Maximum Average-Power (MAP) Tests

The objective of this article is to propose and study frequentist tests that have maximum average power, averaging with respect to some specified weight function. First, some relationships between these tests, called maximum average-power (MAP) tests, and most powerful or uniformly most powerful tests are presented. Second, the existence of a maximum average-power test for any hypothesis testing problem is shown. Third, an MAP test for any hypothesis testing problem with a simple null hypothesis is constructed, including some interesting classical examples. Fourth, an MAP test for a hypothesis testing problem with a composite null hypothesis is discussed. From any one-parameter exponential family, a commonly used UMPU test is shown to be also an MAP test with respect to a rich class of weight functions. Finally, some remarks are given to conclude the article.     

Transform orders and stochastic monotonicity of statistical functionals

In some inferential statistical methods, such as tests and confidence intervals, it is important to describe the stochastic behavior of statistical functionals, aside from their large sample properties. We study such a behavior in terms of the usual stochastic order. For this purpose, we introduce a generalized family of stochastic orders, which is referred to as transform orders, showing that it provides a flexible framework for deriving stochastic monotonicity results. Given that our general definition makes it possible to obtain some well known ordering relations as particular cases, we can easily apply our method to different families of functionals. These include some prominent inequality measures, such as the generalized entropy, the Gini index, and its generalizations. We also illustrate the applicability of our approach by determining the least favorable distribution, and the behavior of some bootstrap statistics, in some goodness-of-fit testing procedures.     

Estimation Of A Survival Function With Doubly Censored Data And Dirichlet Process Prior Knowledge On The Observable Variable

This article presents a nonparametric Bayesian procedure for estimating a survival curve in a proportional hazard model when some of the data are censored on the left and some are censored on the right. The method works under the assumption that there is a Dirichlet process prior knowledge on the observable variable. Strong consistency of the estimator is proved and an example is given. To finish some simulation is presented to analyze the estimator.     

Massively parallel computing: a statistical application

Massively parallel computing is a computing environment with thousands of subprocessors. It requires some special programming methods, but is well suited to certain imaging problems. One such statistical example is discussed in this paper. In addition there are other natural statistical problems for which this technology is well suited. This paper describes our experience, as statisticians, with a massively parallel computer in a problem of image correlation spectroscopy. Even with this computing environment some direct computations would still take in the order of a year to finish. It is shown that some of the algorithms of interest can be made parallel.     

Strong laws of large numbers for sub-linear expectation without independence

In this paper, we investigate some strong laws of large numbers for sub-linear expectation without independence which generalize the classical ones. We give some strong laws of large numbers for sub-linear expectation on some moment conditions with respect to the partial sum and some conditions similar to Petrov’s. We can reduce the conclusion to a simple form when the the sequence of random variables is i.i.d. We also show a strong law of large numbers for sub-linear expectation with assumptions of quasi-surely.     

Performances of the Positive-Rule Stein-Type Ridge Estimator in a Linear Regression Model with Spherically Symmetric Error Distributions

In this article, the positive-rule Stein-type ridge estimator (PSRE) is introduced for the parameters in a multiple linear regression model with spherically symmetric error distributions when it is suspected that the parameter vector may be restricted to a linear manifold. The bias and quadratic risk functions of the PSRE are derived and compared with some related competing estimators in literatures. Particularly, some sufficient conditions are derived for superiority of the PSRE over the ordinary ridge estimator, the restricted ridge estimator and the preliminary test ridge estimator, respectively. Furthermore, some graphical results are provided to illustrate some of the theoretical results.     

Parameter estimation in a verhulst stochastic model

Summary Maximum likelihood estimation is a well-known statistical tool. When applied to the study of dynamical continuous-time phenomena, it requires some specific assumptions. In this paper we discuss some properties of the maximum likelihood estimator for a stochastic Verhulst model and we show some simulation results on the behaviour of the corresponding discrete estimator.     

<Emphasis Type="Italic">Z</Emphasis>-process method for change point problems with applications to discretely observed diffusion processes

The aim of this paper is to develop a general, unified approach, based on some partial estimation functions which we call “Z-process”, to some change point problems in mathematical statistics. The method proposed can be applied not only to ergodic models but also to some models where the Fisher information matrix is random. Applications to some concrete models, including a parametric model for volatilities of diffusion processes are presented. Simulations for randomly time-transformed Brownian bridge process appearing as the limit of the proposed test statistics are performed with computer intensive use.     

