Multiple hypothesis test for parameter constancy based on recursive residuals

This article presents a multiple hypothesis test procedure that combines two well known tests for structural change in the linear regression model, the CUSUM test and the recursive t test. The CUSUM test is run through the sequence of recursive residuals as usual; if the CUSUM plot does not violate the critical lines, one more step is taken to perform the t test for hypothesis of zero mean based on all recursive residuals. The asymptotic size of this multiple hypothesis test is derived; power simulation results suggest that it outperforms the traditional CUSUM test and complements other tests that are currently stressed in econometrics.     

Multiple hypothesis test for parameter constancy based on recursive residuals

This article presents a multiple hypothesis test procedure that combines two well known tests for structural change in the linear regression model, the CUSUM test and the recursive t test. The CUSUM test is run through the sequence of recursive residuals as usual; if the CUSUM plot does not violate the critical lines, one more step is taken to perform the t test for hypothesis of zero mean based on all recursive residuals. The asymptotic size of this multiple hypothesis test is derived; power simulation results suggest that it outperforms the traditional CUSUM test and complements other tests that are currently stressed in econometrics.     

Examination and analysis of residuals: A test for detecting a monotonic relation between mean and variance in regression through the origin

This paper presents a simple and exact test for detecting a monotonic relation between the mean and variance in linear regression through the origin. This test resulted from utilizing uncorrelated Theil-residuals and the Goldfeld-Quandt peak test. A numerical example is provided to elucidate the method. A simulation experiment was performed to compare the empirical power of this test with those of the existing tests.     

On hypothesis tests in misspecified change-point problems for a Poisson process

Consider an inhomogeneous Poisson process X on [0, T] whose unk-nown intensity function “switches” from a lower function g_* to an upper function h_* at some unknown point ?_* that has to be identified. We consider two known continuous functions g and h such that g_*(t) ? g(t) < h(t) ? h_*(t) for 0 ? t ? T. We describe the behavior of the generalized likelihood ratio and Wald’s tests constructed on the basis of a misspecified model in the asymptotics of large samples. The power functions are studied under local alternatives and compared numerically with help of simulations. We also show the following robustness result: the Type I error rate is preserved even though a misspecified model is used to construct tests.     

Two non parametric methods for change-point detection in distribution

The change-point detection problem is determining whether a change has taken place. Two non parametric methods based on empirical likelihood and the likelihood ratio are proposed for detecting a change-point problem in distributions for independent observations. Numerical studies are carried out to evaluate the performance of the proposed methods. The simulation results demonstrate that the proposed methods are robust, that is, they perform well regardless of whether the observations are from the same distribution family.     

A Bayesian finite mixture change-point model for assessing the risk of novice teenage drivers

The driving risk during the initial period after licensure for novice teenage drivers is typically the highest but decreases rapidly right after. The change-point of driving risk is a critical parameter for evaluating teenage driving risk, which also varies substantially among drivers. This paper presents latent class recurrent-event change-point models for detecting the change-points. The proposed model is applied to the Naturalist Teenage Driving Study, which continuously recorded the driving data of 42 novice teenage drivers for 18 months using advanced in-vehicle instrumentation. We propose a hierarchical BFMM to estimate the change-points by clusters of drivers with similar risk profiles. The model is based on a non-homogeneous Poisson process with piecewise-constant intensity functions. Latent variables which identify the membership of the subjects are used to detect potential clusters among subjects. Application to the Naturalistic Teenage Driving Study identifies three distinct clusters with change-points at 52.30, 108.99 and 150.20?hours of driving after first licensure, respectively. The overall intensity rate and the pattern of change also differ substantially among clusters. The results of this research provide more insight in teenagers' driving behaviour and will be critical to improve young drivers' safety education and parent management programs, as well as provide crucial reference for the GDL regulations to encourage safer driving.     

Uncorrelated residuals and an exact test for two variance components in experimental design

The error contrasts from an experimental design can be constructed from uncorrelated residuals normally associated with the linear model. In this paper uncorrelated residuals are defined for the linear model that has a design matrix which is less than full rank, typical of many experimental design representations. It transpires in this setting, that for certain choices of uncorrelated residuals, corresponding to recursive type residuals, there is a natural partition of information when two variance components are known to be present. Under an assumtion of normality of errors this leads to construction of appropriate F-tests for testing heteroscedasticity. The test, which can be optimal, is applied to two well known data sets to illustrate its usefullness.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x ²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

Convergence rates of change-point estimators and tail probabilities of the first-passage-time process

In the classical setting of the change-point problem, the maximum-likelihood estimator and the traditional confidence region for the change-point parameter are considered. It is shown that the probability of the correct decision, the coverage probability and the expected size of the confidence set converge exponentially fast as the sample size increases to infinity. For this purpose, the tail probabilities of the first passage times are studied. General inequalities are established, and exact asymptotics are obtained for the case of Bernoulli distributions. A closed asymptotic form for the expected size of the confidence set is derived for this case via the conditional distribution of the first passage times.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

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.     

Asymptotic study of the change-point mle in multivariate Gaussian families under contiguous alternatives

The problem of estimating an unknown change-point in the mean vector or covariance matrix of a sequence of independent multivariate Gaussian random variables is considered. Adapting the estimation methodology that Hinkley pursued for the case of abrupt changes, we develop theory for deriving the asymptotic distribution of the maximum likelihood estimator of the change-point when the amount of change is a function of the sample size and goes to zero in a smooth fashion as the sample size goes to infinity, yielding a contiguous change-point model. Simulations have been performed to illustrate the closeness of the asymptotic distribution with the empirical distribution, and to evaluate its robustness to departures from normality for reasonable sample sizes as well as parameter changes. Finally, we apply the methodology to estimate the change-point in the daily log-returns data of BLS (BellSouth) and VZ (Verizon) from NYSE.     

