Confidence Interval Simulation for Systems of Random Variables

2005 ◽  
Vol 129 (3) ◽  
pp. 836-842 ◽  
Author(s):  
Thomas A. Cruse ◽  
Jeffrey M. Brown
Keyword(s):  
Experimental Data ◽  
Confidence Interval ◽  
Bayesian Network ◽  
Upper Bound ◽  
Random Variables ◽  
Network Models ◽  
Probability Of Failure ◽  
Cumulative Distribution ◽  
Probabilistic Design ◽  
Design Assessment

Bayesian network models are seen as important tools in probabilistic design assessment for complex systems. Such network models for system reliability analysis provide a single probability of failure value whether the experimental data used to model the random variables in the problem are perfectly known or derive from limited experimental data. The values of the probability of failure for each of those two cases are not the same, of course, but the point is that there is no way to derive a Bayesian type of confidence interval from such reliability network models. Bayesian confidence (or belief) intervals for a probability of failure are needed for complex system problems in order to extract information on which random variables are dominant, not just for the expected probability of failure but also for some upper bound, such as for a 95% confidence upper bound. We believe that such confidence bounds on the probability of failure will be needed for certifying turbine engine components and systems based on probabilistic design methods. This paper reports on a proposed use of a two-step Bayesian network modeling strategy that provides a full cumulative distribution function for the probability of failure, conditioned by the experimental evidence for the selected random variables. The example is based on a hypothetical high-cycle fatigue design problem for a transport aircraft engine application.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

Estimating the cumulative distribution function for the linear combination of gamma random variables

2017 ◽  
Vol 20 (5) ◽  
pp. 939-951
Author(s):  
Amal Almarwani ◽  
Bashair Aljohani ◽  
Rasha Almutairi ◽  
Nada Albalawi ◽  
Alya O. Al Mutairi
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Cumulative Distribution

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

Measuring Pump Efficiency: Uncertainty Considerations

2005 ◽  
Vol 127 (4) ◽  
pp. 280-284 ◽  
Author(s):  
Noah D. Manring
Keyword(s):  
Experimental Data ◽  
Confidence Interval ◽  
Confidence Intervals ◽  
Academic Research ◽  
Pump Efficiency ◽  
The Past ◽  
Industrial Sales ◽  
Efficiency Data

The objective of this paper is to analyze the uncertainty associated with pump efficiency measurements and to determine reasonable confidence intervals for these data. In the past, many industrial sales and some pieces of academic research have been based upon the experimental data of pump efficiencies; yet few have questioned the accuracy of the experimental data and no one has provided a confidence interval which reflects the range of uncertainty in the measurement. In this paper, a method for calculating this confidence interval is presented and it is shown that substantially large confidence intervals exist within the testing results of a pump. Furthermore, it is recommended that these confidence intervals be included with the efficiency data whenever it is reported.

Probability Bounds Analysis Applied to the Sandia Verification and Validation Challenge Problem

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Aniruddha Choudhary ◽  
Ian T. Voyles ◽  
Christopher J. Roy ◽  
William L. Oberkampf ◽  
Mayuresh Patil
Keyword(s):  
Experimental Data ◽  
Physical Nature ◽  
Distribution Functions ◽  
Verification And Validation ◽  
Cumulative Distribution ◽  
Probability Bounds ◽  
Model Form ◽  
Interval Representation ◽  
Model Predictions ◽  
True System

Our approach to the Sandia Verification and Validation Challenge Problem is to use probability bounds analysis (PBA) based on probabilistic representation for aleatory uncertainties and interval representation for (most) epistemic uncertainties. The nondeterministic model predictions thus take the form of p-boxes, or bounding cumulative distribution functions (CDFs) that contain all possible families of CDFs that could exist within the uncertainty bounds. The scarcity of experimental data provides little support for treatment of all uncertain inputs as purely aleatory uncertainties and also precludes significant calibration of the models. We instead seek to estimate the model form uncertainty at conditions where the experimental data are available, then extrapolate this uncertainty to conditions where no data exist. The modified area validation metric (MAVM) is employed to estimate the model form uncertainty which is important because the model involves significant simplifications (both geometric and physical nature) of the true system. The results of verification and validation processes are treated as additional interval-based uncertainties applied to the nondeterministic model predictions based on which the failure prediction is made. Based on the method employed, we estimate the probability of failure to be as large as 0.0034, concluding that the tanks are unsafe.

and Var[νˆ ] = 4σ + l ) (l −2(1 + c . Similarly, when σˆ ≤ 0.04, νˆ = δˆ + σˆ − 0.04(c ) (7.12) is an estimate for the constant-scaled metric in accordance with FDA Guidance (2001) using a REML UN model. This estimate is asymptoti-cally normally distributed and unbiased with E[νˆ ] = δ +σ −σ − 0.04(c ) and Var[νˆ ] = 4σ . To assess PBE we ‘plug-in’ estimates of δ and the variance components and calculate the upper bound of an asymptotic 90% confidence interval. If this upper bound is below zero we declare that PBE has been shown. Using the code in Appendix B and the data in Section 7.2, we obtain the value −1.90 for log(AUC) and the value −0.95 for log(Cmax). As both of these are below zero, we can declare that T and R are PBE. 7.5.2 PBE using a replicate design Here we fit the same REML UN model as defined in Section 7.4. Let νˆ = δˆ + σˆ + σˆ − (1 + c )(σˆ + σˆ ) (7.13) be an estimate for the reference-scaled metric in accordance with FDA Guidance (2001) when (σˆ + σˆ > 0.04 and using a REML UN model. Then, this estimate is asymptotically normally distributed, un-biased with E[νˆ ] = δ +σ − (1 + c ) and has variance of Var[νˆ ] = 4σ + l + (1 + c ) (l )+ 2l −2(1+c −2(1 + c + 2(1 + c ) (l ) When σˆ + σˆ ≤ 0.04, let νˆ = δˆ + σˆ + σˆ − (σˆ + σˆ )− 0.04(c ) (7.14) be an estimate for the constant-scaled metric in accordance with FDA

2003 ◽  
pp. 369-369
Keyword(s):  
Confidence Interval ◽  
Upper Bound ◽  
Variance Components ◽  
Fda Guidance ◽  
Replicate Design ◽  
Normally Distributed

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed
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