Modelling of unexpected shift in SPC

Optimal statistical process control (SPC) requires models of both in-control and out-of-control process states. Whereas a normal distribution is the generally accepted model for the in-control state, there is a doubt as to the existence of reliable models for out-of-control cases. Various process models, available in the literature, for discrete manufacturing systems (parts industry) can be treated as bounded discrete-space Markov chains, completely characterized by the original in-control state and a transition matrix for shifts to an out-of-control state. The present work extends these models by using a continuous-state Markov chain, incorporating non-random corrective actions. These actions are to be realized according to the SPC technique and should substantially affect the model. The developed stochastic model yields a Laplace distribution of a process mean. An alternative approach, based on the Information theory, also results in a Laplace distribution. Real-data tests confirm the applicability of a Laplace distribution for the parts industry and show that the distribution parameter is mainly controlled by the SPC sample size.     

Generalized Additive Modelling of Mixed Distribution Markov Models with Application to Melbourne's Rainfall

The paper considers the modelling of time series using a generalized additive model with first-order Markov structure and mixed transition density having a discrete component at zero and a continuous component with positive sample space. Such models have application, for example, in modelling daily occurrence and intensity of rainfall, and in modelling numbers and sizes of insurance claims. The paper shows how these methods extend the usual sinusoidal seasonal assumption in standard chain-dependent models by assuming a general smooth pattern of occurrence and intensity over time. These models can be fitted using standard statistical software. The methods of Grunwald & Jones (2000) can be used to combine these separate occurrence and intensity models into a single model for amount. The models are used to investigate the relationship between the Southern Oscillation Index and Melbourne's rainfall, illustrated with 36 years of rainfall data from Melbourne, Australia.     

Estimation with right‐censored observations under a semi‐Markov model

The semi‐Markov process often provides a better framework than the classical Markov process for the analysis of events with multiple states. The purpose of this paper is twofold. First, we show that in the presence of right censoring, when the right end‐point of the support of the censoring time is strictly less than the right end‐point of the support of the semi‐Markov kernel, the transition probability of the semi‐Markov process is nonidentifiable, and the estimators proposed in the literature are inconsistent in general. We derive the set of all attainable values for the transition probability based on the censored data, and we propose a nonparametric inference procedure for the transition probability using this set. Second, the conventional approach to constructing confidence bands is not applicable for the semi‐Markov kernel and the sojourn time distribution. We propose new perturbation resampling methods to construct these confidence bands. Different weights and transformations are explored in the construction. We use simulation to examine our proposals and illustrate them with hospitalization data from a recent cancer survivor study. The Canadian Journal of Statistics 41: 237–256; 2013 © 2013 Statistical Society of Canada     

A fast distance‐based approach for determining the number of components in mixtures

The authors propose a procedure for determining the unknown number of components in mixtures by generalizing a Bayesian testing method proposed by Mengersen & Robert (1996). The testing criterion they propose involves a Kullback‐Leibler distance, which may be weighted or not. They give explicit formulas for the weighted distance for a number of mixture distributions and propose a stepwise testing procedure to select the minimum number of components adequate for the data. Their procedure, which is implemented using the BUGS software, exploits a fast collapsing approach which accelerates the search for the minimum number of components by avoiding full refitting at each step. The performance of their method is compared, using both distances, to the Bayes factor approach.     

On cross‐validation of Bayesian models

The authors examine several aspects of cross‐validation for Bayesian models. In particular, they propose a computational scheme which does not require a separate posterior sample for each training sample.     

Smooth Semi‐nonparametric Analysis for Mixture Cure Models and Its Application to Breast Cancer

Mixture cure models are widely used when a proportion of patients are cured. The proportional hazards mixture cure model and the accelerated failure time mixture cure model are the most popular models in practice. Usually the expectation–maximisation (EM) algorithm is applied to both models for parameter estimation. Bootstrap methods are used for variance estimation. In this paper we propose a smooth semi‐nonparametric (SNP) approach in which maximum likelihood is applied directly to mixture cure models for parameter estimation. The variance can be estimated by the inverse of the second derivative of the SNP likelihood. A comprehensive simulation study indicates good performance of the proposed method. We investigate stage effects in breast cancer by applying the proposed method to breast cancer data from the South Carolina Cancer Registry.     

