On Mixture Failure Rates Ordering

Mixtures of increasing failure rate distributions (IFR) can decrease at least in some intervals of time. Usually, this property can be observed asymptotically as t → ∞. This is due to the fact that the mixture failure rate is “bent down” compared with the corresponding unconditional expectation of the baseline failure rate, which was proved previously for some specific cases. We generalize this result and discuss the “weakest populations are dying first” property, which leads to the change in the failure rate shape. We also consider the problem of mixture failure rate ordering for the ordered mixing distributions. Two types of stochastic ordering are analyzed: ordering in the likelihood ratio sense and ordering in variances when the means are equal.     

Characterization of a mixture of gamma distributions via conditional finite moments

A necessary and sufficient condition that a continuous, positive random variable follow a gamma distribution is given in terms of any one of its conditional finite moments and an expression involving its failure rate. The results are then used to develop a characterization for a mixture of two gamma distributions. The general results about characterization of a mixture of gamma distributions yield several special cases that have appeared separately in recent literature, including characterization of a single exponential distribution, characterization of a single gamma distribution (in terms of either first or second moments) and a sufficient condition for a mixture of two exponential distributions (in terms of first moments). The condition in this last result is shown to be necessary also. Numerous other cases are possible, using different choices for distribution parameters along with a selection of the mixing parameter, for either individual or mixtures of distributions. Various characterizations can be expressed using higher order moments, too.     

Some results on discrete mixture failure rates

Some properties of the discrete mixture failure rates are studied. Specifically, similar to the continuous case, it is shown that the population mixture failure rate is always smaller than the unconditional expectation in the family of subpopulations failure rates. The analog of the multiplicative and the additive frailty models is introduced via the corresponding survival function. Another approach via the alternative discrete failure rate is also discussed. Stochastic comparisons for two mixed distributions with equal and different mixing distributions are studied.     

On a threshold representation for complex load-sharing systems

Complex load-sharing systems are studied to incorporate dependencies among components through a load-sharing rule. As the load on the system increases, a series of cycles of Phase I/II failures occur where Phase I failure is a single component failure, which then causes a cascade of component failures (Phase II) due to the load transfer as these components fail. A threshold representation for the process of system failure is given. This representation is a gamma-type mixture representation when the component strengths are independent exponentials. In this case, for a given breaking pattern the mixture is over the gamma scale parameter and is based on a convolution of uniforms defined by the load-sharing parameters. Such convolutions can be approximated by normal densities which reduces the dimension of the parameter space. This representation can be generalized to independent component strengths with arbitrary distributions by transforming the strength and load-sharing to pseudo-strength and pseudo-load-sharing rules.     

DETERMINATION OF CHANGE POINTS OF NON-MONOTONIC FAILURE RATES

This paper was motivated by the problem of the determination of the change points of the failure rate of a mixture of two gamma distributions. For certain values of the parameters the existing methods are not applicable since, in this case, there are two turning points of the failure rate. Thus, we extend the results to models having two or more turning points of the failure rates. The extended procedure helps us to identify failure rates of more complex forms. Finally, the mixture gamma case is completely resolved employing theoretical, graphical and numerical techniques wherever necessary.     

A comparative study of the K-means algorithm and the normal mixture model for clustering: Bivariate homoscedastic case

The K-means algorithm and the normal mixture model method are two common clustering methods. The K-means algorithm is a popular heuristic approach which gives reasonable clustering results if the component clusters are ball-shaped. Currently, there are no analytical results for this algorithm if the component distributions deviate from the ball-shape. This paper analytically studies how the K-means algorithm changes its classification rule as the normal component distributions become more elongated under the homoscedastic assumption and compares this rule with that of the Bayes rule from the mixture model method. We show that the classification rules of both methods are linear, but the slopes of the two classification lines change in the opposite direction as the component distributions become more elongated. The classification performance of the K-means algorithm is then compared to that of the mixture model method via simulation. The comparison, which is limited to two clusters, shows that the K-means algorithm provides poor classification performances consistently as the component distributions become more elongated while the mixture model method can potentially, but not necessarily, take advantage of this change and provide a much better classification performance.     

Multivariate Life Testing in Variably Scaled Environments

This paper examines modeling and inference questions for experiments in which different subsets of a set of k possibly dependent components are tested in r different environments. In each environment, the failure times of the set of components on test is assumed to be governed by a particular type of multivariate exponential (MVE) distribution. For any given component tested in several environments, it is assumed that its marginal failure rate varies from one environment to another via a change of scale between the environments, resulting in a joint MVE model which links in a natural way the applicable MVE distributions describing component behavior in each fixed environment. This study thus extends the work of Proschan and Sullo (1976) to multiple environments and the work of Kvam and Samaniego (1993) to dependent data. The problem of estimating model parameters via the method of maximum likelihood is examined in detail. First, necessary and sufficient conditions for the identifiability of model parameters are established. We then treat the derivation of the MLE via a numerically-augmented application of the EM algorithm. The feasibility of the estimation method is demonstrated in an example in which the likelihood ratio test of the hypothesis of equal component failure rates within any given environment is carried out.     

