Failure Inference From a Marker Process Based on a Bivariate Wiener Model

Many models have been proposed that relate failure times and stochastic time-varying covariates. In some of these models, failure occurs when a particular observable marker crosses a threshold level. We are interested in the more difficult, and often more realistic, situation where failure is not related deterministically to an observable marker. In this case, joint models for marker evolution and failure tend to lead to complicated calculations for characteristics such as the marginal distribution of failure time or the joint distribution of failure time and marker value at failure. This paper presents a model based on a bivariate Wiener process in which one component represents the marker and the second, which is latent (unobservable), determines the failure time. In particular, failure occurs when the latent component crosses a threshold level. The model yields reasonably simple expressions for the characteristics mentioned above and is easy to fit to commonly occurring data that involve the marker value at the censoring time for surviving cases and the marker value and failure time for failing cases. Parametric and predictive inference are discussed, as well as model checking. An extension of the model permits the construction of a composite marker from several candidate markers that may be available. The methodology is demonstrated by a simulated example and a case application.     

Characterizations of the poisson process using,the variance

Let γ(t) be the residual life at time t of the renewal process {A(t), t > 0}, which has F as the common distribution function of the inter-arrival times. In this article we prove that if Var(γ(t)) is constant, then F will be exponentially or geometrically distributed under the assumption F is continuous or discrete respectively. An application and a related example also are given.     

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.     

Testing the equality of two independent regression models

In some situations, for example in agriculture, biology, hydrology, and psychology, researchers wish to determine whether the relationship between response variable and predictor variables differs in two populations. In other words, we are interested in comparing two regression models for two independent datasets. In this work, we will use the parametric and nonparametric methods to establish hypothesis testing for the equality of two independent regression models. Then the simulation study is provided to investigate the performance of the proposed method.     

A recursive approach to parameter estimation in regression and time series models

In this paper we discuss the recursive (or on line) estimation in (i) regression and (ii) autoregressive integrated moving average (ARIMA) time series models. The adopted approach uses Kalman filtering techniques to calculate estimates recursively. This approach is used for the estimation of constant as well as time varying parameters. In the first section of the paper we consider the linear regression model. We discuss recursive estimation both for constant and time varying parameters. For constant parameters, Kalman filtering specializes to recursive least squares. In general, we allow the parameters to vary according to an autoregressive integrated moving average process and update the parameter estimates recursively. Since the stochastic model for the parameter changes will "be rarely known, simplifying assumptions have to be made. In particular we assume a random walk model for the time varying parameters and show how to determine whether the parameters are changing over time. This is illustrated with an example.     

Interval estimation of process capability index cpk

Several sampling distribution properties of the estimator for C_pk. are presented under the assumption that the data are normal, independent and identically distributed. In particular, the expectation, variance and skewness are derived. Since the sampling distribution is only weakly skewed, we concluded that a symmetric interval estimator for C_pk . might be reasonable. We developed such a symmetric interval estimator and conducted a simulation study to explore its coverage probabilities.     

The rate of convergence to stationarity for M/G/1 models with admission controls via coupling

We study the workload processes of two M/G/1 queueing systems with restricted capacity: in Model 1 any service requirement that would exceed a certain capacity threshold is truncated; in Model 2 new arrivals do not enter the system if they have to wait more than a fixed threshold time in line. For Model 1 we obtain several results concerning the rate of convergence to equilibrium. In particular, we derive uniform bounds for geometric ergodicity with respect to certain subclasses. For Model 2 geometric ergodicity follows from the finiteness of the moment-generating function of the service time distribution. We derive bounds for the convergence rates in special cases. The proofs use the coupling method.     

Hidden Markov model for discrete circular–linear wind data time series

ABSTRACT

In this work, we deal with a bivariate time series of wind speed and direction. Our observed data have peculiar features, such as informative missing values, non-reliable measures under a specific condition and interval-censored data, that we take into account in the model specification. We analyse the time series with a non-parametric Bayesian hidden Markov model, introducing a new emission distribution, suitable to model our data, based on the invariant wrapped Poisson, the Poisson and the hurdle density. The model is estimated on simulated datasets and on the real data example that motivated this work.     

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.     