Joint Modelling of Survival and Longitudinal Data with Informative Observation Times

In this paper, we consider the joint modelling of survival and longitudinal data with informative observation time points. The survival model and the longitudinal model are linked via random effects, for which no distribution assumption is required under our estimation approach. The estimator is shown to be consistent and asymptotically normal. The proposed estimator and its estimated covariance matrix can be easily calculated. Simulation studies and an application to a primary biliary cirrhosis study are also provided.     

Modelling and simulation for one‐day cricket

This article is concerned with the simulation of one‐day cricket matches. Given that only a finite number of outcomes can occur on each ball that is bowled, a discrete generator on a finite set is developed where the outcome probabilities are estimated from historical data involving one‐day international cricket matches. The probabilities depend on the batsman, the bowler, the number of wickets lost, the number of balls bowled and the innings. The proposed simulator appears to do a reasonable job at producing realistic results. The simulator allows investigators to address complex questions involving one‐day cricket matches. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Therapeutic Hypothermia: Quantification of the Transition of Core Body Temperature Using the Flexible Mixture Bent‐Cable Model for Longitudinal Data

Hypothermia which is induced by reducing core body temperature is a therapeutic tool used to prevent brain damage resulting from physical trauma. However, all physiological systems begin to slow down due to hypothermia and this can result in increased risk of mortality. Therefore quantification of the transition of core body temperature to early hypothermia is of great clinical interest. Conceptually core body temperature may exhibit an either gradual or abrupt transition. Bent‐cable regression is an appealing statistical tool to model such data due to the model's flexibility and readily interpretable regression coefficients. It handles more flexibly models that traditionally have been handled by low‐order polynomial models (for gradual transition) or piecewise linear changepoint models (for abrupt change). We consider a rat model to quantify the temporal trend of core body temperature primarily to address the question: What is the critical time point associated with a breakdown in the compensatory mechanisms following the start of hypothermia therapy? To this end, we develop a Bayesian modelling framework for bent‐cable regression of longitudinal data to simultaneously account for gradual and abrupt transitions. Our analysis reveals that: (i) about 39% of rats exhibit a gradual transition in core body temperature; (ii) the critical time point is approximately the same regardless of transition type; and (iii) both transition types show a significant increase of core body temperature followed by a significant decrease.     

Covariates in multipath change‐point problems: Modelling and consistency of the MLE

Although the single‐path change‐point problem has been extensively treated in the statistical literature, its multipath counterpart has largely been ignored. In the multipath change‐point setting, it is often of interest to assess the impact of covariates on the change point itself as well as on the parameters before and after the change point. This paper is concerned only with the inclusion of covariates in the change‐point distribution. This is achieved through the hazard of change. Maximum likelihood estimation is discussed and consistency of the maximum likelihood estimators established.     

Non‐Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise‐Deterministic Markov Processes

This paper presents a non‐parametric method for estimating the conditional density associated to the jump rate of a piecewise‐deterministic Markov process. In our framework, the estimation needs only one observation of the process within a long time interval. Our method relies on a generalization of Aalen's multiplicative intensity model. We prove the uniform consistency of our estimator, under some reasonable assumptions related to the primitive characteristics of the process. A simulation study illustrates the behaviour of our estimator.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

Asymptotic Properties of QML Estimators for VARMA Models with Time‐dependent Coefficients

This paper is about vector autoregressive‐moving average models with time‐dependent coefficients to represent non‐stationary time series. Contrary to other papers in the univariate case, the coefficients depend on time but not on the series' length n. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is almost surely consistent and asymptotically normal. The theoretical results are illustrated by means of two examples of bivariate processes. It is shown that the assumptions underlying the theoretical results apply. In the second example, the innovations are marginally heteroscedastic with a correlation ranging from ?0.8 to 0.8. In the two examples, the asymptotic information matrix is obtained in the Gaussian case. Finally, the finite‐sample behaviour is checked via a Monte Carlo simulation study for n from 25 to 400. The results confirm the validity of the asymptotic properties even for short series and the asymptotic information matrix deduced from the theory.     

