Venkata K. Jandhyala

Inference for single and multiple change-points in time series

The article reviews methods of inference for single and multiple change‐points in time series, when data are of retrospective (off‐line) type. The inferential methods reviewed for a single change‐point in time series include likelihood, Bayes, Bayes‐type and some relevant non‐parametric methods. Inference for multiple change‐points requires methods that can handle large data sets and can be implemented efficiently for estimating the number of change‐points as well as their locations. Our review in this important area focuses on some of the recent advances in this direction. Greater emphasis is placed on multivariate data while reviewing inferential methods for a single change‐point in time series. Throughout the article, more attention is paid to estimation of unknown change‐point(s) in time series, and this is especially true in the case of multiple change‐points. Some specific data sets for which change‐point modelling has been carried out in the literature are provided as illustrative examples under both single and multiple change‐point scenarios.

Tests for parameter changes at unknown times in linear regression models

Abstract Statistics are derived for tests of changes at unknown times in the parameters of a general linear regression model. Asymptotic distribution theory for the tests is discussed. Simulations are carried out to compare power of the statistics derived in this paper with that of other statistics. The derived statistics are shown to have good power properties as compared to other statistics, particularly for the difficult problem of detecting small changes. The change-point methodology is then applied to data on the incidence of AIDS in the United States.

Maximum likelihood estimation of a change-point for exponentially distributed random variables

We consider the problem of estimating the unknown change-point in the parameter of a sequence of independent and exponentially distributed random variables. An exact expression for the asymptotic distribution of the maximum likelihood estimate of the change-point is derived. The analysis is based on the application of Weiner-Hopf factorization identity involving the distribution of ascending and descending ladder heights, and the renewal measure in random walks.

Exact asymptotic distribution of change-point mle for change in the mean of Gaussian sequences

We derive exact computable expressions for the asymptotic distribution of the change-point mle when a change in the mean occurred at an unknown point of a sequence of time-ordered independent Gaussian random variables. The derivation, which assumes that nuisance parameters such as the amount of change and variance are known, is based on ladder heights of Gaussian random walks hitting the half-line. We then show that the exact distribution easily extends to the distribution of the change-point mle when a change occurs in the mean vector of a multivariate Gaussian process. We perform simulations to examine the accuracy of the derived distribution when nuisance parameters have to be estimated as well as robustness of the derived distribution to deviations from Gaussianity. Through simulations, we also compare it with the well-known conditional distribution of the mle, which may be interpreted as a Bayesian solution to the change-point problem. Finally, we apply the derived methodology to monthly averages of water discharges of the Nacetinsky creek, Germany.

Change‐point methods for Weibull models with applications to detection of trends in extreme temperatures

We develop change-point methodology for identifying dynamic trends in the scale and shape parameters of a Weibull distribution. The methodology includes asymptotics of the likelihood ratio statistic for detecting unknown changes in the parameters as well as asymptotics of the maximum likelihood estimate of the unknown change-point. The developed methodology is applied to detect dynamic changes in the minimum temperatures of Uppsala, Sweden. Copyright

Change‐point methods and their applications: contributions of Ian MacNeill

The present paper reviews the important contributions of Ian MacNeill to the theory and methodology of change-point analysis and environmental statistics. The review concentrates on four areas of change-point analysis: sequences of independent random variables; linear regression models with independent as well as serially correlated random errors; regression models with continuity constraints and spatial models of change-points.

A comparison of unconditional and conditional solutions to the maximum likelihood estimation of a change-point

In this paper, we compare the performance of unconditional and conditional solutions to the problem of estimating an unknown change-point by the method of maximum likelihood estimation. We begin by presenting some asymptotic results for the first two moments of the unconditional solution. Then, considering the cases of normal and exponential distributions, we evaluate through simulations the accuracy of large sample distribution theory results for the above two solutions through bias, mean square error, and graphical displays. In both normal and exponential cases, the unconditional solution shows better performance whenever the amount of change is small and when the true parameters are unknown. The differences are particular sharp when the true change-point is towards the tail of the data. When the amount of change is large, both methods perform at similar levels of accuracy. Finally, we illustrate the two solutions through the data on British coal mine explosions.

A property of partial sums of regression least squares residuals and its applications

Abstract For a class of regression models, the sum of the sequence of partial sums of least squares residuals is shown to be zero. Simple and higher order polynomial regression models are included in the class for which this property holds. The result is applied to derive properties of the sample path behavior of residual partial sum processes. It is also applied to show the singularity of the Bayes-type statistic derived to test for one-sided change at unknown time in the intercept parameter of a regression model.

Non-linear mixture models for cross-sectional financial log returns

High-frequency financial log returns are known to exhibit sharp departures from the usual normality assumption. This article carries out an in-depth evaluation of the effectiveness of multidimensional non-linear mixed models that arise from considering a scale mixture-based approach to modeling non-normal log returns over a cross-sectional period of time. Simulation-based first-order and second-order comparison measures that compare the non-linear mixed model with the linear model show strong preference in favour of the non-linear mixed models. The methodology is illustrated through a thorough analysis of quarterly intra-day log returns from CISCO, DELL, COKE, and S&P500 for the years 1998–2000.

CAD Model Assembly Hierarchy Reorganization for Application in Virtual Assembly: A Hybrid Approach Using the CAD System and a Visualization Tool

The hierarchy of assembly components in a CAD assembly model is rarely a true representation of the sequence of assembly of these components during manufacturing. Thus, any assembly planning or evaluation software system needs to re-order and re-group the various components of the CAD assembly model to reflect the sequence of component assembly. Although all parametric CAD systems allow reorganization of the assembly tree, it is a difficult and timeconsuming process due to the relationships and constraints between the various components. We propose an alternative hybrid method that couples the CAD system and a visualization tool that supports reorganization, while preserving data, to allow fast and easy rearranging of the assembly hierarchy. Also, after the reorganization, polygonal representations of the new sub-assemblies are created and the original constraints are also transformed in a consistent manner. As a next logical step, we compare the time required to rearrange the assembly hierarchy using both methods — the CAD system alone and the hybrid system. A statistical analysis using three treatment factors indicates that if the number of components is more than 15, then it is more efficient to use the hybrid method over the CAD system. The overarching goal was to allow fast and efficient creation of different assembly hierarchies to allow the corresponding assembly sequences to be verified in a virtual assembly application that derives its models and constraints from the assembly hierarchy in the CAD system. We have implemented the method to allow the successful reorganization and virtual assembly verification of many industry models, some with several hundred components, provided by various industry partners.Copyright