A proportional hazards regression model with change-points in the baseline function

In this article, we consider a new regression model for counting processes under a proportional hazards assumption. This model is motivated by the need of understanding the evolution of the booking process of a railway company. The main novelty of the approach consists in assuming that the baseline hazard function is piecewise constant, with unknown times of jump (these times of jump are estimated from the data as model parameters). Hence, the parameters of the model can be separated into two different types: parameters that measure the influence of the covariates, and parameters from a multiple change-point model for the baseline. Cox??s semiparametric regression can be seen as a limit case of our model. We develop an iterative procedure to estimate the different parameters, and a test procedure that allows to perform change-point detection in the baseline. Our technique is supported by simulation studies and a real data analysis, which show that our model can be a reasonable alternative to Cox??s regression model, particularly in the presence of tied event times.     

Estimation in the piece-wise constant hazard rate model

The piece-wise constant hazard rate is presented along with the resulting piece-wise constant exponential model for the life times. Maximum likelihood estimation is considered for the complete life test and a life test censored at a preset time. The estimators are found along with their expected values and variances. An example from industry illustrates the estimation procedure in the special case where only two pieces are used.     

Detecting multiple change points in piecewise constant hazard functions

The National Cancer Institute (NCI) suggests a sudden reduction in prostate cancer mortality rates, likely due to highly successful treatments and screening methods for early diagnosis. We are interested in understanding the impact of medical breakthroughs, treatments, or interventions, on the survival experience for a population. For this purpose, estimating the underlying hazard function, with possible time change points, would be of substantial interest, as it will provide a general picture of the survival trend and when this trend is disrupted. Increasing attention has been given to testing the assumption of a constant failure rate against a failure rate that changes at a single point in time. We expand the set of alternatives to allow for the consideration of multiple change-points, and propose a model selection algorithm using sequential testing for the piecewise constant hazard model. These methods are data driven and allow us to estimate not only the number of change points in the hazard function but where those changes occur. Such an analysis allows for better understanding of how changing medical practice affects the survival experience for a patient population. We test for change points in prostate cancer mortality rates using the NCI Surveillance, Epidemiology, and End Results dataset.     

Bayesian outlier testing using the predictive distribution for a linear model op constant intraclass form

The problem of testing suspected outliers from a linear model with constant intraclass correlation is considered from a Bayesian viewpoint. The main objective of this paper is to develop an outlier test procedure based on the predictive distribution of suspected outlier observations given a set of existing inlier observations. The test procedure is easily performed with the usual F and t distributions.     

An Objective Bayesian Criterion to Determine Model Prior Probabilities

We discuss the problem of selecting among alternative parametric models within the Bayesian framework. For model selection problems, which involve non‐nested models, the common objective choice of a prior on the model space is the uniform distribution. The same applies to situations where the models are nested. It is our contention that assigning equal prior probability to each model is over simplistic. Consequently, we introduce a novel approach to objectively determine model prior probabilities, conditionally, on the choice of priors for the parameters of the models. The idea is based on the notion of the worth of having each model within the selection process. At the heart of the procedure is the measure of this worth using the Kullback–Leibler divergence between densities from different models.     

Additive transformation models for clustered failure time data

We propose a class of additive transformation risk models for clustered failure time data. Our models are motivated by the usual additive risk model for independent failure times incorporating a frailty with mean one and constant variability which is a natural generalization of the additive risk model from univariate failure time to multivariate failure time. An estimating equation approach based on the marginal hazards function is proposed. Under the assumption that cluster sizes are completely random, we show the resulting estimators of the regression coefficients are consistent and asymptotically normal. We also provide goodness-of-fit test statistics for choosing the transformation. Simulation studies and real data analysis are conducted to examine the finite-sample performance of our estimators.     

Statistical inference for the accelerated failure time model under two-stage generalized case–cohort design

Abstract

In this article, we propose a two-stage generalized case–cohort design and develop an efficient inference procedure for the data collected with this design. In the first-stage, we observe the failure time, censoring indicator and covariates which are easy or cheap to measure, and in the second-stage, select a subcohort by simple random sampling and a subset of failures in remaining subjects from the first-stage subjects to observe their exposures which are different or expensive to measure. We derive estimators for regression parameters in the accelerated failure time model under the two-stage generalized case–cohort design through the estimated augmented estimating equation and the kernel function method. The resulting estimators are shown to be consistent and asymptotically normal. The finite sample performance of the proposed method is evaluated through the simulation studies. The proposed method is applied to a real data set from the National Wilm’s Tumor Study Group.     