Large Deviations for Empirical Estimators of the Stationary Distribution of a Semi-Markov Process with Finite State Space

We prove the large deviation principle for empirical estimators of stationary distributions of semi-Markov processes with finite state space, irreducible embedded Markov chain, and finite mean sojourn time in each state. We consider on/off Gamma sojourn processes as an illustrative example, and, in particular, continuous time Markov chains with two states. In the second case, we compare the rate function in this article with the known rate function concerning another family of empirical estimators of the stationary distribution.     

Sampling from a Max‐Stable Process Conditional on a Homogeneous Functional with an Application for Downscaling Climate Data

Conditional simulation of max‐stable processes allows for the analysis of spatial extremes taking into account additional information provided by the conditions. Instead of observations at given sites as usually done, we consider a single condition given by a more general functional of the process as may occur in the context of climate models. As the problem turns out to be intractable analytically, we make use of Markov chain Monte Carlo methods to sample from the conditional distribution. Simulation studies indicate fast convergence of the Markov chains involved. In an application to precipitation data, the utility of the procedure as a tool to downscale climate data is demonstrated.     

A Markov regime‐switching model for crude‐oil markets: Comparison of composite likelihood and full likelihood

We use the two‐state Markov regime‐switching model to explain the behaviour of the WTI crude‐oil spot prices from January 1986 to February 2012. We investigated the use of methods based on the composite likelihood and the full likelihood. We found that the composite‐likelihood approach can better capture the general structural changes in world oil prices. The two‐state Markov regime‐switching model based on the composite‐likelihood approach closely depicts the cycles of the two postulated states: fall and rise. These two states persist for on average 8 and 15 months, which matches the observed cycles during the period. According to the fitted model, drops in oil prices are more volatile than rises. We believe that this information can be useful for financial officers working in related areas. The model based on the full‐likelihood approach was less satisfactory. We attribute its failure to the fact that the two‐state Markov regime‐switching model is too rigid and overly simplistic. In comparison, the composite likelihood requires only that the model correctly specifies the joint distribution of two adjacent price changes. Thus, model violations in other areas do not invalidate the results. The Canadian Journal of Statistics 41: 353–367; 2013 © 2013 Statistical Society of Canada     

Building and Fitting Non‐Gaussian Latent Variable Models via the Moment‐Generating Function

Abstract. For certain classes of hierarchical models, it is easy to derive an expression for the joint moment‐generating function (MGF) of data, whereas the joint probability density has an intractable form which typically involves an integral. The most important example is the class of linear models with non‐Gaussian latent variables. Parameters in the model can be estimated by approximate maximum likelihood, using a saddlepoint‐type approximation to invert the MGF. We focus on modelling heavy‐tailed latent variables, and suggest a family of mixture distributions that behaves well under the saddlepoint approximation (SPA). It is shown that the well‐known normalization issue renders the ordinary SPA useless in the present context. As a solution we extend the non‐Gaussian leading term SPA to a multivariate setting, and introduce a general rule for choosing the leading term density. The approach is applied to mixed‐effects regression, time‐series models and stochastic networks and it is shown that the modified SPA is very accurate.     

The Skew‐normal Distribution and Related Multivariate Families*

Abstract. This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so‐called skew‐normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew‐elliptical families, and some forms of semi‐parametric formulations, extending the relevance of the field much beyond the original theme of ‘skewness’. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non‐linear time series, especially in connection with financial data, and more.