On the use of priors in goodness‐of‐fit tests

Priors are introduced into goodness‐of‐fit tests, both for unknown parameters in the tested distribution and on the alternative density. Neyman–Pearson theory leads to the test with the highest expected power. To make the test practical, we seek priors that make it likely a priori that the power will be larger than the level of the test but not too close to one. As a result, priors are sample size dependent. We explore this procedure in particular for priors that are defined via a Gaussian process approximation for the logarithm of the alternative density. In the case of testing for the uniform distribution, we show that the optimal test is of the U‐statistic type and establish limiting distributions for the optimal test statistic, both under the null hypothesis and averaged over the alternative hypotheses. The optimal test statistic is shown to be of the Cramér–von Mises type for specific choices of the Gaussian process involved. The methodology when parameters in the tested distribution are unknown is discussed and illustrated in the case of testing for the von Mises distribution. The Canadian Journal of Statistics 47: 560–579; 2019 © 2019 Statistical Society of Canada     

A new class of survival regression models with heavy-tailed errors: robustness and diagnostics

Birnbaum-Saunders models have largely been applied in material fatigue studies and reliability analyses to relate the total time until failure with some type of cumulative damage. In many problems related to the medical field, such as chronic cardiac diseases and different types of cancer, a cumulative damage caused by several risk factors might cause some degradation that leads to a fatigue process. In these cases, BS models can be suitable for describing the propagation lifetime. However, since the cumulative damage is assumed to be normally distributed in the BS distribution, the parameter estimates from this model can be sensitive to outlying observations. In order to attenuate this influence, we present in this paper BS models, in which a Student-t distribution is assumed to explain the cumulative damage. In particular, we show that the maximum likelihood estimates of the Student-t log-BS models attribute smaller weights to outlying observations, which produce robust parameter estimates. Also, some inferential results are presented. In addition, based on local influence and deviance component and martingale-type residuals, a diagnostics analysis is derived. Finally, a motivating example from the medical field is analyzed using log-BS regression models. Since the parameter estimates appear to be very sensitive to outlying and influential observations, the Student-t log-BS regression model should attenuate such influences. The model checking methodologies developed in this paper are used to compare the fitted models.     

Forecasting future values of changing sequences

Sequences of independent random variables are observed and on the basis of these observations future values of the process are forecast. The Bayesian predictive density of k future observations for normal, exponential, and binomial sequences which change exactly once are analyzed for several cases. It is seen that the Bayesian predictive densities are mixtures of standard probability distributions. For example, with normal sequences the Bayesian predictive density is a mixture of either normal or t-distributions, depending on whether or not the common variance is known. The mixing probabilities are the same as those occurring in the corresponding posterior distribution of the mean(s) of the sequence. The predictive mass function of the number of future successes that will occur in a changing Bernoulli sequence is computed and point and interval predictors are illustrated.     

Testing the difference between two independent regression models

ABSTRACT

In some situations, for example, in biology or psychology studies, we wish to determine whether the linear relationship between response variable and predictor variables differs in two populations. The analysis of the covariance (ANCOVA) or, equivalently, the partial F-test approaches are the commonly used methods. In this study, the asymptotic distribution for the difference between two independent regression coefficients was established. The proposed method was used to derive the asymptotic confidence set for the difference between coefficients and hypothesis testing for the equality of the two regression models. Then a simulation study was conducted to compare the proposed method with the partial F method. The performance of the new method was comparable with that of the partial F method.     

Expert knowledge elicitation using item response theory

In an expert knowledge elicitation exercise, experts face a carefully constructed list of questions that they answer according to their knowledge. The elicitation process concludes when a probability distribution is found that adequately captures the experts' beliefs in the light of those answers. In many situations, it is very difficult to create a set of questions that will efficiently capture the experts' knowledge, since experts might not be able to make precise probabilistic statements about the parameter of interest. We present an approach for capturing expert knowledge based on item response theory, in which a set of binary response questions is proposed to the expert, trying to capture responses directly related to the quantity of interest. As a result, the posterior distribution of the parameter of interest will represent the elicited prior distribution that does not assume any particular parametric form. The method is illustrated by a simulated example and by an application involving the elicitation of rain prophets' predictions for the rainy season in the north-east of Brazil.     

Robust estimation of extremes

We consider the problem of making inferences about extreme values from a sample. The underlying model distribution is the generalized extreme-value (GEV) distribution, and our interest is in estimating the parameters and quantiles of the distribution robustly. In doing this we find estimates for the GEV parameters based on that part of the data which is well fitted by a GEV distribution. The robust procedure will assign weights between 0 and 1 to each data point. A weight near 0 indicates that the data point is not well modelled by the GEV distribution which fits the points with weights at or near 1. On the basis of these weights we are able to assess the validity of a GEV model for our data. It is important that the observations with low weights be carefully assessed to determine whether diey are valid observations or not. If they are, we must examine whether our data could be generated by a mixture of GEV distributions or whether some other process is involved in generating the data. This process will require careful consideration of die subject matter area which led to the data. The robust estimation techniques are based on optimal B-robust estimates. Their performance is compared to the probability-weighted moment estimates of Hosking et al. (1985) in both simulated and real data.     

