Central composite designs for estimating the optimum conditions for a second-order model

Central composite designs which maximize both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters are constructed. Two optimality criteria are developed, the one relating to precision and based on the sum of the first-order approximations to the asymptotic variances and the other to accuracy and based on the sum of squares of the second-order approximations to the asymptotic biases of the estimates of the coordinates of the extremal point. Exact and continuous central composite designs are introduced and in particular designs which place no restriction on the pattern of the weights, termed benchmark designs, and designs which comprise equally weighted factorial and equally weighted axial points, termed axial-factorial designs, are explored. Algebraic results proved somewhat elusive and the requisite designs are obtained by a mix of algebra and numeric calculation or simply numerically. An illustrative example is presented and some interesting features which emerge from that example are discussed.     

A simple least squares method for estimating a change in mean

A simple least squares method for estimating a change in mean of a sequence of independent random variables is studied. The method first tests for a change in mean based on the regression principle of constrained and unconstrained sums of squares. Conditionally on a decision by this test that a change has occurred, least squares estimates are used to estimate the change point, the initial mean level (prior to the change point) and the change itself. The estimates of the initial level and change are functions of the change point estimate. All estimates are shown to be consistent, and those for the initial level and change are shown to be asymptotically jointly normal. The method performs well for moderately large shifts (one standard deviation or more), but the estimates of the initial level and change are biased in a predictable way for small shifts. The large sample theory is helpful in understanding this problem. The asymptotic distribution of the change point estimator is obtained for local shifts in mean, but the case of non-local shifts appears analytically intractable.     

Identifying the time of a step change with multivariate single control charts

Change point estimation procedures simplify the efforts to search for and identify special causes in multivariate statistical process monitoring. After a signal is generated by the simultaneously used control charts or a single control chart, add-on change point procedure estimates the time of the change. In this study, multivariate joint change point estimation performance for simultaneous monitoring of both location and dispersion is compared under the assumption that various single charts are used to monitor the process. The change detection performance for several structural changes for the mean vector and covariance matrix is also discussed. It is concluded that choice of the control chart to obtain a signal may affect the change point detection performance.     

Partially linear models and their applications to change point detection of chemical process data

In many chemical data sets, the amount of radiation absorbed (absorbance) is related to the concentration of the element in the sample by Lambert–Beer's law. However, this relation changes abruptly when the variable concentration reaches an unknown threshold level, the so-called change point. In the context of analytical chemistry, there are many methods that describe the relationship between absorbance and concentration, but none of them provide inferential procedures to detect change points. In this paper, we propose partially linear models with a change point separating the parametric and nonparametric components. The Schwarz information criterion is used to locate a change point. A back-fitting algorithm is presented to obtain parameter estimates and the penalized Fisher information matrix is obtained to calculate the standard errors of the parameter estimates. To examine the proposed method, we present a simulation study. Finally, we apply the method to data sets from the chemistry area. The partially linear models with a change point developed in this paper are useful supplements to other methods of absorbance–concentration analysis in chemical studies, for example, and in many other practical applications.     

A semiparametric maximum likelihood ratio test for the change point in copula models

In the present paper, a semiparametric maximum-likelihood-type test statistic is proposed and proved to have the same limit null distribution as the classical parametric likelihood one. Under some mild conditions, the limiting law of the proposed test statistic, suitably normalized and centralized, is shown to be double exponential, under the null hypothesis of no change in the parameter of copula models. We also discuss the Gaussian-type approximations for the semiparametric likelihood ratio. The asymptotic distribution of the proposed statistic under specified alternatives is shown to be normal, and an approximation to the power function is given. Simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the double exponential and Gaussian-type approximations.     

Finite sample distributions of statistics in change point analysis

This paper is concerned with derivation of finite sampling distributions of some statistics which appear frequently in change point analysis. The exact distribution of cusum test statistic is approximated by two methods. Approximations are presented and their accuracies are measured. We first consider the change point in mean problem and we study the exact distribution of change point estimator. Finally, we consider the change point in variance case.     