On Estimation and Tests of Time-Varying Effects in the Proportional Hazards Model

Abstract.  Cox's proportional hazards model is routinely used in many applied fields, some times, however, with too little emphasis on the fit of the model. In this paper, we suggest some new tests for investigating whether or not covariate effects vary with time. These tests are a natural and integrated part of an extended version of the Cox model. An important new feature of the suggested test is that time constancy for a specific covariate is examined in a model, where some effects of other covariates are allowed to vary with time and some are constant; thus making successive testing of time-dependency possible. The proposed techniques are illustrated with the well-known Mayo liver disease data, and a small simulation study investigates the finite sample properties of the tests.     

A Distribution for Multivariate Frailty Based on the Compound Poisson Distribution with Random Scale

Frailty models are often used to model heterogeneity in survival analysis. The most common frailty model has an individual intensity which is a product of a random factor and a basic intensity common to all individuals. This paper uses the compound Poisson distribution as the random factor. It allows some individuals to be non-susceptible, which can be useful in many settings. In some diseases, one may suppose that a number of families have an increased susceptibility due to genetic circumstances. Then, it is logical to use a frailty model where the individuals within each family have some shared factor, while individuals between families have different factors. This can be attained by randomizing the Poisson parameter in the compound Poisson distribution. To our knowledge, this is a new distribution. The power variance function distributions are used for the Poisson parameter. The subsequent appearing distributions are studied in some detail, both regarding appearance and various statistical properties. An application to infant mortality data from the Medical Birth Registry of Norway is included, where the model is compared to more traditional shared frailty models.     

Influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes

In this paper, we propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest follows the Negative Binomial distribution and the time to event follows a Weibull distribution. Indeed, we introduce the Weibull-Negative-Binomial (WNB) distribution, which can be used in order to model survival data when the hazard rate function is increasing, decreasing and some non-monotonous shaped. Another advantage of the proposed model is that it has some distributions commonly used in lifetime analysis as particular cases. Moreover, the proposed model includes as special cases some of the well-know cure rate models discussed in the literature. We consider a frequentist analysis for parameter estimation of a WNB model with cure rate. Then, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. Finally, the methodology is illustrated on a medical data.     

Distribution-free test in Tobit mean regression model

This paper discusses the problem of fitting a parametric model in Tobit mean regression models. The proposed test is based on the supremum of the Khamaladze-type transformation of a partial sum process of calibrated residuals. The asymptotic null distribution of this transformed process is shown to be the same as that of a time-transformed standard Brownian motion. Consistency of this sequence of tests against some fixed alternatives and asymptotic power under some local nonparametric alternatives are also discussed. Simulation studies are conducted to assess the finite sample performance of the proposed test. The power comparison with some existing tests shows some superiority of the proposed test at the chosen alternatives.     

General mixed Poisson regression models with varying dispersion

A general class of mixed Poisson regression models is introduced. This class is based on a mixing between the Poisson distribution and a distribution belonging to the exponential family. With this, we unified some overdispersed models which have been studied separately, such as negative binomial and Poisson inverse gaussian models. We consider a regression structure for both the mean and dispersion parameters of the mixed Poisson models, thus extending, and in some cases correcting, some previous models considered in the literature. An expectation–maximization (EM) algorithm is proposed for estimation of the parameters and some diagnostic measures, based on the EM algorithm, are considered. We also obtain an explicit expression for the observed information matrix. An empirical illustration is presented in order to show the performance of our class of mixed Poisson models. This paper contains a Supplementary Material.     

Multi-sample inference for the simple-tree alternative based on one-sample confidence intervals

The problem of testing homogeneity of several group means is considered against some patterned alternatives for the one-way classified data. The patterns of interest include the simple-tree and the trend alternatives. The approach is to begin with some suitably defined one-sample confidence intervals for the groups in a graphical display. Depending on the pattern of interest, orientation features of the display are examined, more formally, using proposed overall tests or rules. In the classical setup under normality, the case of known common variance is treated in detail; extensions to the case of unknown variance are indicated. When normality is in doubt, a nonparametric procedure based on the sign test is proposed. The necessary critical values are percentiles of either a multivariate normal distribution or a multivariate t-distribution. Although some existing tables can be used for the critical values (or the P-values) in some special cases, in general, the use of simulations is recommended and the steps are detailed in the appendix. An illustrative numerical example is provided.     

