An Introduction to Order Statistics

Ahsanullah, Mohammad; Nevzorov, V. B.; Shakil, Mohammad

doi:10.2991/978-94-91216-83-1

Cited by 113 publications

(108 citation statements)

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“…and Φ(x) = as shown in [6]. Using this, we now show that the CMOS with 2-OS yields the same Bayesian accuracy.…”

Section: Theoretical Analysis: Gaussian Distributionsupporting

confidence: 54%

“…One has to resort to tabulated cumulative error functions or to numerical methods to obtain precise percentile values. However, a lot of work has been done in this area for certain OS, and can be found in [6,8,10,11], from which we can make some interesting conclusions.…”

Section: Theoretical Analysis: Gaussian Distributionmentioning

confidence: 99%

“…provided that both sides of the equality exist [6,7]. The fundamental theorem concerning the OS that we invoke is found in many papers [7,8,9].…”

Section: Relevant Background Areas Regarding Order Statisticsmentioning

confidence: 99%

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Optimal “Anti-Bayesian” Parametric Pattern Classification for the Exponential Family Using Order Statistics Criteria

Thomas

Oommen

2012

Lecture Notes in Computer Science

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Abstract. This paper reports some pioneering results in which optimal parametric classification is achieved in a counter-intuitive manner, quite opposed to the Bayesian paradigm. The paper, which builds on the results of [1], demonstrates (with both theoretical and experimental results) how this can be done for some distributions within the exponential family. To be more specific, within a Bayesian paradigm, if we are allowed to compare the testing sample with only a single point in the feature space from each class, the optimal Bayesian strategy would be to achieve this based on the (Mahalanobis) distance from the corresponding means, which in one sense, is the most central point in the respective distribution. In this paper, we shall show that we can obtain optimal results by operating in a diametrically opposite way, i.e., a socalled "anti-Bayesian" manner. Indeed, we shall show that by working with a very few (sometimes as small as two) points distant from the mean, one can obtain remarkable classification accuracies. These points, in turn, are determined by the Order Statistics of the distributions, and the accuracy of our method, referred to as Classification by Moments of Order Statistics (CMOS), attains the optimal Bayes' bound! In this paper, we shall show the claim for two uni-dimensional members of the exponential family. The theoretical results, which have been verified by rigorous experimental testing, also present a theoretical foundation for the families of Border Identification (BI) reported algorithms.

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“…and Φ(x) = as shown in [6]. Using this, we now show that the CMOS with 2-OS yields the same Bayesian accuracy.…”

Section: Theoretical Analysis: Gaussian Distributionsupporting

confidence: 54%

Section: Theoretical Analysis: Gaussian Distributionmentioning

confidence: 99%

See 1 more Smart Citation

Optimal “Anti-Bayesian” Parametric Pattern Classification for the Exponential Family Using Order Statistics Criteria

Thomas

Oommen

2012

Lecture Notes in Computer Science

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“…Thus, without losing generality, we can safely assume that all m random values are different. Therefore, the whole range [0,1] m is divided into m! sub-ranges corresponding to different orders between v i .…”

Section: Explaining the Second Example Of Seemingly Irrational Human mentioning

confidence: 99%

“…. < v m by an appropriate permutation in which v 1 is swapped with the smallest v (1) of m values, v 2 is swapped with the second smallest v (2) , etc.…”

Section: Explaining the Second Example Of Seemingly Irrational Human mentioning

confidence: 99%

Granularity Helps Explain Seemingly Irrational Features of Human Decision Making

Lorkowski

Kreinovich

2015

Studies in Big Data

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Starting from well-known studies by Kahmenan and Tarsky, researchers have found many examples when our decision making -and our decision making -seem to be irrational. In this chapter, we show that this seemingly irrational decision making can be explained if we take into account that human abilities to process information are limited; as a result, instead of the exact values of different quantities, we operate with granules that contain these values. On several examples, we show that optimization under such granularity restriction indeed leads to observed human decision making. Thus, granularity helps explain seemingly irrational human decision making. Known Examples of Seemingly Irrational Human Decision Making: Formulation of the ProblemIn the ideal world, people should make perfect decisions. In many real-life situations, we know what is best for us, and we know the exact consequences of each of our actions. In this case, a rational person should select an action that leads to the best possible outcome. This assumption underlies basic (idealized) economic models: in these models, our decision making may hurt others but every person is interesting in selecting a decision which is the best for him/herself.In the real world, people's decisions are not perfect. In the perfect world, people should make perfect decisions. It is well known, however, that our world is not perfect, and that many people make decisions which are not in their own best interests. People eat unhealthy food, fail to exercise, get drunk, smoke, take drugs, gamble, and do many other things which -as they perfectly know -are bad for their health and bad for their wallets.Such imperfect decisions can still be described in optimization terms. People engage in all kinds of unhealthy and asocial decision making because they get a lot of positive emotions from this engagement. A drug addict may lose his money, his family, his job, his health -but he gets so much pleasure from his drugs that he cannot stop. A gambler may lose all his money, but the pleasure of gambling is so high that he continues gambling (and losing) money until no money is left.These examples of bad decisions are bad from the viewpoint of a person's health or wealth or social status. In all these examples, people clearly know what they want -e.g., more pleasure from drugs or from gambling -and they select a decision which is the "best" from this viewpoint.On top of this well-known abnormal decision making, there are also many examples when a seemingly rational decision making is actually irrational. It is well known that people make 1

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Different Estimation Methods for Constant Stress Accelerated Life Test under the Family of the Exponentiated Distributions

Ghaly

Aly

Salah

2015

Quality & Reliability Eng

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The Accelerated Life Testing (ALT) has been used for a long time in several fields to obtain information on the reliability of product components and materials under operating conditions in a much shorter time. One of the main purposes of applying ALT is to estimate the failure time functions and reliability performance under normal conditions. This paper concentrates on the estimation procedures under ALT and how to select the best estimation method that gives accurate estimates for the reliability function. For this purpose, different estimation methods are used, such as maximum likelihood, least squares (LS), weighted LS, and probability weighted moment. Moreover, the reliability function under usual conditions is predicted. The estimation procedures are applied under the family of the exponentiated distributions in general, and for the exponentiated inverted Weibull (EIW) as a special case. Numerical analysis including simulated data and a real life data set is conducted to compare the performances between these four methods. It is found that the ML method gives the best results among other estimation methods. Finally, a comparison between the EIW and the Inverted Weibull (IW) distributions based on a real life data set is made using a likelihood ratio test. It is observed that the EIW distribution can provide better fitting than the IW in case of ALT. Copyright © 2015 John Wiley & Sons, Ltd.

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An Introduction to Order Statistics

Cited by 113 publications

References 0 publications

Optimal “Anti-Bayesian” Parametric Pattern Classification for the Exponential Family Using Order Statistics Criteria

Optimal “Anti-Bayesian” Parametric Pattern Classification for the Exponential Family Using Order Statistics Criteria

Granularity Helps Explain Seemingly Irrational Features of Human Decision Making

Different Estimation Methods for Constant Stress Accelerated Life Test under the Family of the Exponentiated Distributions

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