On conditional correlations defined by a class of nonlinearly truncated distributions and their applications

This article derives and studies several types of conditional correlations. The correlations are obtained by a class of two-piece scale mixture skew-normal distributions. The class is obtained by applying a set of nonlinear constraints to the bivariate scale mixture of normal distributions. The correlations of the class are invariant with respect to the choice of the scale mixing function, however, they are dependent upon the type of the nonlinear truncation. Moreover, their respective upper and lower limits are no longer 1.00 and?1.00. They are useful for the truncated data analysis, the multivariate interdependence methods (such as the principal component analysis and the factor analysis), and the random truncation modelling. Some distributional properties and the Bayesian computation of the correlations are considered when developing necessary theories and providing illustrative examples, respectively. Two applications are also given to demonstrate the usefulness of the conditional correlations in a multivariate analysis.     

A model for k-out-of-m load-sharing systems

ABSTRACT

In a load-sharing system, the failure of a component affects the residual lifetime of the surviving components. We propose a model for the load-sharing phenomenon in k-out-of-m systems. The model is based on exponentiated conditional distributions of the order statistics formed by the failure times of the components. For an illustration, we consider two component parallel systems with the initial lifetimes of the components having Weibull and linear failure rate distributions. We analyze one data set to show that the proposed model may be a better fit than the model based on sequential order statistics.     

Bayesian analysis for mixtures of discrete distributions with a non-parametric component

Bayesian finite mixture modelling is a flexible parametric modelling approach for classification and density fitting. Many areas of application require distinguishing a signal from a noise component. In practice, it is often difficult to justify a specific distribution for the signal component; therefore, the signal distribution is usually further modelled via a mixture of distributions. However, modelling the signal as a mixture of distributions is computationally non-trivial due to the difficulties in justifying the exact number of components to be used and due to the label switching problem. This paper proposes the use of a non-parametric distribution to model the signal component. We consider the case of discrete data and show how this new methodology leads to more accurate parameter estimation and smaller false non-discovery rate. Moreover, it does not incur the label switching problem. We show an application of the method to data generated by ChIP-sequencing experiments.     

Dynamic mixing probability measures of mixtures

ABSTRACT

Lifetime of heterogeneous population can be modeled as mixture of a family of lifetime distributions according to a mixing probability measure. With the help of dynamic mixing measure, the hazard rate of the mixture can also be expressed as the mixture of the hazard rates of the lifetime distributions. Various local stochastic orderings are defined in this article. Applying these local stochastic orderings, we can explore the behavior of the dynamic mixing measures locally and then compare the hazard rates of two heterogeneous populations in both the local and global ways.     

Reliability Evaluation of a Load-sharing Parallel System with Failure Dependence

In a parallel structure load-sharing system, the failure rate of the operating components will usually increase, due to the additional loading induced by the other components' failure. Hence failure dependency exists among components. To quantify the failure dependency, a dependence function is introduced. Under the assumptions that the repair time distributions of components are arbitrary and life times are exponential distributions whose failure rates vary with the number of operating components, a new load-sharing parallel system with failure dependency is proposed. To model the stochastic behavior of the system, the Semi-Markov process induced by it is given. The Semi-Markov kernel associated with the process is also presented. The availability and the time to the first system failure are obtained by employing Markov renewal theory. A numerical example is presented to illustrate the results obtained in the paper. The impact of the failure dependence on the system is also considered.     

The aligned rank transform and discrete variables: A warning

For two-way layouts in a between subjects ANOVA design the aligned rank transform (ART) is compared with the parametric F-test as well as six other nonparametric methods: rank transform (RT), inverse normal transform (INT), a combination of ART and INT, Puri & Sen's L statistic, van der Waerden and Akritas & Brunners ATS. The type I error rates are computed for the uniform and the exponential distributions, both as continuous and in several variations as discrete distribution. The computations had been performed for balanced and unbalanced designs as well as for several effect models. The aim of this study is to analyze the impact of discrete distributions on the error rate. And it is shown that this scaling impact is restricted to the ART- as well as the combination of ART- and INT-method. There are two effects: first with increasing cell counts their error rates rise beyond any acceptable limit up to 20 percent and more. And secondly their rates rise when the number of distinct values of the dependent variable decreases. This behavior is more severe for underlying exponential distributions than for uniform distributions. Therefore there is a recommendation not to apply the ART if the mean cell frequencies exceed 10.     

