Life Testing

Epstein, Benjamin; Sobel, Milton

doi:10.1080/01621459.1953.10483488

Cited by 349 publications

(104 citation statements)

References 5 publications

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“…It is a matter of debate whether power law distributions, which are valid descriptions of the numerous small and intermediate events, can be extrapolated to large events; the largest events are, almost by definition, undersampled. Based on analytical calculations and synthetic tests, we demonstrate that the observation of a few tens of the largest events is sufficient to qualify their distribution with good precision, using the rank-ordering technique, initially introduced in linguistics [Zipf, 1949] and later largely used in statistics [Epstein and Sobel, 1953;Gumbel, 1960;Deemer and Votaw, 1955].…”

Section: Introductionmentioning

confidence: 99%

Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes

Sornette¹,

Knopoff²,

Kagan³

et al. 1996

J. Geophys. Res.

183

148

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Rank-ordering statistics provides a perspective on the rare, largest elements of a population, whereas the statistics of cumulative distributions are dominated by the more numerous small events. The exponent of a power law distribution can be determined with good accuracy by rank-ordering statistics from the observation of only a few tens of the largest events. Using analytical results and synthetic tests, we quantify the systematic and the random errors.We also study the case of a distribution defined by two branches, each having a power law distribution, one defined for the largest events and the other for smaller events, with application to the World-Wide (Harvard) and Southern California earthquake catalogs. In the case of the Harvard moment catalog, we make more precise earlier claims of the existence of a transition of the earthquake magnitude distribution between small and large earthquakes; the b-values are b 2 = 2.3 ± 0.3 for large shallow earthquakes and b 1 = 1.00 ± 0.02 for smaller shallow earthquakes. However, the cross-over magnitude between the two distributions is ill-defined. The data available at present do not provide a strong constraint on the cross-over which has a 50% probability of being between magnitudes 7.1 and 7.6 for shallow earthquakes; this interval may be too conservatively estimated. Thus, any influence of a universal geometry of rupture on the distribution of earthquakes world-wide is ill-defined at best. We caution that there is no direct evidence to confirm the hypothesis that the 1 large-moment branch is indeed a power law. In fact, a gamma distribution fits the entire suite of earthquake moments from the smallest to the largest satisfactorily.There is no evidence that the earthquakes of the Southern California catalog have a distribution with two branches, or that a rolloff in the distribution is needed; for this catalog b = 1.00 ± 0.02 up to the largest magnitude observed, M W ≃ 7.5; hence we conclude that the thickness of the seismogenic layer has no observable influence whatsoever on the frequency distribution in this region.2

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Section: Introductionmentioning

confidence: 99%

Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes

Sornette¹,

Knopoff²,

Kagan³

et al. 1996

J. Geophys. Res.

183

148

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“…For comparison purposes we shall consider from the Bayesian viewpoint the same problem Epstein and Sobel treated classically in their celebrated 1953 paper [7].…”

Section: Exponential Time-to-failure Distributionmentioning

confidence: 99%

The Bayesian Inference Method and Its Application to Reliability Problems

Smith¹

1982

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“…For the exponential model we can use the expression for the expected value of the generic ordered valuation provided in Epstein and Sobel (1953): Introducing this expression in the general expression of the expected profit (12) we obtain…”

Section: Exponential Modelmentioning

confidence: 99%

The value of location in keyword auctions

Naldi

D'Acquisto²,

Italiano

2010

Electronic Commerce Research and Applications

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a b s t r a c tSponsored links on search engines are an emerging advertising tool, whereby a number of slots are put on sale through keyword auctions. This is also known as contextual advertising. Slot assignment and pricing in keyword auctions are then essential for the search engine's management since provide the main stream of revenues, and are typically accomplished by the Generalized Second Price (GSP) mechanism. In GSP the price of slots is a monotone function of the slot location, being larger for the highest slots. Though a higher location is associated with larger revenues, the lower costs associated with the lowest slots may make them more attractive for the advertiser. The contribution of this research is to show, by analytical and simulation results based on the theory of order statistics, that advertisers may not get the optimal slot they aim at (the slot maximizing their expected profit) and that the GSP mechanism may be unfair to all the winning bidders but the one who submitted the lowest bid.

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Life Testing

Cited by 349 publications

References 5 publications

Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes

Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes

The Bayesian Inference Method and Its Application to Reliability Problems

The value of location in keyword auctions

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