On conditional stochastic ordering of distributions

Conditional stochastic ordering is concerned with the stochastic ordering of a pair of probability measures conditional on certain subsets or sub-σ-algebras. Some basic results of conditional stochastic ordering were proved by Whitt. We extend some of Whitt's results and prove a basic relation between stochastic ordering conditional on subsets and stochastic ordering conditional onσ-algebras. In the second part of the paper we consider the ordering of conditional expectations. There are several different formulations of this problem motivated by different types of applications.

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Conditional coherent risk measures and regime-switching conic pricing

Probability Uncertainty and Quantitative Risk ◽

10.3934/puqr.2021014 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Engel John C Dela Vega ◽

Robert J Elliott

Keyword(s):

Regime Switching ◽

Convex Set ◽

Risk Measures ◽

Distortion Function ◽

Probability Measures ◽

Coherent Risk Measures ◽

Coherent Risk ◽

Price Process ◽

Conditional Expectations ◽

And Diffusion

<p style='text-indent:20px;'>This paper introduces and represents conditional coherent risk measures as essential suprema of conditional expectations over a convex set of probability measures and as distorted expectations given a concave distortion function. A model is then developed for the bid and ask prices of a European-type asset by a conic formulation. The price process is governed by a modified geometric Brownian motion whose drift and diffusion coefficients depend on a Markov chain. The bid and ask prices of a European-type asset are then characterized using conic quantization.</p>

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Characterisation of a multivariate stochastic ordering

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700027751 ◽

1988 ◽

Vol 38 (3) ◽

pp. 387-392 ◽

Cited By ~ 1

Author(s):

Colm Art O'Cinneide

Keyword(s):

Stochastic Ordering ◽

Probability Measures ◽

Random Vectors ◽

Nondecreasing Functions

The multivariate stochastic ordering induced by the convex nondecreasing functions compares a combination of size and variability of random vectors. Closely following methods developed by Strassen, we show that two probability measures are ordered in this way if and only if they are the marginals of some submartingale. The implications of this in majorisation theory are discussed.

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Characterization theorems using conditional expectations

Journal of Applied Probability ◽

10.1017/s0021900200048130 ◽

1975 ◽

Vol 12 (02) ◽

pp. 400-406

Author(s):

Ignacy I. Kotlarski ◽

William E. Hinds

Keyword(s):

Real Line ◽

Probability Measures ◽

The Real ◽

Conditional Expectations ◽

Arbitrary Space ◽

Characterization Theorems ◽

Special Case

Recent characterizations of probability measures by use of conditional expectations have included many probability measures defined on subsets of the real line. This paper contains results which allow similar characterizations by use of conditional expectations but are valid for probability measures defined on subsets of an arbitrary space. In the special case of a measure defined on subsets of R' the result is not always as sharp as known results, but these theorems can be applied to measures defined on subsets of Rn with n ≧ 1.

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Characterization theorems using conditional expectations

Journal of Applied Probability ◽

10.2307/3212458 ◽

1975 ◽

Vol 12 (2) ◽

pp. 400-406 ◽

Cited By ~ 2

Author(s):

Ignacy I. Kotlarski ◽

William E. Hinds

Keyword(s):

Real Line ◽

Probability Measures ◽

The Real ◽

Conditional Expectations ◽

Arbitrary Space ◽

Characterization Theorems ◽

Special Case

Recent characterizations of probability measures by use of conditional expectations have included many probability measures defined on subsets of the real line. This paper contains results which allow similar characterizations by use of conditional expectations but are valid for probability measures defined on subsets of an arbitrary space. In the special case of a measure defined on subsets of R' the result is not always as sharp as known results, but these theorems can be applied to measures defined on subsets of Rn with n ≧ 1.