Estimating the Number of Change Points in Exponential Families Distributions

A simple method is proposed to detect the number of change points in a sequence of independent exponential family random variables. An estimator to maximize some criterion, say SC ( k ), which is to maximize the log- likelihood function with some penalty term, is used in detection. Under some mild assumptions, the consistency of the estimator for the true number of change points and the boundedness between the estimated change locations and the true change location are obtained. Some simulated results are given, and the Nile problem is investigated by this method.     

A Bayesian approach for non-homogeneous gamma degradation processes

A Bayesian approach based on the Markov Chain Monte Carlo technique is proposed for the non-homogeneous gamma process with power-law shape function. Vague and informative priors, formalized on some quantities having a “physical” meaning, are provided. Point and interval estimation of process parameters and some functions thereof are developed, as well as prediction on some observable quantities that are useful in defining the maintenance strategy is proposed. Some useful approximations are derived for the conditional and unconditional mean and median of the residual life to reduce computational time. Finally, the proposed approach is applied to a real dataset.     

Bayesian bootstrap prediction

In this paper, bootstrap prediction is adapted to resolve some problems in small sample datasets. The bootstrap predictive distribution is obtained by applying Breiman's bagging to the plug-in distribution with the maximum likelihood estimator. The effectiveness of bootstrap prediction has previously been shown, but some problems may arise when bootstrap prediction is constructed in small sample datasets. In this paper, Bayesian bootstrap is used to resolve the problems. The effectiveness of Bayesian bootstrap prediction is confirmed by some examples. These days, analysis of small sample data is quite important in various fields. In this paper, some datasets are analyzed in such a situation. For real datasets, it is shown that plug-in prediction and bootstrap prediction provide very poor prediction when the sample size is close to the dimension of parameter while Bayesian bootstrap prediction provides stable prediction.     

Inference for the Birnbaum–Saunders Lifetime Regression Model with Applications

The Birnbaum–Saunders distribution is a positively skewed distribution that is frequently used for analyzing lifetime data. Regression analysis is widely used in this context when some covariates are involved in the life-test. In this article, we discuss the maximum likelihood estimation of the model parameters and associated inference. We discuss the likelihood-ratio tests for some hypotheses of interest as well as some interval estimation methods. A Monte Carlo simulation study is then carried out to examine the performance of the proposed estimators and the interval estimation methods. Finally, some numerical data analyses are done for illustrating all the inferential methods developed here.     

Improvements to the Escalation with Overdose Control design and a comparison with the restricted Continual Reassessment Method

The Escalation with Overdose Control (EWOC) design for cancer dose finding clinical trials is a variation of the Continual Reassessment Method (CRM) that was proposed to overcome the limitation of the original CRM of exposing patients to high toxic doses. The properties of EWOC have been studied to some extent, but some aspects of the design are not well studied, and its performance is not fully understood. Comparisons of the EWOC design to the most commonly used modified CRM designs have not yet been performed, and the advantages of EWOC over the modified CRM designs are unclear. In this paper, we assess the properties of the EWOC design and of the restricted CRM and some variations of these designs. We show that EWOC has several weaknesses that CRM does not have that make it impractical to use in its original formulation. We propose modified EWOC designs that address some of the weaknesses and that have some desirable statistical properties compared with the original EWOC design, the restricted CRM design, and the 3 + 3 design. However, their statistical properties are sensitive to correct specification of the prior distribution of their parameters and hence nevertheless will need to be used with some caution. The restricted CRM design is shown to have more stable performance across a wider family of dose‐toxicity curves than EWOC and therefore may be a preferable general choice in cancer clinical research.     

Some aspects of the mean past lifetime of a parallel system under double regularly checking

In this paper, a parallel system consisting of a finite number of identical components with independent lifetimes having a common distribution function is considered, when the failure time of the system is restricted to a finite interval (double regularly checking). Under these conditions, the mean past lifetime (MPL) of the system is presented and some of its properties are derived. It is shown that the underlying distribution function can be recovered from the proposed MPL. Then, a consistent estimator for MPL is presented and some of its properties are studied. This estimator also could be used for the single monitoring case or ordinary MPL. Finally, some properties of the MPL of a parallel system with nonidentical components are discussed.