Likelihood Ratio Order Research Articles

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Let Xλ1, Xλ2, …, Xλn be independent Weibull random variables with Xλi ∼ W(α, λi), where λi > 0 for i = 1, …, n. Let Xn:nλ denote the lifetime of the parallel system formed from Xλ1, Xλ2, …, Xλn. We investigate the effect of the changes in the scale parameters (λ1, …, λn) on the magnitude of Xn:nλ according to reverse hazard rate and likelihood ratio orderings.

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

On likelihood ratio ordering of parallel system with two exponential components

This paper considers stochastic comparison for parallel systems with two exponential components. For a given such system, we identify a region, such that, if the hazard rate pair of another parallel system lies in that region, then there exists likelihood ratio ordering between the two systems. The new results presented in this paper extend most existing ones in the literature.

On the Likelihood Ratio Order for Convolution of Independent Generalized Gamma Distribution

On the Likelihood Ratio Order for Convolution of Independent Generalized Gamma Distribution

Contract design and non-cooperative renegotiation

We study a contract design setting in which the contracting parties cannot commit not to renegotiate previous contract agreements. In particular, we characterize the outcome functions that are implementable for an uninformed principal and an informed agent if, having observed the agent's contract choice, the principal can offer a new menu of contracts in its place. An outcome function can be implemented in this setting if and only if it is optimal for the principal for some belief over agent types which is more pessimistic, in the sense of the likelihood ratio order, than the prior. Furthermore, the outcome function cannot be too sensitive to variations in the agent's type. We show that the direct revelation mechanism which implements such a function when renegotiation can be prevented will also implement it in any equilibrium when it cannot, so the standard contract is robust to renegotiation.

Comparing two treatments in terms of the likelihood ratio order

In this paper new families of test-statistics are introduced and studied for the problem of comparing two treatments in terms of the likelihood ratio order. The considered families are based on φ-divergence measures and arise as natural extensions of the classical likelihood ratio test and Pearson test-statistics. It is proven that their asymptotic distribution is a common chi-bar random variable. An illustrative example is presented and the performance of these statistics is analysed through a simulation study. Through a simulation study it is shown that, for most of the proposed scenarios adjusted to be small or moderate, some members of this new family of test-statistic display clearly better performance with respect to the power in comparison to the classical likelihood ratio and the Pearson's chi-square test while the exact size remains closed to the nominal size. In view of the exact powers and significance levels, the study also shows that the Wilcoxon test-statistic is not as good as the two classical test-statistics.

Likelihood ratio order of sample minimum from heterogeneous Weibull random variables

For heterogeneous Weibull samples with a common shape parameter and weakly majorized scale parameters, we study the likelihood ratio order and the stochastic order between minimums of independent and dependent samples, respectively. Also, an application in fabric strength is presented.

Stochastic comparisons of parallel systems with exponentiated Weibull components

The exponentiated Weibull (EW) distribution is the first generalization of the two-parameter Weibull distribution to accommodate nonmonotone hazard rates, including the bathtub shaped hazard rate. In this paper, we discuss stochastic comparisons of parallel systems with exponentiated Weibull components in terms of the usual stochastic order, dispersive order and the likelihood ratio order. We give some sufficient conditions for stochastic comparisons between lifetimes of parallel systems.

Ordering properties of order statistics from random variables of Archimedean copulas with applications

Ordering properties on order statistics from heterogeneous observations have been paid lots of attention. This article devotes to investigating the ordering properties on order statistics from statistically dependent observations. We build the likelihood ratio order and the (reversed) hazard rate order of the extremal values and their adjacent order statistics. Also, some applications of our comparison results in TTE model and its dual version, auction theory and k-out-of-n reliability systems are presented.

Optimal insurance design of ambiguous risks

We examine the characteristics of the optimal insurance contract under linear transaction costs and an ambiguous distribution of losses. Under the standard expected utility model, we know from Arrow (1965) that it contains a straight deductible. In this paper, we assume that the policyholder is ambiguity averse in the sense of Klibanoff et al. (Econometrica 73(6):1849–1892, 2005). The optimal contract depends upon the structure of the ambiguity. For example, if the set of possible priors can be ranked according to the monotone likelihood ratio order, the optimal contract contains a disappearing deductible. We also show that the policyholder’s ambiguity aversion may have the counterintuitive effect to reduce the optimal insurance coverage of an ambiguous risk.

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

In this paper we compare the residual lifetime of a used coherent system of age t>0 with the lifetime of the similar coherent system made up of used components of age t. Here ‘similar’ means that the system has the same structure and the component lifetimes have the same dependence (joint reliability copula). Some comparison results are obtained for the likelihood ratio order, failure rate order, reversed failure rate order and the usual stochastic order. Similar results are reported for comparing inactivity time of a coherent system with lifetime of similar coherent system having component lifetimes same as inactivity times of failed components.