Testing and dating structural changes in copula-based dependence measures

This paper is concerned with testing and dating structural breaks in the dependence structure of multivariate time series. We consider a cumulative sum (CUSUM) type test for constant copula-based dependence measures, such as Spearman''s rank correlation and quantile dependencies. The asymptotic null distribution is not known in closed form and critical values are estimated by an i.i.d. bootstrap procedure. We analyze size and power properties in a simulation study under different dependence measure settings, such as skewed and fat-tailed distributions. To date breakpoints and to decide whether two estimated break locations belong to the same break event, we propose a pivot confidence interval procedure. Finally, we apply the test to the historical data of 10 large financial firms during the last financial crisis from 2002 to mid-2013.     

Optimal sequential Bayesian analysis for degradation tests

Degradation tests are especially difficult to conduct for items with high reliability. Test costs, caused mainly by prolonged item duration and item destruction costs, establish the necessity of sequential degradation test designs. We propose a methodology that sequentially selects the optimal observation times to measure the degradation, using a convenient rule that maximizes the inference precision and minimizes test costs. In particular our objective is to estimate a quantile of the time to failure distribution, where the degradation process is modelled as a linear model using Bayesian inference. The proposed sequential analysis is based on an index that measures the expected discrepancy between the estimated quantile and its corresponding prediction, using Monte Carlo methods. The procedure was successfully implemented for simulated and real data.     

The Method to Identify a Biomarker and to Evaluate Its Efficiency for Survival by Using the Joint Model of the Accelerate Failure Time and Longitudinal Data

In this article, we discuss how to identify longitudinal biomarkers in survival analysis under the accelerated failure time model and also discuss the effectiveness of biomarkers under the accelerated failure time model. Two methods proposed by Shcemper et al. are deployed to measure the efficacy of biomarkers. We use simulations to explore how the factors can influence the power of a score test to detect the association of a longitudinal biomarker and the survival time. These factors include the functional form of the random effects from the longitudinal biomarkers, in the different number of individuals, and time points per individual. The simulations are used to explore how the number of individuals, the number of time points per individual influence the effectiveness of the biomarker to predict survival at the given endpoint under the accelerated failure time model. We illustrate our methods using a prothrombin index as a predictor of survival in liver cirrhosis patients.     

Failure Inference and Optimization for Step Stress Model Based on Bivariate Wiener Model

In this article, we consider the situation under a life test, in which the failure time of the test units are not related deterministically to an observable stochastic time varying covariate. In such a case, the joint distribution of failure time and a marker value would be useful for modeling the step stress life test. The problem of accelerating such an experiment is considered as the main aim of this article. We present a step stress accelerated model based on a bivariate Wiener process with one component as the latent (unobservable) degradation process, which determines the failure times and the other as a marker process, the degradation values of which are recorded at times of failure. Parametric inference based on the proposed model is discussed and the optimization procedure for obtaining the optimal time for changing the stress level is presented. The optimization criterion is to minimize the approximate variance of the maximum likelihood estimator of a percentile of the products’ lifetime distribution.     

Bayesian binary segmentation procedure for detecting streakiness in sports

Summary.  When an individual player or team enjoys periods of good form, and when these occur, is a widely observed phenomenon typically called 'streakiness'. It is interesting to assess which team is a streaky team, or who is a streaky player in sports. Such competitors might have a large number of successes during some periods and few or no successes during other periods. Thus, their success rate is not constant over time. We provide a Bayesian binary segmentation procedure for locating changepoints and the associated success rates simultaneously for these competitors. The procedure is based on a series of nested hypothesis tests each using the Bayes factor or the Bayesian information criterion. At each stage, we only need to compare a model with one changepoint with a model based on a constant success rate. Thus, the method circumvents the computational complexity that we would normally face in problems with an unknown number of changepoints. We apply the procedure to data corresponding to sports teams and players from basketball, golf and baseball.     

