazard rate.…”

Section: Our Contributionsmentioning

confidence: 99%

Prophet Inequalities for Cost Minimization

Vasilis¹,

Mehta²

2022

Preprint

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Prophet inequalities for rewards maximization are fundamental results from optimal stopping theory with several applications to mechanism design and online optimization. We study the cost minimization counterpart of the classical prophet inequality due to Krengel, Sucheston, and Garling [KS77], where one is facing a sequence of costs X 1 , X 2 , . . . , X n in an online manner and must "stop" at some point and take the last cost seen. 1 Given that the X i 's are independent random variables drawn from known distributions, the goal is to devise a stopping strategy S (online algorithm) that minimizes the expected cost. The best cost possible is E [min i X i ] (offline optimum), achievable only by a prophet who can see the realizations of all X i 's upfront. We say that strategy S is an α-approximation to the prophet (α ≥ 1) ifWe first observe that if the X i 's are not identically distributed, then no strategy can achieve a bounded approximation, no matter if the arrival order is adversarial or random, even when restricted to n = 2 and distributions with support size at most two. 2 This leads us to consider the case where the X i 's are independent and identically distributed (I.I.D.). For the I.I.D. case, we give a complete characterization of the optimal stopping strategy. We show that it achieves a (distribution-dependent) constant-factor approximation to the prophet's cost for almost all distributions and that this constant is tight. In particular, for distributions for which the integral of the hazard rate 3 is a polynomial H(x) = k i=1 a i x di , where d 1 < • • • < d k , the approximation factor is λ(d 1 ), a decreasing function of d 1 , and is the best possible for H(x) = x d1 . Furthermore, when the hazard rate is monotonically increasing (i.e. the distribution is MHR), we show that this constant is at most 2, and this again is the best possible for the MHR distributions.For the classical prophet inequality for reward maximization, single-threshold strategies have been powerful enough to achieve the best possible approximation factor. Motivated by this, we analyze single-threshold strategies for the cost prophet inequality problem. We design a threshold that achieves a O (polylog n)-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound.We believe that our results may be of independent interest for analyzing approximately optimal (posted price-style) mechanisms for procuring items.

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Section: Our Contributionsmentioning

confidence: 99%

Prophet Inequalities for Cost Minimization

Vasilis¹,

Mehta²

2022

Preprint

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“…The reversed hazard rate condition requires that the density does not increase too rapidly. The condition holds for many commonly considered probability distributions such as uniform, Weibull, gamma, lognormal and exponential distributions (Kayid et al 2011). 19 Furthermore, we assume that Eγ = 0.…”

Section: Incomplete Informationmentioning

confidence: 99%

The better route to global tax coordination: Gradualism or multilateralism?

Konrad

Thum

2021

Canadian J of Economics

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We compare the success probability of multilateral negotiations and sequential negotiations over international tax cooperation. To make this difference relevant, we introduce incomplete information as a friction that can lead to bargaining failure. We find plausible conditions for when multilateral negotiations are more likely to achieve full global tax coordination than a gradual/sequential approach. We also compare different routes of sequential bargaining. We ask whether negotiations should start with countries that are the most unpredictable candidates or the most difficult negotiation should be preserved for the final round. We find that sequencing along this dimension matters. Under plausible conditions, full cooperation is least likely to emerge if the negotiations with the most unpredictable country take place last.Résumé. Quelle approche pour une meilleure coordination fiscale internationale : gradualiste ou multilatéraliste? En matière de coordination fiscale internationale, nous comparons les chances de réussite des négociations multilatérales par rapport aux négociations séquentielles. Pour illustrer la pertinence de cette différence, nous introduisons des informations incomplètes comme points de friction pouvant conduire à l'échec des négociations. Comparativement à une approche graduelle/séquentielle, et sous certaines conditions vraisemblables, nous constatons que les négociations multilatérales ont davantage de chances d'aboutir à une coordination fiscale internationale complète. Nous comparons également les différentes façons d'envisager les négociations séquentielles. Nous cherchons à savoir si les négociations doivent commencer avec les pays les plus imprévisibles, ou si les négociations les plus ardues doivent être réservées pour la phase finale. Dans cette optique, nous constatons qu'une approche séquencée prend toute son importance. Dans des conditions vraisemblables, si les négociations avec les pays imprévisibles se déroulent lors de la phase finale, alors il est très peu probable qu'une coopération totale n'émerge.

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“…Irrespective of the shape of the hazard rate function, the RHR cannot increase on (0, ∞) , as shown byBlock et al (1998). Testing the behaviour of the RHR is dealt with inKayid et al (2011).…”

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confidence: 99%

Some More Properties of the Unit-Gompertz Distribution

Anis¹,

De²

2020

Preprint

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In a recent paper, Mazucheli et al. (2019) introduced the unit-Gompertz (UG) distribution and studied some of its properties. It is a continuous distribution with bounded support, and hence may be useful for modelling life-time phenomena. We present counter-examples to point out some subtle errors in their work, and subsequently correct them. We also look at some other interesting properties of this new distribution. Further, we also study some important reliability measures and consider some stochastic orderings associated with this new distribution.

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Testing behavior of the reversed hazard rate

Cited by 8 publications

References 17 publications

Prophet Inequalities for Cost Minimization

Prophet Inequalities for Cost Minimization

The better route to global tax coordination: Gradualism or multilateralism?

Some More Properties of the Unit-Gompertz Distribution

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