Discrete Likelihood Ratio Order for Power Series Distribution

It is well-known that some discrete distributions belong to the power series distribution (PSD) family, so it seems useful to study conditions to establish the discrete likelihood ratio order for this family. In this paper, conditions to some cases of PSD family under which the discrete likelihood ratio order we have looked at the holds. Also, we study the discrete version of the proportional likelihood ratio as an extension of the likelihood ratio order. Then we compare some members of the PSD family by discrete proportional likelihood ratio order.

Phi-divergence statistics for the likelihood ratio order: An approach based on log-linear models

When some treatments are ordered according to the categories of an ordinal categorical variable (e.g., extent of side effects) in a monotone order, one might be interested in knowing whether the treatments are equally effective or not. One way to do that is to test if the likelihood ratio order is strictly verified. A method based on log-linear models is derived to make statistical inference and phi-divergence test-statistics are proposed for the test of interest. Focused on log-linear modeling, the theory associated with the asymptotic distribution of the phi-divergence test-statistics is developed. An illustrative example motivates the procedure and a simulation study for small and moderate sample sizes shows that it is possible to find phi-divergence test-statistic with an exact size closer to nominal size and higher power in comparison with the classical likelihood ratio.

Stochastic comparisons of component and system redundancies with dependent components

Under the assumption of dependent identically distributed components and redundant (spares) components, the problem of stochastic comparison of component and system redundancies have been considered. This study has been carried out under the criteria of the likelihood ratio ordering, the reversed failure rate ordering, the failure rate ordering and the usual stochastic ordering.

Likelihood ratio comparisons among spacings related to both one or two samples

In this paper, we investigate some stochastic comparisons in terms of likelihood ratio ordering between spacings from independent random variables exponentially distributed with different scale parameters. We partially solve some open problems in [Wen S, Lu Q, Hu T. Likelihood ratio orderings of spacings of heterogeneous exponential random variables. J Multivariate Anal. 2007;98:743–756] for a one-sample problem and in [Hu T, Lu Q, Wen S. Stochastic comparisons and dependence of spacings from two samples of exponential random variables. Commun Stat – Theory Methods 2006;35:979–988] for a two-sample problem. Specifically, we prove that the second spacing is always smaller than the third spacing in terms of the likelihood ratio order and we provide the ordering among all spacings in the case n=4. In the two-sample case, we establish comparisons between the second spacings related to each sample under certain conditions.

Characterizations of Bivariate Models Using Some Dynamic Conditional Information Divergence Measures

In this article, we study some relevant information divergence measures viz. Renyi divergence and Kerridge’s inaccuracy measures. These measures are extended to conditionally specified models and they are used to characterize some bivariate distributions using the concepts of weighted and proportional hazard rate models. Moreover, some bounds are obtained for these measures using the likelihood ratio order.

A stochastic inequality for the largest order statistics from heterogeneous gamma variables

Abstract In this paper, we compare the largest order statistics arising from independent heterogeneous gamma random variables based on the likelihood ratio order. Let X 1 , … , X n be independent gamma random variables with X i having shape parameter r ∈ ( 0 , 1 ] and scale parameter λ i , i = 1 , … , n , and let X n : n denote the corresponding largest order statistic. Let Y n : n denote the largest order statistic arising from independent and identically distributed gamma random variables Y 1 , … , Y n with Y i having shape parameter r and scale parameter λ = ∑ i = 1 n λ i / n , the arithmetic mean of λ i ’s. It is shown here that X n : n is stochastically greater than Y n : n in terms of the likelihood ratio order. The result established here answers an open problem posed by Balakrishnan and Zhao (2013), and strengthens and generalizes some of the results known in the literature. Numerical examples are also provided to illustrate the main result established here.

On allocating redundancies to k‐out‐of‐ n reliability systems

AbstractFor two components in series and one redundancy with their lifetimes following the proportional hazard models, we build the likelihood ratio order and the hazard rate order for lifetimes of the redundant systems. Also, for k‐out‐of‐ n system with components’ lifetimes having the arrangement increasing joint density and the redundancies having identically distributed lifetimes, allocating more redundancies to weaker components is shown to help improve the system's reliability. Copyright © 2014 John Wiley & Sons, Ltd.

On stochastic comparisons of residual life time at random time

Abstract Let X 1 , X 2 , Θ and Θ ′ be independent non-negative random variables. The residual life of X i at random time Θ , that is, X i Θ = X i − Θ ∣ X i > Θ is considered. Some sufficient conditions which lead to the likelihood ratio ordering, the failure rate ordering, the reverse failure rate ordering and the mean residual life ordering between X 1 Θ and X 2 Θ are obtained and an application in queuing theory is explained. A set of conditions which lead to the same stochastic orderings between X 1 Θ and X 1 Θ ′ are also derived.