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Characterization Results Based on a Functional Derivative Approach

Combinatorics Probability Computing ◽

10.1017/s0963548301004825 ◽

2001 ◽

Vol 10 (5) ◽

pp. 417-434 ◽

Cited By ~ 2

Author(s):

R. E. LILLO ◽

M. MARTIN

Keyword(s):

Poisson Process ◽

Differential Calculus ◽

Goodness Of Fit ◽

Inverse Operator ◽

Functional Derivative ◽

Unified Approach ◽

Homogeneous Poisson Process ◽

Conditional Expectations ◽

Different Types ◽

Linear Differential

We introduce a notion of the derivative with respect to a function, not necessarily related to a probability distribution, which generalizes the concept of derivative as proposed by Lebesgue [14]. The differential calculus required to solve linear differential equations using this notion of the derivative is included in the paper. The definition given here may also be considered as the inverse operator of a modified notion of the Riemann–Stieltjes integral. Both this unified approach and the results of differential calculus allow us to characterize distributions in terms of three different types of conditional expectations. In applying these results, a test of goodness of fit is also indicated. Finally, two characterizations of a general Poisson process are included. Specifically, a useful result for the homogeneous Poisson process is generalized.

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Visual Representation of SARS-CoV-2 Genomes in Multiple Regions on Integrated Maps

10.21203/rs.3.rs-68270/v1 ◽

2020 ◽

Author(s):

Jeffrey Zheng ◽

Minghan Zhu

Keyword(s):

Theoretical Foundation ◽

Probability Measures ◽

Proper Selection ◽

Systematic Expansion ◽

Different Types ◽

Multiple Regions ◽

Multiple Levels ◽

Virus Genomes ◽

Visual Results ◽

Initial Exploration

Abstract This paper represents visual results for the B2 module of the MAS. Using four meta genetic variables, multiple probability measures are extracted. Variations between pairs of virus genomes could be measured. From a macroscopic viewpoint, measures can be organized in a set of 1616 with (m+1)2 density matrix in combinations with supersymmetric properties. In view of the different types of coronary virus samples, various pairs of genomes make different projections under multiple levels of hierarchical forms of quantization matrix. Under proper selection, huge numbers of variations could be performed. Applying this transformation, a list of sample results are generated to illustrate intrinsic symmetric maps associated with selected parameters. Since this is an initial exploration, further explorations on theoretical foundation and specific applications are essential to support applicable theory and the systematic expansion on medical applications for COVID-19 patients.

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Visual Representation of SARS-CoV-2 Genomes in Multiple Regions on Integrated Maps

10.21203/rs.3.rs-68270/v2 ◽

2021 ◽

Author(s):

Minghan Zhu ◽

Jeffrey Zheng

Keyword(s):

Theoretical Foundation ◽

Probability Measures ◽

Proper Selection ◽

Systematic Expansion ◽

Different Types ◽

Multiple Regions ◽

Multiple Levels ◽

Virus Genomes ◽

Visual Results ◽

Initial Exploration

Abstract This paper represents visual results for the B2 module of the MAS. Using four meta genetic variables, multiple probability measures are extracted. Variations between pairs of virus genomes could be measured. From a macroscopic viewpoint, measures can be organized in a set of 16 × 16 with (m+ 1)2 density matrix in combinations with supersymmetric properties. In view of the different types of coronary virus samples, various pairs of genomes make different projections under multiple levels of hierarchical forms of quantization matrix. Under proper selection, huge numbers of variations could be performed. Applying this transformation, a list of sample results are generated to illustrate intrinsic symmetric maps associated with selected parameters. Since this is an initial exploration, further explorations on theoretical foundation and specific applications are essential to support applicable theory and the systematic expansion on medical applications for COVID-19 patients.

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The extinction time of a general birth and death process with catastrophes

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell

Keyword(s):

Laplace Transform ◽

Size Distribution ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Death Process ◽

Extinction Time ◽

Birth And Death Process ◽

Different Types ◽

The Laplace Transform ◽

Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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Differentiating between different forms of moral obligations

Behavioral and Brain Sciences ◽

10.1017/s0140525x19002589 ◽

2020 ◽

Vol 43 ◽

Author(s):

Rajen A. Anderson ◽

Benjamin C. Ruisch ◽

David A. Pizarro

Keyword(s):

Moral Character ◽

Moral Obligations ◽

Different Types

Abstract We argue that Tomasello's account overlooks important psychological distinctions between how humans judge different types of moral obligations, such as prescriptive obligations (i.e., what one should do) and proscriptive obligations (i.e., what one should not do). Specifically, evaluating these different types of obligations rests on different psychological inputs and has distinct downstream consequences for judgments of moral character.

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