Detecting structural change in linear models

Consider a sequence of independent observations which change their marginal distribution at most once somewhere in the sequence and one is not certain where the change has occurred. One would be interested in detecting the change and determining the two distributions which would describe the sequence. On the other hand if no change had occurred, one would want to know the common distribution of the observations. This study develops a Bayesian test for detecting a switch from one linear model to another. The test is based on the marginal posterior mass function of the switch point and the posterior probability of a stable model. This test and an informal sequential procedure of Smith are illustrated with data generated from an unstable linear regression model, which changes the linear relationship between the dependent and independent variables     

Bootstrap methods for single structural change tests: power versus corrected size and empirical illustration

This paper assesses the performance of tests for a single structural change at unknown date when regressors are stationary, trending and when they have a break in mean. Size and power of the test procedures are compared in a simulation setup particularly aimed at autoregressive models using their limiting distribution and some bootstrap approximations. The comparisons are performed using graphical methods, namely P value discrepancy plots and size–power curves. The simulation study gives some interesting insights to the test procedures. Indeed, it documents that tests based on the conventional asymptotic distribution are oversized in small samples. The size correction is achieved by some bootstrap methods which appear to possess reasonable size properties. For the power study, the proposed bootstrap method improves on the asymptotic approximations of some tests for heteroskedastic regression errors especially when there is a mean-shift in the regressors. This result has not been found for the case of i.i.d. errors where the bootstrap tests have the same power properties as the tests based on the asymptotic approximations. We finally study the relationship between two monthly US interest rates. The results show that such relationship has been altered by a regime-shift located in May 1981.     

Two-sided tests for change in level for correlated data

The classical problem of change point is considered when the data are assumed to be correlated. The nuisance parameters in the model are the initial level μ and the common variance σ². The four cases, based on none, one, and both of the parameters are known are considered. Likelihood ratio tests are obtained for testing hypotheses regarding the change in level, δ, in each case. Following Henderson (1986), a Bayesian test is obtained for the two sided alternative. Under the Bayesian set up, a locally most powerful unbiased test is derived for the case μ=0 and σ²=1. The exact null distribution function of the Bayesian test statistic is given an integral representation. Methods to obtain exact and approximate critical values are indicated.     

The relationship between crime, punishment and economic conditions: is reliable inference possible when crimes are under-recorded?

We investigate the estimation of dynamic models of criminal activity, when there is significant under-recording of crime. We give a theoretical analysis and use simulation techniques to investigate the resulting biases in conventional regression estimates. We find the biases to be of little practical significance. We develop and apply a new simulated maximum likelihood procedure that estimates simultaneously the measurement error and crime processes, using extraneous survey data. This also confirms that measurement error biases are small. Our estimation results for data from England and Wales imply a significant response of crime to both the economic and the enforcement environment.     

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.     

Local Fourier tests for structural change based on residuals

We propose a structural change test based on the recursive residuals with the local Fourier series estimators. The statistical properties of the proposed test are derived and the empirical properties are shown via simulation. We also consider other structural change tests based on CUSUM, MOSUM, moving estimates (ME), and empirical distribution functions with the recursive residuals and the ordinary residuals. Empirical powers are calculated in various structural change models for the comparison of those tests. These structural change tests are applied to South Korea's gross domestic product (GDP), South Korean Won to US Dollar currency exchange rates, and South Korea's Okun's law.     

Empirical likelihood ratio test for a mean change point model with a linear trend followed by an abrupt change

In this paper, a change point model with the mean being constant up to some unknown point, and increasing linearly to another unknown point, then dropping back to the original level is studied. A nonparametric method based on the empirical likelihood test is proposed to detect and estimate the locations of change points. Under some mild conditions, the asymptotic null distribution of an empirical likelihood ratio test statistic is shown to have the extreme distribution. The consistency of the test is also proved. Simulations of the powers of the test indicate that it performs well under different assumptions of the data distribution. The test is applied to the aircraft arrival time data set and the Stanford heart transplant data set.     

The Effects of Price Aggregation Bias in Systems of Demand Equations

This article illustrates the importance of maintaining consistent levels of aggregation between prices and quantities when estimating consumer demand functions. The impact of violating this condition is quantified by using summary performance measures and estimates of demand elasticity biases. Results derived from an application of 1972–1973 Consumer Expenditure Survey data and supported with a quasi-Monte Carlo experiment consistently indicate that the point estimates of demand elasticities are significantly biased. Thus the study indicates the importance of developing and maintaining price indexes disaggregated to the same level as the expenditure data in consumer expenditure and budget surveys.     

An ANOVA-type test for multiple change points

We consider the problem of testing the null hypothesis of no change against the alternative of multiple change points in a series of independent observations. We propose an ANOVA-type test statistic and obtain its asymptotic null distribution. We also give approximations of its limiting critical values. We report the results of Monte Carlo studies conducted to compare the power of the proposed test against a number of its competitors. As illustrations we analyzed three real data sets.     