Nonparametric Estimation of Distribution Functions of Nonstandard Mixtures

ABSTRACT

Nonstandard mixtures are those that result from a mixture of a discrete and a continuous random variable. They arise in practice, for example, in medical studies of exposure. Here, a random variable that models exposure might have a discrete mass point at no exposure, but otherwise may be continuous. In this article we explore estimating the distribution function associated with such a random variable from a nonparametric viewpoint. We assume that the locations of the discrete mass points are known so that we will be able to apply a classical nonparametric smoothing approach to the problem. The proposed estimator is a mixture of an empirical distribution function and a kernel estimate of a distribution function. A simple theoretical argument reveals that existing bandwidth selection algorithms can be applied to the smooth component of this estimator as well. The proposed approach is applied to two example sets of data.     

Two-phase outcome-dependent studies for failure times and testing for effects of expensive covariates

Two- or multi-phase study designs are often used in settings involving failure times. In most studies, whether or not certain covariates are measured on an individual depends on their failure time and status. For example, when failures are rare, case–cohort or case–control designs are used to increase the number of failures relative to a random sample of the same size. Another scenario is where certain covariates are expensive to measure, so they are obtained only for selected individuals in a cohort. This paper considers such situations and focuses on cases where we wish to test hypotheses of no association between failure time and expensive covariates. Efficient score tests based on maximum likelihood are developed and shown to have a simple form for a wide class of models and sampling designs. Some numerical comparisons of study designs are presented.     

A simple method for estimating a regression model for κ between a pair of raters

Agreement studies commonly occur in medical research, for example, in the review of X-rays by radiologists, blood tests by a panel of pathologists and the evaluation of psychopathology by a panel of raters. In these studies, often two observers rate the same subject for some characteristic with a discrete number of levels. The κ-coefficient is a popular measure of agreement between the two raters. The κ-coefficient may depend on covariates, i.e. characteristics of the raters and/or the subjects being rated. Our research was motivated by two agreement problems. The first is a study of agreement between a pastor and a co-ordinator of Christian education on whether they feel that the congregation puts enough emphasis on encouraging members to work for social justice (yes versus no). We wish to model the κ-coefficient as a function of covariates such as political orientation (liberal versus conservative) of the pastor and co-ordinator. The second example is a spousal education study, in which we wish to model the κ-coefficient as a function of covariates such as the highest degree of the father of the wife and the father of the husband. We propose a simple method to estimate the regression model for the κ-coefficient, which consists of two logistic (or multinomial logistic) regressions and one linear regression for binary data. The estimates can be easily obtained in any generalized linear model software program.     

On correlation analysis of many‐to‐many observations: an alternative to Pearson's correlation coefficient and its application to an ecotoxicological study

Correlation studies are an important hypothesis‐generating and testing tool, and have a wide range of applications in many scientific fields. In ecological studies in particular, multiple environmental variables are often measured in an attempt to determine relationships between chemical, physical and biological factors. For example, one may wish to know whether and how soil properties correlate with plant physiology. Although correlation coefficients are widely used, their properties and limitations are often imperfectly understood. This is especially the case when one is interested in correlations between, say, trace element content in sediments and in marine organisms, where no one‐to‐one correspondence exists. We show that evaluating Pearson's correlation coefficient for either site‐specific means or composite samples results in biased estimates, and we propose an alternative estimator. We use simulation studies to demonstrate that our estimator generally has a much smaller bias and mean squared error. We further illustrate its use in a case study of the correlation between trace element content in sediments and in mussels in Lyttelton Harbour, New Zealand.     

How to choose a time zero for patients in external control arms

When a sponsor carries out a single-arm trial of a novel oncology compound, it may wish to assess the efficacy of the compound via comparison of overall survival to an external control arm, constructed using patients included in some retrospective registry. If efficacy of the novel compound is compared to efficacy of physician's choice of chemotherapy, patients in the retrospective registry might qualify for inclusion in the external control arm at multiple different points in time, when they receive different chemotherapy treatments. For example, a patient might qualify at the start of their second, third and fourth lines of therapy. From the start of which line of therapy should this patient's survival be compared to survival of participants in the single-arm trial? Some sponsors have elected to include patients in the external control arm from the last available line of therapy in the retrospective database. Another possibility is to randomly select a line of therapy for each external control arm patient from among those available. In this paper, we show, via probabilistic arguments and also via simulation based on real data, that both of these methods give rise to a bias in favor of the single-arm trial. We further show that this bias can be avoided by instead including external control arm patients multiple times in the external control arm, once for each time they receive qualifying treatment.