More bounds on the expectation of a convex function of a random variable

1972 ◽  
Vol 9 (04) ◽  
pp. 803-812 ◽  
Author(s):  
Ben-Tal A. ◽  
E. Hochman
Keyword(s):  
Convex Function ◽  
Upper Bound ◽  
Random Variable ◽  
Independent Random Variables ◽  
Multivariate Distributions ◽  
Absolute Deviation ◽  
Additional Information ◽  
Wait And See ◽  
The Mean ◽  
Moment Spaces

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T). The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.

More bounds on the expectation of a convex function of a random variable

1972 ◽  
Vol 9 (4) ◽  
pp. 803-812 ◽  
Author(s):  
Ben-Tal A. ◽  
E. Hochman
Keyword(s):  
Convex Function ◽  
Upper Bound ◽  
Random Variable ◽  
Independent Random Variables ◽  
Multivariate Distributions ◽  
Absolute Deviation ◽  
Additional Information ◽  
Wait And See ◽  
The Mean ◽  
Moment Spaces

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T).The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.

A MEAN–VARIANCE BOUND FOR A THREE-PIECE LINEAR FUNCTION

2007 ◽  
Vol 21 (4) ◽  
pp. 611-621 ◽  
Author(s):  
Karthik Natarajan ◽  
Zhou Linyi
Keyword(s):  
Convex Function ◽  
Closed Form ◽  
Linear Function ◽  
Upper Bound ◽  
Random Variable ◽  
Expected Value ◽  
Variance Bound ◽  
The Mean ◽  
Mean Variance

In this article, we derive a tight closed-form upper bound on the expected value of a three-piece linear convex function E[max(0, X, mX − z)] given the mean μ and the variance σ2 of the random variable X. The bound is an extension of the well-known mean–variance bound for E[max(0, X)]. An application of the bound to price the strangle option in finance is provided.

scholarly journals Risk Assessment of Power Generated from a Wind Turbine in Different Climate Cities in Indonesia.

2019 ◽  
Vol 2 (1) ◽  
pp. 97-102
Author(s):  
Irvan Lumban Gaol ◽  
Charles O. P. Marpaung
Keyword(s):  
Risk Assessment ◽  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Wind Speed ◽  
Wind Turbine ◽  
Power Density ◽  
Upper Bound ◽  
Random Variable ◽  
The Mean ◽  
Software Risk

In this paper, a risk analysis based on Monte Carlo Simulation has been used to examine the power generated from a wind turbine. There are five cities are selected based on the wind speed to be examined the power density generated from the wind turbine. The cities are Kupang, Tanjung Pinang, Krinci, Kotabaru, and Pontianak. Among the five cities, Kupang has the highest wind speed, while Pontianak has the lowest wind speed. In this study, the wind speed is assumed to be an unspecified parameter or a random variable. The Monte Carlo Simulation is run by using a software @RISK. The results show that the mean of power density generated from the wind turbine is found 171.23, 113.97, 71.28, 28.67, and 12.49 W/m2 for Kupang, Tanjung Pinang, Krinci, Kotabaru, and Pontianak respectively. The width of the confidence interval with the level of probability 90% is 110.30, 75.00, 69.10, 19.64, and 7.34 W/m2 for Kupang, Tanjung Pinang, Krinci, Kotabaru, and Pontianak respectively. The upper bound of the confidence intervals are 230.1, 154.7, 113.3, 39.27, and 16.30 W/m2 for Kupang, Tanjung Pinang, Krinci, Kotabaru, and Pontianak respectively, while the lower bounds are 119.8, 79.7, 44.2, and 19.63 W/m2 for Kupang, Tanjung Pinang, Krinci, Kotabaru, and Pontianak respectively. The probability of the power density will exceed the upper bound or will below the lower bound is 5%.

