Stochastic Orders Generated by Integrals: a Unified Study

1997 ◽  
Vol 29 (2) ◽  
pp. 414-428 ◽  
Author(s):  
Alfred Müller
Keyword(s):  
Functional Analysis ◽  
Weak Convergence ◽  
Likelihood Ratio ◽  
Stochastic Orders ◽  
Probability Measures ◽  
Likelihood Ratio Order ◽  
Unified Study ◽  
Maximal Cone ◽  
Class Of Functions

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

Stochastic Orders Generated by Integrals: a Unified Study

1997 ◽  
Vol 29 (02) ◽  
pp. 414-428 ◽  
Author(s):  
Alfred Müller
Keyword(s):  
Functional Analysis ◽  
Weak Convergence ◽  
Likelihood Ratio ◽  
Stochastic Orders ◽  
Probability Measures ◽  
Likelihood Ratio Order ◽  
Unified Study ◽  
Maximal Cone ◽  
Class Of Functions

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

New Stochastic Orders Based on Double Truncation

1997 ◽  
Vol 11 (3) ◽  
pp. 395-402 ◽  
Author(s):  
Jorge Navarro ◽  
Felix Belzunce ◽  
Jose M. Ruiz
Keyword(s):  
Likelihood Ratio ◽  
Random Variable ◽  
Stochastic Orders ◽  
Likelihood Ratio Order ◽  
Reliability Measures ◽  
Life Distributions ◽  
The Relationship ◽  
Double Truncation

The purpose of this paper is to study definitions and characterizations of orders based on reliability measures related with the doubly truncated random variable X[x, y] = (X|x ≤ X ≤ y). The relationship between these orderings and various existing orderings of life distributions are discussed. Moreover, we give two new characterizations of the likelihood ratio order based on double truncation. These new orders complete a general diagram between orders defined from truncation.

Glaser’s function and stochastic orders for mixture distributions

2016 ◽  
Vol 33 (8) ◽  
pp. 1230-1238
Author(s):  
Jalil Jarrahiferiz ◽  
G.R. Mohtashami Borzadaran ◽  
A.H. Rezaei Roknabadi
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Exponential Family ◽  
Random Variable ◽  
Moment Generating Function ◽  
Stochastic Orders ◽  
Weight Functions ◽  
Likelihood Ratio Order ◽  
Content Type ◽  
Weighted Distributions

Purpose The purpose of this paper is to study likelihood ratio order for mixture and its components via their Glaser’s functions for weighted distributions. So, some theoretical examples using exponential family and their mixtures are presented. Design/methodology/approach First, Glaser’s functions of mixture and its components for weighted distributions in different scenarios are computed. Then by them the likelihood ratio order is investigated between mixture and its components. Findings The authors find conditions for weight functions under which the mixture random variable is between of its components in likelihood ratio order. Originality/value Results are obtained for weight function in general. It is well known that the some special weights are order statistics, up and down records, hazard rate, reversed hazard rate, moment generating function, etc. So, the results are valid for all of them.

Integral Probability Metrics and Their Generating Classes of Functions

1997 ◽  
Vol 29 (2) ◽  
pp. 429-443 ◽  
Author(s):  
Alfred Müller
Keyword(s):  
Maximal Class ◽  
Probability Measures ◽  
Probability Metrics ◽  
Unified Study ◽  
Kolmogorov Metric ◽  
Stop Loss ◽  
Class Of Functions ◽  
Integral Probability

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

Integral Probability Metrics and Their Generating Classes of Functions

1997 ◽  
Vol 29 (02) ◽  
pp. 429-443 ◽  
Author(s):  
Alfred Müller
Keyword(s):  
Maximal Class ◽  
Probability Measures ◽  
Probability Metrics ◽  
Unified Study ◽  
Kolmogorov Metric ◽  
Stop Loss ◽  
Class Of Functions ◽  
Integral Probability

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

DERIVING ROBUST BAYESIAN PREMIUMS UNDER BANDS OF PRIOR DISTRIBUTIONS WITH APPLICATIONS

2018 ◽  
Vol 49 (1) ◽  
pp. 147-168 ◽  
Author(s):  
M. Sánchez-Sánchez ◽  
M.A. Sordo ◽  
A. Suárez-Llorens ◽  
E. Gómez-Déniz
Keyword(s):  
Bayesian Analysis ◽  
Likelihood Ratio ◽  
Requirements Elicitation ◽  
Likelihood Ratio Order ◽  
Prior Distributions ◽  
Propagation Of Uncertainty ◽  
Distortion Functions ◽  
Wide Family ◽  
Class Of Priors ◽  
Quantities Of Interest

AbstractWe study the propagation of uncertainty from a class of priors introduced by Arias-Nicolás et al. [(2016) Bayesian Analysis, 11(4), 1107–1136] to the premiums (both the collective and the Bayesian), for a wide family of premium principles (specifically, those that preserve the likelihood ratio order). The class under study reflects the prior uncertainty using distortion functions and fulfills some desirable requirements: elicitation is easy, the prior uncertainty can be measured by different metrics, and the range of quantities of interest is easily obtained from the extremal members of the class. We illustrate the methodology with several examples based on different claim counts models.

scholarly journals Value-distribution of twisted L-functions of normalized cusp forms

2010 ◽  
Vol 51 ◽  
Author(s):  
Alesia Kolupayeva
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Complex Plane ◽  
Dirichlet Character ◽  
Value Distribution ◽  
Cusp Forms ◽  
Probability Measures ◽  
Convergence Of Probability Measures

A limit theorem in the sense of weak convergence of probability measures on the complex plane for twisted with Dirichlet character L-functions of holomorphic normalized Hecke eigen cusp forms with an increasing modulus of the character is proved.

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