The Three Sigma Rule

Pukelsheim, Friedrich

doi:10.1080/00031305.1994.10476030

Cited by 770 publications

(268 citation statements)

References 5 publications

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“…by a statistical rule of thumb for unimodal distributions known as the 3-Sigma Rule (Pukelsheim, 1994). Possibly the same value will do for all SSTs, or maybe it will need to be modified by latitude.…”

Section: Specifying the Diagonal Componentsmentioning

confidence: 99%

Probabilistic Inference for Future Climate Using an Ensemble of Climate Model Evaluations

2007

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This paper describes an approach to computing probabilistic assessments of future climate, using a climate model. It clarifies the nature of probability in this context, and illustrates the kinds of judgements that must be made in order for such a prediction to be consistent with the probability calculus. The climate model is seen as a tool for making probabilistic statements about climate itself, necessarily involving an assessment of the model's imperfections. A climate event, such as a 2 • C increase in global mean temperature, is identified with a region of 'climate-space', and the ensemble of model evaluations is used within a numerical integration designed to estimate the probability assigned to that region.

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Section: Specifying the Diagonal Componentsmentioning

confidence: 99%

Probabilistic Inference for Future Climate Using an Ensemble of Climate Model Evaluations

2007

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“…For the error term, it is usual to use a symmetric representation centred on zero. Pukelsheim [23] suggests setting error bounds as v t1 = −3σ and v tJ = 3σ , where σ is the standard deviation of e. Since σ is unknown, it can be replaced by an estimate. Campbell and Carter Hill [5] consider two possibilities: (1)σ from the OLS regression and (2) the sample standard deviation of y.…”

Section: Background Theorymentioning

confidence: 99%

Estimating utility functions using generalized maximum entropy

Pires

Dionísio

Coelho

2012

Journal of Applied Statistics

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This paper estimates von Neumann and Morgenstern utility functions using the generalized maximum entropy (GME), applied to data obtained by utility elicitation methods. Given the statistical advantages of this approach, we provide a comparison of the performance of the GME estimator with ordinary least square (OLS) in a real data small sample setup. The results confirm the ones obtained for small samples through Monte Carlo simulations. The difference between the two estimators is small and it decreases as the width of the parameter support vector increases. Moreover, the GME estimator is more precise than the OLS one. Overall, the results suggest that GME is an interesting alternative to OLS in the estimation of utility functions when data are generated by utility elicitation methods.

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“…A criterion of the implausibility threshold I(x)  3 is applied [1] following the Pukelsheim three sigma rule [28] (around 99%, or to say nearly all values in a distribution lie within a three-fold standard variance band on either hand of the mean). Sample points that fail the criterion are considered implausible.…”

Section: History Matchingmentioning

confidence: 99%

Sampling Schemes for History Matching Using Subset Simulation

Gong¹,

O²,

Beer³

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Abstract. History Matching (HM) is a form of model calibration suitable for high-dimensional and computationally expensive numerical models. It sequentially cuts down the input space to find the non-implausible domain that provides a reasonable match between the output and experimental data. The non-implausible domain can be orders of magnitude smaller than the original input space and it can have a complex topology. This leads to one of the most challenging open problems in implementing HM, namely, the efficient generation of samples in the non-implausible set. Previous work has shown that Subset Simulation can be used to solve this problem. Unlike Direct Monte Carlo, Subset Simulation progressively decomposes a rare event (here is the non-implausible set), which has very small failure probabilities, into sequential less rare nested events. The original Subset Simulation uses a Modified Metropolis algorithm to generate the conditional samples that belong to intermediate less rare failure events. Generating samples moving forwards to the target space is the heart for Subset Simulation. This work considers different sampling strategies to generate samples and compares their performance in the context of expensive model calibration. A numerical example is provided to show the potential of HM using different Subset Simulation sampling schemes.

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The Three Sigma Rule

Abstract: For random variables with a unimodal Lebesgue density the 3σ rule is proved by elementary calculus. It emerges as a special case of the Vysochanskiȋ-Petunin inequality, which in turn is based on the Gauss inequality.

Cited by 770 publications

References 5 publications

Probabilistic Inference for Future Climate Using an Ensemble of Climate Model Evaluations

Probabilistic Inference for Future Climate Using an Ensemble of Climate Model Evaluations

Estimating utility functions using generalized maximum entropy

Sampling Schemes for History Matching Using Subset Simulation

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