scholarly journals Improved Visual Localization via Graph Filtering

2021 ◽  
Vol 7 (2) ◽  
pp. 20
Author(s):  
Carlos Lassance ◽  
Yasir Latif ◽  
Ravi Garg ◽  
Vincent Gripon ◽  
Ian Reid
Keyword(s):  
Image Retrieval ◽  
Large Scale ◽  
Visual Localization ◽  
Localization Accuracy ◽  
Additional Information ◽  
Feature Representations ◽  
The Mean ◽  
Classical Image ◽  
Vision Based Localization ◽  
Improve Reliability

Vision-based localization is the problem of inferring the pose of the camera given a single image. One commonly used approach relies on image retrieval where the query input is compared against a database of localized support examples and its pose is inferred with the help of the retrieved items. This assumes that images taken from the same places consist of the same landmarks and thus would have similar feature representations. These representations can learn to be robust to different variations in capture conditions like time of the day or weather. In this work, we introduce a framework which aims at enhancing the performance of such retrieval-based localization methods. It consists in taking into account additional information available, such as GPS coordinates or temporal proximity in the acquisition of the images. More precisely, our method consists in constructing a graph based on this additional information that is later used to improve reliability of the retrieval process by filtering the feature representations of support and/or query images. We show that the proposed method is able to significantly improve the localization accuracy on two large scale datasets, as well as the mean average precision in classical image retrieval scenarios.

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

scholarly journals The Ancestry of a Sample of Sequences Subject to Recombination

Genetics ◽  
1999 ◽  
Vol 151 (3) ◽  
pp. 1217-1228 ◽  
Author(s):  
Carsten Wiuf ◽  
Jotun Hein
Keyword(s):  
Genetic Distance ◽  
Population Size ◽  
Upper Bound ◽  
Recombination Rate ◽  
Common Ancestor ◽  
Recent Common Ancestor ◽  
Sequence Length ◽  
Most Recent Common Ancestor ◽  
The Mean ◽  
Mean Time

Abstract In this article we discuss the ancestry of sequences sampled from the coalescent with recombination with constant population size 2N. We have studied a number of variables based on simulations of sample histories, and some analytical results are derived. Consider the leftmost nucleotide in the sequences. We show that the number of nucleotides sharing a most recent common ancestor (MRCA) with the leftmost nucleotide is ≈log(1 + 4N Lr)/4Nr when two sequences are compared, where L denotes sequence length in nucleotides, and r the recombination rate between any two neighboring nucleotides per generation. For larger samples, the number of nucleotides sharing MRCA with the leftmost nucleotide decreases and becomes almost independent of 4N Lr. Further, we show that a segment of the sequences sharing a MRCA consists in mean of 3/8Nr nucleotides, when two sequences are compared, and that this decreases toward 1/4Nr nucleotides when the whole population is sampled. A measure of the correlation between the genealogies of two nucleotides on two sequences is introduced. We show analytically that even when the nucleotides are separated by a large genetic distance, but share MRCA, the genealogies will show only little correlation. This is surprising, because the time until the two nucleotides shared MRCA is reciprocal to the genetic distance. Using simulations, the mean time until all positions in the sample have found a MRCA increases logarithmically with increasing sequence length and is considerably lower than a theoretically predicted upper bound. On the basis of simulations, it turns out that important properties of the coalescent with recombinations of the whole population are reflected in the properties of a sample of low size.

Visualizing the Variance of a Random Variable

2011 ◽  
Vol 18 (01) ◽  
pp. 71-85
Author(s):  
Fabrizio Cacciafesta
Keyword(s):  
Stochastic Dominance ◽  
Mean Absolute Error ◽  
Random Variable ◽  
Absolute Error ◽  
Second Order ◽  
Taylor Formula ◽  
Order Stochastic Dominance ◽  
The Mean ◽  
Second Order Stochastic Dominance ◽  
Options Theory

We provide a simple way to visualize the variance and the mean absolute error of a random variable with finite mean. Some application to options theory and to second order stochastic dominance is given: we show, among other, that the "call-put parity" may be seen as a Taylor formula.

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