Some characterization results for parallel systems with two heterogeneous exponential components

In this paper, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the likelihood ratio order (reversed hazard rate order) and the hazard rate order (stochastic order). We establish, among others, that the weakly majorization order between two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between lifetimes of two parallel systems, and that the p-larger order between two hazard rate vectors is equivalent to the hazard rate order (stochastic order) between lifetimes of two parallel systems. Moreover, we extend the results to the proportional hazard rate models. The results derived here strengthen and generalize some of the results known in the literature.     

Further results for parallel systems with two heterogeneous exponential components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components with respect to likelihood ratio and hazard rate orders. Two sufficient conditions are provided for likelihood ratio and hazard rate orders to hold between the lifetimes of two parallel systems, respectively. Moreover, we extend the results from exponential case to the proportional hazard rate models. The results established here strength some of the results known in the literature. Finally, some numerical examples are given to illustrate the theoretical results derived here as well.     

On stochastic behaviors of load-sharing parallel systems

Abstract

This paper mainly investigates a general load-sharing parallel system having two units. First, we construct some comparisons among a load standby system, a warm standby system, a hot standby system and a cold standby system. Moreover, some stochastic comparisons between the load-sharing parallel system and one of its two components are obtained in the sense of the usual stochastic order. Finally, the residual life of this system and its properties are examined.     

Exponentiated models preserve stochastic orderings of parallel and series systems

Abstract

In this paper, we establish that the usual stochastic, hazard rate, reversed hazard rate, likelihood ratio, dispersive and star orders are all preserved for parallel systems under exponentiated models for lifetimes of components. We then use the multiple-outlier exponentiated gamma models to illustrate this result. Finally, we consider the dual family with exponentiated survival function and establish similar results for series systems. The results established here extend some well-known results for series and parallel systems arising from different exponentiated distributions such as generalized exponential and exponentiated Weibull, established previously in the literature.     

Likelihood ratio tests for testing for multiple contaminants in the shocks and labelled slippage models

In the literature related to the study of lifelengths of experimental units, little attention has been paid to the models where shocks to the units generate outliers. In the present article, we consider a situation where n experimental units under investigation receive shocks at several time points. The parameter values of the lifelength distribution may change due to each shock, resulting in the generation of outliers. We derive the likelihood ratio test statistic to investigate if the shocks have significantly altered the parameter values. We also derive a likelihood ratio test under the labelled slippage alternative with multiple contaminations. Monte Carlo studies have been carried out to investigate the power of the proposed test statistics.     

Ordering results for series and parallel systems comprising heterogeneous exponentiated Weibull components

In this paper, we discuss the usual stochastic and reversed hazard rate orders between the series and parallel systems from two sets of independent heterogeneous exponentiated Weibull components. We also obtain the results concerning the convex transform orders between parallel systems and obtain necessary and sufficient conditions under which the dispersive and usual stochastic orders, and the right spread and increasing convex orders between the lifetimes of the two systems are equivalent. Finally, in the multiple-outlier exponentiated Weibull models, based on weak majorization and p-larger orders between the vectors of scale and shape parameters, some characterization results for comparing the lifetimes of parallel and series systems are also established, respectively. The results of this paper can be used in practical situations to find various bounds for the important aging characteristics of these systems.     

A discussion of some inference issues in order restricted models

Traditional inference methodology in order restricted models oftentimes yields procedures with unexpected properties. There are many scattered results exhibiting anomalies and there is disagreement on which inference procedures are appropriate in particular specific models. The authors contend here that these anomalies are not surprising and are to be expected. They air some of the issues that arise in an effort to shed some light on these problems.     

Statistical inference for a relaxation index of stochastic dominance under density ratio model

Stochastic dominance is usually used to rank random variables by comparing their distributions, so it is widely applied in economics and finance. In actual applications, complete stochastic dominance is too demanding to meet, so relaxation indexes of stochastic dominance have attracted more attention. The π index, the biggest gap between two distributions, can be a measure of the degree of deviation from complete dominance. The traditional estimation method is to use the empirical distribution functions to estimate it. Considering the populations under comparison are generally of the same nature, we can link the populations through density ratio model under certain condition. Based on this model, we propose a new estimator and establish its statistical inference theory. Simulation results show that the proposed estimator substantially improves estimation efficiency and power of the tests and coverage probabilities satisfactorily match the confidence levels of the tests, which show the superiority of the proposed estimator. Finally we apply our method to a real example of the Chinese household incomes.     

Stochastic comparisons of sample spacings in single- and multiple-outlier exponential models

This article studies some ordering results for the sample spacings arising from the single- and multiple-outlier exponential models. In the single-outlier exponential models, it is shown that the weak majorization order between the two hazard rate vectors implies the hazard rate order as well as the dispersive order between the corresponding sample spacings. We also extend this result from the single-outlier model to the multiple-outlier model for the special case of the second sample spacing. Furthermore, we obtain some necessary and sufficient conditions such that, on the one hand, the hazard rate, dispersive and usual stochastic orders, and on the other hand, the likelihood ratio and reversed hazard rate orders of the second sample spacings from two independent heterogeneous exponential random variables are equivalent.     