Implementation of recursive nonparametric kernel estimation and a monte carlo study on its finite sample properties

The recursive estimator for the conditional mean of a nonparametric regression model with independent observations was thoroughly explored by Ahmad and Lin (1976), and Singh and Ullah (1986). Their studies are mainly concerned with the estimator's asymptotic behaviour. However, they do not include much discussion on the strategy of computing the estimates. In this paper, we provide a convenient implementation of the recursive estimator and examine its finite sample properties through simulation studies. Our study has demonstrated that for relatively short length of recursive updating, the estimates are generally equivalent to their fixed window width counterparts However, we found that substantial recursive updating can seriously lower the estimator's efficiency even though it is a consistent estimator.     

Implementation of recursive nonparametric kernel estimation and a monte carlo study on its finite sample properties

The recursive estimator for the conditional mean of a nonparametric regression model with independent observations was thoroughly explored by Ahmad and Lin (1976), and Singh and Ullah (1986). Their studies are mainly concerned with the estimator's asymptotic behaviour. However, they do not include much discussion on the strategy of computing the estimates. In this paper, we provide a convenient implementation of the recursive estimator and examine its finite sample properties through simulation studies. Our study has demonstrated that for relatively short length of recursive updating, the estimates are generally equivalent to their fixed window width counterparts However, we found that substantial recursive updating can seriously lower the estimator's efficiency even though it is a consistent estimator.     

Detection of intrusions in information systems by sequential change-point methods

Sequential multi-chart detection procedures for detecting changes in multichannel sensor systems are developed. In the case of complete information on pre-change and post-change distributions, the detection algorithm represents a likelihood ratio-based multichannel generalization of Page’s cumulative sum (CUSUM) test that is applied to general stochastic models that may include correlated and nonstationary observations. There are many potential application areas where it is necessary to consider multichannel generalizations and general statistical models. In this paper our main motivation for doing so is network security: rapid anomaly detection for an early detection of attacks in computer networks that lead to changes in network traffic. Moreover, this kind of application encourages the development of a nonparametric multichannel detection test that does not use exact pre-change (legitimate) and post-change (attack) traffic models. The proposed nonparametric method can be effectively applied to detect a wide variety of attacks such as denial-of-service attacks, worm-based attacks, port-scanning, and man-in-the-middle attacks. In addition, we propose a multichannel CUSUM procedure that is based on binary quantized data; this procedure turns out to be more efficient than the previous two algorithms in certain scenarios. All proposed detection algorithms are based on the change-point detection theory. They utilize the thresholding of test statistics to achieve a fixed rate of false alarms, while allowing changes in statistical models to be detected “as soon as possible”. Theoretical frameworks for the performance analysis of detection procedures, as well as results of Monte Carlo simulations for a Poisson example and results of detecting real flooding attacks, are presented.     

A recursive approach for estimating missing observations in an univariate time series

A linear recursive technique that does not use the Kalman filter approach is proposed to estimate missing observations in an univariate time series. It is assumed that the series follows an invertible ARIMA model. The procedure is based on the restricted forecasting approach, and the recursive linear estimators are optimal in terms of minimum mean-square error.     

A nonparametric test for detecting changes in location

Procedure for the changepoint problem based on Mann-Whitney-Wilcoxon statistics is studied in Schechtman and Wolfe (1981). In this paper we give tables for the null distributions of the statistics for the one-sided and two-sided alternatives. We also report on some Monte Carlo power comparisons involving another nonparametric competitor, proposed by Pettitt (1979).     

A Bayesian analysis of the change-point problem for directional data

In this paper, we discuss a simple fully Bayesian analysis of the change-point problem for the directional data in the parametric framework with von Mises or circular normal distribution as the underlying distribution. We first discuss the problem of detecting change in the mean direction of the circular normal distribution using a latent variable approach when the concentration parameter is unknown. Then, a simpler approach, beginning with proper priors for all the unknown parameters – the sampling importance resampling technique – is used to obtain the posterior marginal distribution of the change-point. The method is illustrated using the wind data [E.P. Weijers, A. Van Delden, H.F. Vugts and A.G.C.A. Meesters, The composite horizontal wind field within convective structures of the atmospheric surface layer, J. Atmos. Sci. 52 (1995. 3866–3878]. The method can be adapted for a variety of situations involving both angular and linear data and can be used with profit in the context of statistical process control in Phase I of control charting and also in Phase II in conjunction with control charts.     

A comparison of sampling grids,cut-off distance and type of residuals in parametric variogram estimation

In spatial statistics, the correct identification of a variogram model when fitted to an empirical variogram depends on many factors. Here, simulation experiments show fitting based on the variogram cloud is preferable to that based on Matheron's and Cressie–Hawkins empirical variogram estimators. For correct model specification, a number of models should be fitted to the empirical variogram using a grid of cut-off values, and recommendations are given for best choice. A design where roughly half the maximum distance between points equals the practical range works well for correct variogram identification of any model, with varying nugget sizes and sample sizes.     

A self-normalization test for a change-point in the shape parameter of a gamma distributed sequence

In this paper, a parametric change-point problem encountered in analyzing a gamma distributed sequence is considered. We propose a self-normalization based CUSUM type test statistic to detect the presence of a change-point, and obtain its limiting null distribution. The CUSUM-based procedure for detecting the location of this change-point is also given. At the same time, simulation results are provided, which show that our procedures are effective.