An Efficient Operator for the Change Point Estimation in Partial Spline Model

In bioinformatics application, the estimation of the starting and ending points of drop-down in the longitudinal data is important. One possible approach to estimate such change times is to use the partial spline model with change points. In order to use estimate change time, the minimum operator in terms of a smoothing parameter has been widely used, but we showed that the minimum operator causes large MSE of change point estimates. In this paper, we proposed the summation operator in terms of a smoothing parameter, and our simulation study showed that the summation operator gives smaller MSE for estimated change points than the minimum one. We also applied the proposed approach to the experiment data, blood flow during photodynamic cancer therapy.     

Change Point Estimation of Multivariate Linear Profiles Under Linear Drift

In this paper, maximum likelihood estimators (MLE) for both step and linear drift changes in the regression parameters of multivariate linear profiles are developed. Performance of the proposed estimators is compared under linear drift changes in the regression parameters when a combined MEWMA and Chi-square control charts method signals an out-of-control condition. The effect of smoothing parameter of MEWMA control charts, missing data, and multiple drift changes on the performance of the both estimators is also evaluated. The application of the proposed estimators is also investigated thorough a numerical example resulted from a real case.     

Intensity Estimation for Spatial Point Processes Observed with Noise

Abstract.  This article introduces a kernel estimator of the intensity function of spatial point processes taking into account location errors. The asymptotic properties of the estimator are derived and a bandwidth selection procedure is described. A simulation study compares our results with that of the classical kernel estimator and shows that the edge-corrected deconvoluting kernel estimator is more appropriate.     

Parameter Estimation for a Discretely Observed Integrated Diffusion Process

Abstract.  We consider the estimation of unknown parameters in the drift and diffusion coefficients of a one-dimensional ergodic diffusion X when the observation is a discrete sampling of the integral of X at times i Δ , i  =  1 ,…, n . Assuming that the sampling interval tends to 0 while the total length time interval tends to infinity, we first prove limit theorems for functionals associated with our observations. We apply these results to obtain a contrast function. The associated minimum contrast estimators are shown to be consistent and asymptotically Gaussian with different rates for drift and diffusion coefficient parameters.     

U-Statistic Based Modified Information Criterion for Change Point Problems

The U-statistic based modified information criterion (MIC) is proposed and applied to detect the change point in a sequence of independent random variables. In this article, we show that the method is consistent in selecting the correct model, and the resulting test statistic has a simple limiting distribution. We investigate the method based on both symmetric and anti-symmetric kernel functions. The simulation results indicate that the new method has better power in detecting the changes compared to other methods, such as the likelihood based MIC (Chen et al., 2006 Chen , J. , Gupta , A. K. , Pan , J. ( 2006 ). Information criterion and change point problem for regular models . Sankhyā 68 : 252 – 282 . [Google Scholar]) and the Bayesian information criterion of Schwarz (BIC, Schwarz, 1978 Schwarz , G. ( 1978 ). Estimating the dimension of a model . Ann. Statist. 6 : 461 – 464 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Detection of Change Point in Nonparametric Function with Unit-Root Noise by Wavelet

A wavelet method is proposed to detect jumps in a function which is observed with unit-root noise. We obtain critical values at any scale and prove the consistency of wavelet detection when the nonparametric function is smooth. It shows that the estimation of the number and locations of change points are consistent when there are change points in the nonparametric function. Simulation study supports our method.     

Parameter Estimation for a Bidimensional Partially Observed Ornstein–Uhlenbeck Process with Biological Application

Abstract. We consider a bidimensional Ornstein–Uhlenbeck process to describe the tissue microvascularization in anti‐cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is to estimate the SDE parameters. We use the main advantage of a one‐dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion, which allows to implement an easy numerical maximization of the likelihood. Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the maximum likelihood estimator. We show that this ARMA property can be generalized to a higher dimensional underlying Ornstein–Uhlenbeck diffusion. We compare this estimator with the one obtained by the well‐known expectation maximization algorithm on simulated data. Our estimation methods can be directly applied to other biological contexts such as drug pharmacokinetics or hormone secretions.     

Optimal Detection of a Change Point in a Poisson Process for Different Observation Schemes

Abstract.  Change point problems are considered where at some unobservable time the intensity of a point process ( T_n ), n ∈  N , has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available information. We consider three information levels, namely sequential observation of ( T_n ), ex post decision after observing the point process up to a fixed time t * and a combination of both observation schemes. In all of these cases the detection problem is viewed as an optimal stopping problem which can be solved by deriving a semimartingale representation of the gain process and applying tools from filtering theory.     

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

We consider the problem of parameter estimation for inhomogeneous space‐time shot‐noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality‐reducing techniques to estimate different parts of the model separately. We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes – possible overlapping in the projection process of clusters, which are clearly separated in the original space‐time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process. For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.     

A Likelihood Integrated Method for Exploratory Graphical Analysis of Change Point Problem with Directional Data

In this article we introduce a new likelihood based method, called the likelihood integrated method, which is distinct from the well-known integrated likelihood method. We use the likelihood integrated to propose a simple exploratory graphical analysis for the change point problem in the context of directional data. The method is applied to analysis of two real life data sets. The results obtained by application of this simple method are seen to be quite similar to those obtained earlier by different formal methods in most cases. A small simulation study is conducted to assess the effectiveness of this procedure in indicating presence of change point.     

Asymptotic Inferences for an AR(1) Model with a Change Point and Possibly Infinite Variance

The basic model in this paper is an AR(1) model with a structural break in the AR parameter β at an unknown time k₀. That is, y_t = β₁y_{t ? 1}I{t ? k₀} + β₂y_{t ? 1}I{t > k₀} + ?_t, t = 1, 2, ???, T, where I{ · } denotes the indicator function. Suppose |β₁| < 1, |β₂| < 1, and {?_t, t ? 1} is a sequence of i.i.d. random variables which are in the domain of attraction of the normal law with zero mean and possibly infinite variance, then the limiting distributions for the least squares estimators of β₁ and β₂ are studied in the present paper, which extend some results in Chong (2001 Chong, T.L. (2001). Structural change in AR(1) models. Econometric Theory 17:87–155.[Crossref], [Web of Science ®] , [Google Scholar]).