Likelihood ratio tests based on subglobal optimization: A power comparison in exponential mixture models

The paper compares several versions of the likelihood ratio test for exponential homogeneity against mixtures of two exponentials. They are based on different implementations of the likelihood maximization algorithm. We show that global maximization of the likelihood is not appropriate to obtain a good power of the LR test. A simple starting strategy for the EM algorithm, which under the null hypothesis often fails to find the global maximum, results in a rather powerful test. On the other hand, a multiple starting strategy that comes close to global maximization under both the null and the alternative hypotheses leads to inferior power.     

Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

Let X₁, …, X_n be independent random variables with X_i ～ EWG(α, β, λ_i, p_i), i = 1, …, n, and Y₁, …, Y_n be another set of independent random variables with Y_i ～ EWG(α, β, γ_i, q_i), i = 1, …, n. The results established here are developed in two directions. First, under conditions p₁ = ??? = p_n = q₁ = ??? = q_n = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ₁ = ??? = λ_n = γ₁ = ??? = γ_n = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p₁, …, p_n) and (q₁, …, q_n). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.     

Likelihood ratio tests for fuzzy hypotheses testing

In hypotheses testing, such as other statistical problems, we may confront imprecise concepts. One case is a situation in which hypotheses are imprecise. In this paper, we recall and redefine some concepts about fuzzy hypotheses testing, and then we introduce the likelihood ratio test for fuzzy hypotheses testing. Finally, we give some applied examples.     

Ordering properties of largest order statistics from independent and heterogeneous Dagum populations

Abstract

The Dagum distribution has been extensively used to model income data, and its features have been appreciated in economics and financial studies. In this article, we discuss ordering properties of largest order statistics from independent and heterogeneous Dagum populations. We present some sufficient conditions for stochastic comparisons between largest order statistics in terms of the reversed hazard rate order, the usual stochastic order, the convex order, the likelihood ratio order and the dispersive order. Several numerical examples are presented to illustrate the results established here.     

On the Convergence of the Monte Carlo Maximum Likelihood Method for Latent Variable Models

While much used in practice, latent variable models raise challenging estimation problems due to the intractability of their likelihood. Monte Carlo maximum likelihood (MCML), as proposed by Geyer & Thompson (1992 ), is a simulation-based approach to maximum likelihood approximation applicable to general latent variable models. MCML can be described as an importance sampling method in which the likelihood ratio is approximated by Monte Carlo averages of importance ratios simulated from the complete data model corresponding to an arbitrary value of the unknown parameter. This paper studies the asymptotic (in the number of observations) performance of the MCML method in the case of latent variable models with independent observations. This is in contrast with previous works on the same topic which only considered conditional convergence to the maximum likelihood estimator, for a fixed set of observations. A first important result is that when is fixed, the MCML method can only be consistent if the number of simulations grows exponentially fast with the number of observations. If on the other hand, is obtained from a consistent sequence of estimates of the unknown parameter, then the requirements on the number of simulations are shown to be much weaker.     

Likelihood ratio type unit root tests for ar(1)models with nonconsecutive observations

For the nonconsecutively observed or missing data situation likelihood ratio type unit root tests in AR(1)models containing an intercept or both an intercept and a time trend are proposed and are shown to have the same limiting distributions as the likelihood ratio tests for the complete data case as tabulated by Dickey and Fuller(1981). Some simulation results on our tests in finite samples under A–B sampling schemes are also presented.     

Likelihood ratio test for testing order of continuous time finite markov chains

Anderson and Goodman ( 1957) have obtained the likelihood ratio tests and chi-square tests for testing the hypothesis about the order of discrete time finite Markov chains, On the similar lines we have obtained likeli¬hood ratio tests and chi-square tests (asymptotic) for testing hypotheses about the order of continuous time Markov chains (MC) with finite state space.     

On the right spread ordering of parallel systems with two heterogeneous components

This article discusses the variability ordering of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the right spread order. It is proved, among others, that the reciprocal majorization order between the two hazard rate vectors implies the right spread order between the lifetimes of two parallel systems. The result is then extended to the proportional hazard rate model as well. The results established here extend and enrich those known in the literature.     

Likelihood ratio ordering of order statistics,mixtures and systems

Let X=(_X1,_X2,…,_Xn)$\mathit{X}=(X_1,X_2,\dots ,X_n)$ be an exchangeable random vector, and denote _X1:i=min{_X1,_X2,…,_Xi}$X_{1:i}=\min \{X_1,X_2,\dots ,X_i\}$ and _Xi:i=max{_X1,_X2,…,_Xi}$X_{i:i}=\max \{X_1,X_2,\dots ,X_i\}$, 1?i?n$1?i?n$. These order statistics represent the lifetimes of the series and the parallel systems, respectively, with component lifetimes _Xi$X_i$. In this paper we obtain conditions under which _X1:i$X_{1:i}$ (or _Xi:i$X_{i:i}$) decreases (increases) in i in the likelihood ratio (lr) order. An even more general result involving general (that is, not necessary exchangeable) random vectors is also derived for general series (or parallel) systems. We show that the series (parallel) systems are not necessarily lr-ordered even if the components are independent.