Bayesian approach to change point problems

A Bayesian approach is considered to study the change point problems. A hypothesis for testing change versus no change is considered using the notion of predictive distributions. Bayes factors are developed for change versus no change in the exponential families of distributions with conjugate priors. Under vague prior information, both Bayes factors and pseudo Bayes factors are considered. A new result is developed which describes how the overall Bayes factor has a decomposition into Bayes factors at each point. Finally, an example is provided in which the computations are performed using the concept of imaginary observations.     

A comparison of estimators for regression models with change points

We consider two problems concerning locating change points in a linear regression model. One involves jump discontinuities (change-point) in a regression model and the other involves regression lines connected at unknown points. We compare four methods for estimating single or multiple change points in a regression model, when both the error variance and regression coefficients change simultaneously at the unknown point(s): Bayesian, Julious, grid search, and the segmented methods. The proposed methods are evaluated via a simulation study and compared via some standard measures of estimation bias and precision. Finally, the methods are illustrated and compared using three real data sets. The simulation and empirical results overall favor both the segmented and Bayesian methods of estimation, which simultaneously estimate the change point and the other model parameters, though only the Bayesian method is able to handle both continuous and dis-continuous change point problems successfully. If it is known that regression lines are continuous then the segmented method ranked first among methods.     

Nonparametric two-sample tests for dispersion differences based on placements

Two different two-sample tests for dispersion differences based on placement statistics are proposed. The means and variances of the test statistics are derived, and asymptotic normality is established for both. Variants of the proposed tests based on reversing the X and Y labels in the test statistic calculations are shown to have different small-sample properties; for both pairs of tests, one member of the pair will be resolving, the other nonresolving. The proposed tests are similar in spirit to the dispersion tests of both Mood and Hollander; comparative simulation results for these four tests are given. For small sample sizes, the powers of the proposed tests are approximately equal to the powers of the tests of both Mood and Hollander for samples from the normal, Cauchy and exponential distributions. The one-sample limiting distributions are also provided, yielding useful approximations to the exact tests when one sample is much larger than the other. A bootstrap test may alternatively be performed. The proposed test statistics may be used with lightly censored data by substituting Kaplan-Meier estimates for the empirical distribution functions.     

An Approach for Estimating Causal Effects in Randomized Trials with Noncompliance

We developed methods for estimating the causal risk difference and causal risk ratio in randomized trials with noncompliance. The developed estimator is unbiased under the assumption that biases due to noncompliance are identical between both treatment arms. The biases are defined as the difference or ratio between the expectations of potential outcomes for a group that received the test treatment and that for the control group in each randomly assigned group. Although the instrumental variable estimator yields an unbiased estimate under a sharp null hypothesis but may yield a biased estimate under a non-null hypothesis, the bias of the developed estimator does not depend on whether this hypothesis holds. Then the estimate of the causal effect from the developed estimator may have a smaller bias than that from the instrumental variable estimator when the treatment effect exists. There is not yet a standard method for coping with noncompliance, and thus it is important to evaluate estimates under different assumptions. The developed estimator can serve this purpose. Its application to a field trial for coronary heart disease is provided.     

The sensitivity of estimates of the change in population behaviour to realistic changes in bias in repeated surveys

Summary.  The first British National Survey of Sexual Attitudes and Lifestyles (NATSAL) was conducted in 1990–1991 and the second in 1999–2001. When surveys are repeated, the changes in population parameters are of interest and are generally estimated from a comparison of the data between surveys. However, since all surveys may be subject to bias, such comparisons may partly reflect a change in bias. Typically limited external data are available to estimate the change in bias directly. However, one approach, which is often possible, is to define in each survey a sample of participants who are eligible for both surveys, and then to compare the reporting of selected events that occurred before the earlier survey time point. A difference in reporting suggests a change in overall survey bias between time points, although other explanations are possible. In NATSAL, changes in bias are likely to be similar for groups of sexual experiences. The grouping of experiences allows the information that is derived from the selected events to be incorporated into inference concerning population changes in other sexual experiences. We use generalized estimating equations, which incorporate weighting for differential probabilities of sampling and non-response in a relatively straightforward manner. The results, combined with estimates of the change in reporting, are used to derive minimum established population changes, based on NATSAL data. For some key population parameters, the change in reporting is seen to be consistent with a change in bias alone. Recommendations are made for the design of future surveys.