Inference for post-change mean by a CUSUM procedure

In this paper, the inference problem for the post-change mean is considered after a change is detected by a CUSUM process in a sequence of independent normal variables. The change-point is estimated as the maximum likelihood estimate at the reference value and the post-change mean is estimated as the sample mean after the change-point estimate. By assuming the change-point is large and the monitoring limit approaches infinity, the first-order bias of the post-change mean estimate and a corrected asymptotic normal pivot are derived conditioning on that a change is detected. Local approximations for small reference value and post-change mean are obtained for numerical evaluation.     

An Application of EM Algorithm to a Change-Point Problem

We consider the problem of change-point in a classical framework while assuming a probability distribution for the change-point. An EM algorithm is proposed to estimate the distribution of the change-point. A change-point model for multiple profiles is also proposed, and EM algorithm is presented to estimate the model. Two examples of Illinois traffic data and Dow Jones Industrial Averages are used to demonstrate the proposed methods.     

Maximum likelihood estimation in hazard rate models with a change-point

The problem of estimation of parameters in hazard rate models with a change-point is considered. An interesting feature of this problem is that the likelihood function is unbounded. A maximum likelihood estimator of the change-point subject to a natural constraint is proposed, which is shown to be consistent.The limiting distributions are also derived.     

Estimation in multi-path change-point problems

The current literature deals with the change-point problem only in the context of the obser¬vation of a single sequence. In this paper, inference will be based on the observation of TV sequences of random variables, each sequence containing one change-point. This extension allows the effective use of bootstrap and empirical Bayes methods, both of which are not feasible in the single-path context. Two classes of these “multi-path” change-point problems are considered. If the change-point is assumed to occur at the the same position in each sequence, then the terminology “fixed-tau multi-path change-point” will be used. In other cases, one may expect the change-point to occur at random positions in each sequence, according to some distribution, a “random-tau multi-path change-point” problem. Examples and simulations are given.     

The LAD estimation of the change-point linear model with randomly censored data

Abstract

In this paper, a change-point linear model with randomly censored data is investigated. We propose the least absolute deviation estimation procedure for regression and change-point parameters simultaneously. The asymptotic properties of the change-point and regression parameter estimators are obtained. We show that the resulting regression parameter estimator is asymptotically normal, and the change-point estimator converges weakly to the minimizer of a given random process. The extensive simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.     

Change point-cure models with application to estimating the change-point effect of age of diagnosis among prostate cancer patients

Previous research on prostate cancer survival trends in the United States National Cancer Institute's Surveillance Epidemiology and End Results database has indicated a potential change-point in the age of diagnosis of prostate cancer around age 50. Identifying a change-point value in prostate cancer survival and cure could have important policy and health care management implications. Statistical analysis of this data has to address two complicating features: (1) change-point models are not smooth functions and so present computational and theoretical difficulties; and (2) models for prostate cancer survival need to account for the fact that many men diagnosed with prostate cancer can be effectively cured of their disease with early treatment. We develop a cure survival model that allows for change-point effects in covariates to investigate a potential change-point in the age of diagnosis of prostate cancer. Our results do not indicate that age under 50 is associated with increased hazard of death from prostate cancer.     

Bayesian analysis for a change in the intercept of simple linear regression

A Bayesian approach is considered to detect a change-point in the intercept of simple linear regression. The Jeffreys noninformative prior is employed and compared with the uniform prior in Bayesian analysis. The marginal posterior distributions of the change-point, the amount of shift and the slope are derived. Mean square errors, mean absolute errors and mean biases of some Bayesian estimates are considered by Monte Carlo methad and some numerical results are also shown.     

On computing estimates of a change-point in the Weibull regression hazard model

The hazard function describes the instantaneous rate of failure at a time t, given that the individual survives up to t. In applications, the effect of covariates produce changes in the hazard function. When dealing with survival analysis, it is of interest to identify where a change point in time has occurred. In this work, covariates and censored variables are considered in order to estimate a change-point in the Weibull regression hazard model, which is a generalization of the exponential model. For this more general model, it is possible to obtain maximum likelihood estimators for the change-point and for the parameters involved. A Monte Carlo simulation study shows that indeed, it is possible to implement this model in practice. An application with clinical trial data coming from a treatment of chronic granulomatous disease is also included.     

Testing for a change in repeated measures data

Four nonparametric test statistics for the change-point problem with repeated measures data are proposed. In a Monte Carlo simulation study, critical values for the proposed test statistics are simulated and the performances of the proposed tests are compared with the performances of some competitive tests in terms of asymptotic behavior and power. We provide appropriate recommendations for different occurrences of the change-point and illustrate the testing methods using a set of real data.     

Maximum likelihood estimator in a multi-phase random regression model

We consider a random regression model with several-fold change-points. The results for one change-point are generalized. The maximum likelihood estimator of the parameters is shown to be consistent, and the asymptotic distribution for the estimators of the coefficients is shown to be Gaussian. The estimators of the change-points converge, with n ^?1 rate, to the vector whose components are the left end points of the maximizing interval with respect to each change-point. The likelihood process is asymptotically equivalent to the sum of independent compound Poisson processes.     

