Conditional failure occurrence rates for semi-Markov chains*

ABSTRACT

In the present paper, we aim at providing plug-in-type empirical estimators that enable us to quantify the contribution of each operational or/and non-functioning state to the failures of a system described by a semi-Markov model. In the discrete-time and finite state space semi-Markov framework, we study different conditional versions of an important reliability measure for random repairable systems, the failure occurrence rate, which is based on counting processes. The identification of potential failure contributors through the conditional counterparts of the failure occurrence rate is of paramount importance since it could lead to corrective actions that minimize the occurrence of the more important failure modes and therefore improve the reliability of the system. The aforementioned estimators are characterized by appealing asymptotic properties such as strong consistency and asymptotic normality. We further obtain detailed analytical expressions for the covariance matrices of the random vectors describing the conditional failure occurrence rates. As particular cases we present the failure occurrence rates for hidden (semi-) Markov models. We illustrate our results by means of a simulated study. Different applications are presented based on wind, earthquake and vibration data.     

On the total sojourn time of particles in semi-markov compartmental system with branching particles

We consider a semi-Markov compartroental system with branching particles, Me study the total sojourn time of all the particles and obtain the asymptotic behaviour of its mean. We also discuss some special systems.     

An alternative approach to reliability analysis of cold standby systems

ABSTRACT

An alternative approach is applied for reliability analysis of standby systems on the basis of matrix renewal function. In this regard, a single-server, two identical unit cold standby systems with an imperfect switch is considered as a three-state semi-Markov process. Several important reliability measures such as availability, mean time to failure, expected number of failures, etc., are obtained for general lifetime distributions. Also, the main results have been treated to the case of exponential lifetimes and explicit formulas obtained for this case in addition of some numerical illustrations. This approach can easily be extended to more general standby systems with different configurations.     

Random rounded integer-valued autoregressive conditional heteroskedastic process

The statistical literature on the analysis of discrete variate time series has concentrated mainly on parametric models, that is the conditional probability mass function is assumed to belong to a parametric family. Generally, these parametric models impose strong assumptions on the relationship between the conditional mean and variance. To generalize these implausible assumptions, this paper instead considers a more realistic semiparametric model, called random rounded integer-valued autoregressive conditional heteroskedastic (RRINARCH) model, where there are essentially no assumptions on the relationship between the conditional mean and variance. The new model has several advantages: (a) it provides a coherent semiparametric framework for discrete variate time series, in which the conditional mean and variance can be modeled separately; (b) it allows negative values both for the series and its autocorrelation function; (c) its autocorrelation structure is the same as that of a standard autoregressive (AR) process; (d) standard software for its estimation is directly applicable. For the new model, conditions for stationarity, ergodicity and the existence of moments are established and the consistency and asymptotic normality of the conditional least squares estimator are proved. Simulation experiments are carried out to assess the performance of the model. The analyses of real data sets illustrate the flexibility and usefulness of the RRINARCH model for obtaining more realistic forecast means and variances.     

Bernstein conditional density estimation with application to conditional distribution and regression functions

In this paper we propose a smooth nonparametric estimation for the conditional probability density function based on a Bernstein polynomial representation. Our estimator can be written as a finite mixture of beta densities with data-driven weights. Using the Bernstein estimator of the conditional density function, we derive new estimators for the distribution function and conditional mean. We establish the asymptotic properties of the proposed estimators, by proving their asymptotic normality and by providing their asymptotic bias and variance. Simulation results suggest that the proposed estimators can outperform the Nadaraya–Watson estimator and, in some specific setups, the local linear kernel estimators. Finally, we use our estimators for modeling the income in Italy, conditional on year from 1951 to 1998, and have another look at the well known Old Faithful Geyser data.     

On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.     

Error rates in classification consisting of discrete and continuous variables in the presence of covariates

We consider classifying an object based on mixed continuous and discrete variables between two populations. Mixed discrete and continuous covariates with identical means in both populations are amongst the variables. Under the location model with homogeneous location specific conditional dispersion matrices for both populations, the Bayes rule is given. Classification is implemented by a plug-in version of the Bayes rule with full covariate adjustment. An asymptotic expansion of the overall expected error of the procedure is derived. Our findings generalize several classical results.     

A note on diagnostic checking of the double autoregressive model

The double autoregressive model finds its use in the modelling of conditional heteroscedasticity of time series data. In view of its growing popularity, the goodness-of-fit of the model is examined. The asymptotic distributions of the residual and squared residual autocorrelations are derived. Two test statistics are then constructed which can be used to measure the adequacy of the conditional mean and conditional variance components of a fitted model. Our goodness-of-fit tests out-perform other benchmark tests such as the Ljung–Box test in simulation studies. To illustrate the testing procedure, the model is fitted to the weekly log-return series of the Hang Seng Index.     