On the determination of mixing density in reliability studies

In reliability studies the three quantities (1) the survival function, (2) the failure rate and (3) the mean residual life function are all equivalent in the sense that given one of them, the other two can be determined. In this paper we have considered the class of exponential type distributions and studied its mixture. Given any one of the above mentioned three quantities of the mixture a method is developed for determining the mixing density. Some examples are provided as illustrations. Some well known results follow trivially.     

On Non-parametric Estimation of the Survival Function with Competing Risks

The non-parametric maximum likelihood estimators (MLEs) are derived for survival functions associated with individual risks or system components in a reliability framework. Lifetimes are observed for systems that contain one or more of those components. Analogous to a competing risks model, the system is assumed to fail upon the first instance of any component failure; i.e. the system is configured in series. For any given risk or component type, the asymptotic distribution is shown to depend explicitly on the unknown survival function of the other risks, as well as the censoring distribution. Survival functions with increasing failure rate are investigated as a special case. The order restricted MLE is shown to be consistent under mild assumptions of the underlying component lifetime distributions.     

Finite mixture of Burr type XII distribution and its reciprocal: properties and applications

In this paper, the finite mixture of Burr type XII distribution with its reciprocal, is proposed as a failure model. The failure rate (FR) of the new model covers several types of failure rates. It is shown that depending on the parameter values, the model is capable of covering different combinations of failure rates. A study of the behavior of the FR curve of the model is made.     

Assessing the fit of finite mixture distributions

Mixture distributions have become a very flexible and common class of distributions, used in many different applications, but hardly any literature can be found on tests for assessing their goodness of fit. We propose two types of smooth tests of goodness of fit for mixture distributions. The first test is a genuine smooth test, and the second test makes explicit use of the mixture structure. In a simulation study the tests are compared to some traditional goodness of fit tests that, however, are not customised for mixture distributions. The first smooth test has overall good power and generally outperforms the other tests. The second smooth test is particularly suitable for assessing the fit of each component distribution separately. The tests are applicable to both continuous and discrete distributions and they are illustrated on three medical data sets.     

Flexible mixture modeling via the multivariate t distribution with?the?Box-Cox transformation: an?alternative to?the?skew-t distribution

Cluster analysis is the automated search for groups of homogeneous observations in a data set. A popular modeling approach for clustering is based on finite normal mixture models, which assume that each cluster is modeled as a multivariate normal distribution. However, the normality assumption that each component is symmetric is often unrealistic. Furthermore, normal mixture models are not robust against outliers; they often require extra components for modeling outliers and/or give a poor representation of the data. To address these issues, we propose a new class of distributions, multivariate t distributions with the Box-Cox transformation, for mixture modeling. This class of distributions generalizes the normal distribution with the more heavy-tailed t distribution, and introduces skewness via the Box-Cox transformation. As a result, this provides a unified framework to simultaneously handle outlier identification and data transformation, two interrelated issues. We describe an Expectation-Maximization algorithm for parameter estimation along with transformation selection. We demonstrate the proposed methodology with three real data sets and simulation studies. Compared with a wealth of approaches including the skew-t mixture model, the proposed t mixture model with the Box-Cox transformation performs favorably in terms of accuracy in the assignment of observations, robustness against model misspecification, and selection of the number of components.     

On the asymptotic properties of fuzzy c-means cluster prototypes as estimators of mixture subpopulation centers

Several computational studies suggest that the fuzzy c-means (FCM) clustering scheme may be used successfully in some cases to obtain estimates for the parameters of a statistical mixture (e.g., for a mixture of normal distributions). While these (limited) simulation results for the fuzzy c-means approach support this hypothesis, we provide herein an example that shows that the FCM cluster prototypes cannot generally be statistically consistent estimators of the centers (means) of any univariate mixture having symmetric component distributions     

Mixture cure rate models with accelerated failures and nonparametric form of covariate effects

Two-component mixture cure rate model is popular in cure rate data analysis with the proportional hazards and accelerated failure time (AFT) models being the major competitors for modelling the latency component. [Wang, L., Du, P., and Liang, H. (2012), ‘Two-Component Mixture Cure Rate Model with Spline Estimated Nonparametric Components’, Biometrics, 68, 726–735] first proposed a nonparametric mixture cure rate model where the latency component assumes proportional hazards with nonparametric covariate effects in the relative risk. Here we consider a mixture cure rate model where the latency component assumes AFTs with nonparametric covariate effects in the acceleration factor. Besides the more direct physical interpretation than the proportional hazards, our model has an additional scalar parameter which adds more complication to the computational algorithm as well as the asymptotic theory. We develop a penalised EM algorithm for estimation together with confidence intervals derived from the Louis formula. Asymptotic convergence rates of the parameter estimates are established. Simulations and the application to a melanoma study shows the advantages of our new method.