APPLICATIONS OF LIKELIHOOD RATIO ORDER IN BAYESIAN INFERENCES

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.

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DERIVING ROBUST BAYESIAN PREMIUMS UNDER BANDS OF PRIOR DISTRIBUTIONS WITH APPLICATIONS

Astin Bulletin ◽

10.1017/asb.2018.36 ◽

2018 ◽

Vol 49 (1) ◽

pp. 147-168 ◽

Cited By ~ 1

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.

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Likelihood ratio order of parallel systems under multiple-outlier models

Communication in Statistics- Theory and Methods ◽

10.1080/03610926.2017.1300278 ◽

2017 ◽

Vol 47 (1) ◽

pp. 55-63 ◽

Cited By ~ 1

Author(s):

Jiantian Wang

Keyword(s):

Likelihood Ratio ◽

Parallel Systems ◽

Likelihood Ratio Order

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Bayesian Inference and Testing Any Hypothesis You Can Specify

Advances in Methods and Practices in Psychological Science ◽

10.1177/2515245918773087 ◽

2018 ◽

Vol 1 (2) ◽

pp. 281-295 ◽

Cited By ~ 6

Author(s):

Alexander Etz ◽

Julia M. Haaf ◽

Jeffrey N. Rouder ◽

Joachim Vandekerckhove

Keyword(s):

Bayesian Inference ◽

Model Selection ◽

Hypothesis Testing ◽

Likelihood Ratio ◽

Special Form ◽

Null Hypothesis ◽

Bayes Factor ◽

Alternative Hypotheses ◽

Competing Models

Hypothesis testing is a special form of model selection. Once a pair of competing models is fully defined, their definition immediately leads to a measure of how strongly each model supports the data. The ratio of their support is often called the likelihood ratio or the Bayes factor. Critical in the model-selection endeavor is the specification of the models. In the case of hypothesis testing, it is of the greatest importance that the researcher specify exactly what is meant by a “null” hypothesis as well as the alternative to which it is contrasted, and that these are suitable instantiations of theoretical positions. Here, we provide an overview of different instantiations of null and alternative hypotheses that can be useful in practice, but in all cases the inferential procedure is based on the same underlying method of likelihood comparison. An associated app can be found at https://osf.io/mvp53/ . This article is the work of the authors and is reformatted from the original, which was published under a CC-By Attribution 4.0 International license and is available at https://psyarxiv.com/wmf3r/ .

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On the Convolution of Heterogeneous Bernoulli Random Variables

Journal of Applied Probability ◽

10.1239/jap/1316796922 ◽

2011 ◽

Vol 48 (3) ◽

pp. 877-884 ◽

Cited By ~ 8

Author(s):

Maochao Xu ◽

N. Balakrishnan

Keyword(s):

Likelihood Ratio ◽

Hazard Rate ◽

Random Variables ◽

Likelihood Ratio Order ◽

Hazard Rate Order ◽

Reversed Hazard Rate ◽

Bernoulli Random Variables

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

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Likelihood ratio order of the second spacing in multiple-outlier exponential models

Statistics ◽

10.1080/02331888.2015.1016028 ◽

2015 ◽

Vol 50 (1) ◽

pp. 206-218

Author(s):

Peng Zhao ◽

Jianfei Qiao ◽

N. Balakrishnan

Keyword(s):

Likelihood Ratio ◽

Likelihood Ratio Order ◽

Exponential Models

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STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964816000383 ◽

2016 ◽

Vol 31 (3) ◽

pp. 366-380

Author(s):

Ebrahim Amini-Seresht ◽

Yiying Zhang

Keyword(s):

Distribution Function ◽

Order Statistics ◽

Likelihood Ratio ◽

Monotonicity Property ◽

Independent Random Variables ◽

Likelihood Ratio Order ◽

Stochastic Monotonicity ◽

Common Distribution ◽

Common Distribution Function ◽

Parametric Families

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X1, …, Xn be a set of non-negative independent random variables with Xi, i=1, …, p, having common distribution function F(λ1x), and Xj, j=p+1, …, n, having common distribution function F(λ2x), where F(·) denotes the baseline distribution. Let Xi:n(p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$. Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s1≤s2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤i≤n and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

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STOCHASTIC COMPARISONS OF LARGEST ORDER STATISTICS FROM MULTIPLE-OUTLIER EXPONENTIAL MODELS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964811000313 ◽

2012 ◽

Vol 26 (2) ◽

pp. 159-182 ◽

Cited By ~ 24

Author(s):

Peng Zhao ◽

N. Balakrishnan

Keyword(s):

Order Statistics ◽

Likelihood Ratio ◽

Hazard Rate ◽

Stochastic Order ◽

Likelihood Ratio Order ◽

Stochastic Comparisons ◽

Hazard Rate Order ◽

Reversed Hazard Rate ◽

Exponential Models ◽

Usual Stochastic Order

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

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Stochastic Orders Generated by Integrals: a Unified Study

Advances in Applied Probability ◽

10.1017/s0001867800028068 ◽

1997 ◽

Vol 29 (02) ◽

pp. 414-428 ◽

Cited By ~ 8

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.

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Stochastic Orders Generated by Integrals: a Unified Study

Advances in Applied Probability ◽

10.2307/1428010 ◽

1997 ◽

Vol 29 (2) ◽

pp. 414-428 ◽

Cited By ~ 57

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.

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