Nonparametric estimation of some important indicators in reliability for semi-Markov processes

In this article we present nonparametric estimators of the mean time to failure (MTTF), the mean up time (MUT), and the mean down time (MDT) of an arbitrary finite semi-Markov process. Strong consistency and weak convergence is proved for the estimators of MTTF, MUT, and MDT, as the time interval of observation becomes large.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

On the Variances and Convariances of the Duration State Sizes of Semi-Markov Systems

In the present article we study the asymptotic behaviour of the sequence of vectors with components expectations, variances, and covariances of the state sizes of a semi-Markov system. In this respect, we transform the semi-Markov system into a Markov system with a different though equivalent state space and relate the sequence of the transition probabilities of the respective Markov system as functions of the parameters of the semi-Markov system. Also, we study the asymptotic behavior of the sequence of vectors with components variances and covariances of the duration state sizes, of the related Markov system, under perturbation of the transition probability matrices. We use the results in the study of asymptotic behavior under perturbation of the sequence of vectors with components variances and covariances of the semi-Markov system.     

Conditional Score Approach to Errors‐in‐Variable Current Status Data Under the Proportional Odds Model

Abstract. The conditional score approach is proposed to the analysis of errors‐in‐variable current status data under the proportional odds model. Distinct from the conditional scores in other applications, the proposed conditional score involves a high‐dimensional nuisance parameter, causing challenges in both asymptotic theory and computation. We propose a composite algorithm combining the Newton–Raphson and self‐consistency algorithms for computation and develop an efficient conditional score, analogous to the efficient score from a typical semiparametric likelihood, for building an asymptotic linear expression and hence the asymptotic distribution of the conditional‐score estimator for the regression parameter. Our proposal is shown to perform well in simulation studies and is applied to a zebrafish basal cell carcinoma data involving measurement errors in gene expression levels.     

Non-parametric Maximum-Likelihood Estimation in a Semiparametric Mixture Model for Competing-Risks Data

Abstract.  This paper describes our studies on non-parametric maximum-likelihood estimators in a semiparametric mixture model for competing-risks data, in which proportional hazards models are specified for failure time models conditional on cause and a multinomial model is specified for the marginal distribution of cause conditional on covariates. We provide a verifiable identifiability condition and, based on it, establish an asymptotic profile likelihood theory for this model. We also provide efficient algorithms for the computation of the non-parametric maximum-likelihood estimate and its asymptotic variance. The success of this method is demonstrated in simulation studies and in the analysis of Taiwan severe acute respiratory syndrome data.     

Nonparametric Estimation of Variance Function for Functional Data Under Mixing Conditions

This article investigates nonparametric estimation of variance functions for functional data when the mean function is unknown. We obtain asymptotic results for the kernel estimator based on squared residuals. Similar to the finite dimensional case, our asymptotic result shows the smoothness of the unknown mean function has an effect on the rate of convergence. Our simulation studies demonstrate that estimator based on residuals performs much better than that based on conditional second moment of the responses.     

On semi-markov compartmental models with branching particles under the influence of disasters

A semi-Markov multi-compart mental system in which particles reproduce similar particles as a Markov branching process and being subjected to disasters is studied. Expressions for the mean number of particles alive at time t in each compartment are obtained. The results concerning irreversible, mammillarian and catenary compartmental systems have been discussed.     

Nonparametric estimation of regression models with mixed discrete and continuous covariates by the K-nn method

In this article we consider the problem of estimating a nonparametric conditional mean function with mixed discrete and continuous covariates by the nonparametric k-nearest-neighbor (k-nn) method. We derive the asymptotic normality result of the proposed estimator and use Monte Carlo simulations to demonstrate its finite sample performance. We also provide an illustrative empirical example of our method.     

Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.     

Specification testing in nonparametric AR‐ARCH models

In this paper, an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. Tests for the model assumption of independence of innovations from past time series values are suggested. Tests based on weighted L²‐distances of empirical characteristic functions are considered as well as a Cramér–von Mises‐type test. The asymptotic distributions under the null hypothesis of independence are derived, and the consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested, and its performance is shown in a simulation study.     

Semi-Markov Disability Insurance Models

In this article, we present a stochastic model for disability insurance contracts. The model is based on a discrete time non homogeneous semi-Markov process (DTNHSMP) to which the backward recurrence time process is introduced. This permits a more exhaustive study of disability evolution and a more efficient approach to the duration problem. The use of semi-Markov reward processes facilitates the possibility of deriving equations of the prospective and retrospective mathematical reserves. The model is applied to a sample of contracts drawn at random from a mutual insurance company.