Tests for bivaroate exponentiality against bifra alternatives based on censored samples

We develop a test procedure to test the hypothesis that the distribution of the lifetime is bivariate exponential of Marshall and Olkin against that it is bivariate increasing failure rate average when the sample is of the type univariate or bivariate randomly censored.     

A new test for two-sample location problem based on empirical distribution function

We propose a new test for testing the equality of location parameter of two populations based on empirical distribution function (ECDF). The test statistics is obtained as a power divergence between two ECDFs. The test is shown to be distribution free, and its null distribution is obtained. We conducted empirical power comparison of the proposed test with several other available tests in the literature. We found that the proposed test performs better than its competitors considered here under several population structures. We also used two real datasets to illustrate the procedure.     

Exponentiality test based on Renyi distance between equilibrium distributions

ABSTRACT

On the basis of Csiszar's φ-divergence discrimination information, we propose a measure of discrepancy between equilibriums associated with two distributions. Proving that a distribution can be characterized by associated equilibrium distribution, a Renyi distance of the equilibrium distributions is constructed that made us to propose an EDF-based goodness-of-fit test for exponential distribution. For comparing the performance of the proposed test, some well-known EDF-based tests and some entropy-based tests are considered. Based on the simulation results, the proposed test has better powers than those of competing entropy-based tests for the alternatives with decreasing hazard rate function. The use of the proposed test is evaluated in an illustrative example.     

A Test for Constant Hazard Against a Change-Point Alternative

In this article, we consider the change-point hazard rate model which arises quite commonly in mechanical or biological systems, which experience a high hazard rate early in their lifetime due to infant mortality and then a constant or steady hazard rate after the threshold time. We first derive the corresponding mean residual life function (MRLF) and observe that the MRLF is initially increasing and then constant. Here, we derive a test statistic for exponentiality against Increasing Initially then Constant Mean Residual Life (ICMRL). We also derive the asymptotic distribution of the test statistic and compare the power of the test with other existing tests such as likelihood ratio, Weibull, and Log gamma tests considered in the literature. The test performs quite well as compared to other alternatives studied.     

Sharp bounds for the mean of the total time on test for distributions with increasing generalized failure rate

The results of this paper are the continuation of the research presented by Bieniek [Optimal bounds for the mean of the total time on test for distributions with decreasing generalized failure rate. Statistics. 2016;50:1206–1220]. We consider the remaining total time on test after a given failure in a life test experiment. We derive sharp upper bounds on the mean of the total time on test optimal in the class of distributions with increasing generalized failure rate with respect to generalized Pareto distributions. Specific results are obtained for distributions with increasing density and increasing failure rate. We also provide exemplary numerical values of the obtained bounds and we compare them with the corresponding bounds for distributions with decreasing generalized failure rate.     

Testing hypotheses in truncated samples by means of divergence statistics

In this paper we consider the problem of testing hypotheses in parametric models, when only the first r (of n) ordered observations are known.Using divergence measures, a procedure to test statistical hypotheses is proposed, Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained.Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators for truncated samples are considered.Applications of these results in testing statistical hypotheses, on the basis of truncated data, are presented.The small sample behavior of the proposed test statistics is analyzed in particular cases.A comparative study of power values is carried out by computer simulation.     

Variance Estimation in Heteroscedastic Models by Undecimated Haar Transform

We propose a method in order to maximize the accuracy in the estimation of piecewise constant and piecewise smooth variance functions in a nonparametric heteroscedastic fixed design regression model. The difference-based initial estimates are obtained from the given observations. Then an estimator is constructed by using iterative regularization method with the analysis-prior undecimated three-level Haar transform as regularizer term. We notice that this method shows better results in the mean square sense over an existing adaptive estimation procedure considering all the standard test functions used in addition to the functions that we target. Some simulations and comparisons with other methods are conducted to assess the performance of the proposed method.     

Test for independence between time to failure and cause of failure in competing risks with k causes

In this paper, we develop a simple nonparametric test for testing the independence of time to failure and cause of failure in competing risks set up. We generalise the test to the situation where failure data is right censored. We obtain the asymptotic distribution of the test statistics for complete and censored data. The efficiency loss due to censoring is studied using Pitman efficiency. The performance of the proposed test is evaluated through simulations. Finally we illustrate our test procedure using three real data sets.