A Bayesian wavelet approach to estimation of a change-point in a nonlinear multivariate time series

ABSTRACT

We propose a semiparametric approach to estimate the existence and location of a statistical change-point to a nonlinear multivariate time series contaminated with an additive noise component. In particular, we consider a p-dimensional stochastic process of independent multivariate normal observations where the mean function varies smoothly except at a single change-point. Our approach involves conducting a Bayesian analysis on the empirical detail coefficients of the original time series after a wavelet transform. If the mean function of our time series can be expressed as a multivariate step function, we find our Bayesian-wavelet method performs comparably with classical parametric methods such as maximum likelihood estimation. The advantage of our multivariate change-point method is seen in how it applies to a much larger class of mean functions that require only general smoothness conditions.     

Detection of a change-point in variance by a weighted sum of powers of variances test

This paper proposes a weighted sum of powers of variances test for detecting changes in variance of a data sequence. Asymptotic critical value formulas are derived for this test. The modified weighted sum of powers of variances test is also introduced so that the accuracy of change-point detection is highly improved for a sample of small size. Simulation studies and real data analysis are presented to assess the proposed tests.     

Testing the Existence of Change-Point in NHPP Software Reliability Models

In the software testing process, the nature of the failure data is affected by many factors, such as the testing environment, testing strategy, and resource allocation. These factors are unlikely to all be kept stable during the entire process of software testing. As a result, the statistical structure of the failure data is likely to experience major changes. Recently, some useful non homogeneous Poisson process (NHPP) models with change-point are proposed. However, in many realistic situations, whether a change-point exists is unknown. Furthermore, some real data seem to have two or more change-points. In this article we propose test statistics to test the existence of change-point(s). The experimental results of real data show that our tests perform well.     

Approximations to the p-values of tests for a change-point under non-standard conditions

Three test statistics for a change-point in a linear model, variants of those considered by Andrews and Ploberger [Optimal tests when a nusiance parameter is present only under the alternative. Econometrica. 1994;62:1383–1414]: the sup-likelihood ratio (LR) statistic; a weighted average of the exponential of LR-statistics and a weighted average of LR-statistics, are studied. Critical values for the statistics with time trend regressors, obtained via simulation, are found to vary considerably, depending on conditions on the error terms. The performance of the bootstrap in approximating p-values of the distributions is assessed in a simulation study. A sample approximation to asymptotic analytical expressions extending those of Kim and Siegmund [The likelihood ratio test for a change-point in simple linear regression. Biometrika. 1989;76:409–423] in the case of the sup-LR test is also assessed. The approximations and bootstrap are applied to the Quandt data [The estimation of a parameter of a linear regression system obeying two separate regimes. J Amer Statist Assoc. 1958;53:873–880] and real data concerning a change-point in oxygen uptake during incremental exercise testing and the bootstrap gives reasonable results.     

An evaluation of change-point estimators for a sequence of normal observations with unknown parameters

Performance of maximum likelihood estimators (MLE) of the change-point in normal series is evaluated considering three scenarios where process parameters are assumed to be unknown. Different shifts, sample sizes, and locations of a change-point were tested. A comparison is made with estimators based on cumulative sums and Bartlett's test. Performance analysis done with extensive simulations for normally distributed series showed that the MLEs perform better (or equal) in almost every scenario, with smaller bias and standard error. In addition, robustness of MLE to non-normality is also studied.     

A change-point estimator with the hazard ratio

The problem of estimation of parameters in hazard rate change models with a change-point is considered. A change-point estimator using the hazard ratio is suggested and compared with the previously developed change-point estimators. The proposed estimator is shown to be consistent. The performance of the proposed estimator is checked and compared with other change-point estimators via simulation.     

位置参数变点的统计推断

关于随机变量序列中是否存在变点的问题一直不断地有人研究,对变点进行估计具有很重要的应用价值。为此,研究至多一个变点的位置参数变点问题,对变点是否存在进行假设检验,论证相关统计量的渐进性质,在此基础上提出变点的一种点估计,并证明估计的强相合性。     

Truncating Estimation for the Mean Change-Point in Heavy-Tailed Dependent Observations

ABSTRACT

This article considers the problem of a mean change-point in heavy-tailed dependent observations. A method of change-point estimation by truncating initial process is proposed, which can weaken the affection of outliers. In the infinite variance case, we obtained a generalization Hájek-Rényi type inequality. Consistency and the rate of convergence for the estimated change-point are also established. The results of a simulation study support validity of our method.     

Change-point mle in the rate of exponential sequences with application to Indonesian seismological data

In this paper, we derive explicit computable expressions for the asymptotic distribution of the maximum likelihood estimate of an unknown change-point in a sequence of independently and exponentially distributed random variables. First we state and prove a theorem that shows asymptotic equivalence of the change-point mle for the cases of both known and unknown parameters, respectively. Thereafter, the computational form of the asymptotic distribution of the change-point mle is derived for the case of known parameter situation only. Simulations show that the distribution for the known case applies very well to the case where the parameters are estimated. Further, it is seen from simulations that the derived unconditional mle shows better performance compared to the conditional solution of Cobb. Application of change detection methodology and the derived estimation methodology show strong support in favor the dynamic triggering hypothesis for seismic faults in Sumatra, Indonesia region.     

Information approach for a lifetime change-point model based on the exponential-logarithmic distribution

ABSTRACT

The exponential-logarithmic distribution is a distribution which has a decreasing failure function and various applications such as in biological and engineering fields. In this paper, we study a change-point problem of this distribution. A procedure based on Schwarz information criterion is proposed to detect changes in parameters of this distribution. Simulations are conducted to indicate the performance of the proposed procedure under different scenarios. Applications on two real data are provided to illustrate the detection procedure.