Alternative approaches to implementing Lagrange multiplier tests for serial correlation in dynamic regression models

An approximate F-form of the Lagrange multiplier (LM) test for serial correlation in dynamic regression models is compared with three bootstrap tests. In one bootstrap procedure, residuals from restricted estimation under the null hypothesis are resampled. The other two bootstrap tests use residuals from unrestricted estimation under an alternative hypothesis. A fixed autocorrelation alternative is assumed in one of the two unrestricted bootstrap tests and the other is based upon a Pitman-type sequence of local alternatives. Monte Carlo experiments are used to estimate rejection probabilities under the null hypothesis and in the presence of serial correlation.     

Bootstrap tests for fractional integration and cointegration: A comparison study

There are few methods in the literature to test for integration and cointegration in the traditional framework, i.e. using the I(0)–I(1) paradigm. In the first case, the most known are the Dickey–Fuller (DF), the Augmented Dickey–Fuller (ADF) and the Phillips–Perron (PP) tests, while in the latter case, the Engle and Granger (EG) and Johansen procedures are broadly used. But how well do these methods perform when the underlying process presents the long-memory characteristic? The bootstrap technique is used here to approximate the distribution of integration and cointegration test statistics based on a semiparametric estimator of the fractional parameter of ARFIMA(p,d,q) models. The proposed bootstrap tests, along with the asymptotic test based on the fractional semiparametric estimator, are empirically compared to the standard tests, for testing integration and cointegration in the long-memory context. Monte Carlo simulations are performed to evaluate the size and power of the tests. The results show that the conventional tests, except for the procedures based on the DF approach, loose power when compared to fractional tests. As an illustration, the tests were applied to the series of Ibovespa (Brazil) and Dow Jones (USA) indexes and led to the conclusion that these series do not share a long-run relationship.     

A variance shift model for detection of outliers in the linear mixed model

A variance shift outlier model (VSOM), previously used for detecting outliers in the linear model, is extended to the variance components model. This VSOM accommodates outliers as observations with inflated variance, with the status of the ith observation as an outlier indicated by the size of the associated shift in the variance. Likelihood ratio and score test statistics are assessed as objective measures for determining whether the ith observation has inflated variance and is therefore an outlier. It is shown that standard asymptotic distributions do not apply to these tests for a VSOM, and a modified distribution is proposed. A parametric bootstrap procedure is proposed to account for multiple testing. The VSOM framework is extended to account for outliers in random effects and is shown to have an advantage over case-deletion approaches. A simulation study is presented to verify the performance of the proposed tests. Challenges associated with computation and extensions of the VSOM to the general linear mixed model with correlated errors are discussed.     

A new approach to bootstrap inference in functional coefficient models

To infer on functional dependence of regression parameters, a new, factor based bootstrap approach is introduced, that is robust under various forms of heteroskedastic error terms. Modeling the functional coefficient parametrically, the bootstrap approximation of an F-statistic is shown to hold asymptotically. In simulation studies with both parametric and nonparametric functional coefficients, factor based bootstrap inference outperforms the wild bootstrap and pairs bootstrap approach, according to its rejection frequencies under the null hypothesis. Applying the functional coefficient model to a cross sectional investment regression on savings, the saving retention coefficient is found to depend on third variables as the population growth rate and the openness ratio.     

A study of partial F tests for multiple linear regression models

Partial F tests play a central role in model selections in multiple linear regression models. This paper studies the partial F tests from the view point of simultaneous confidence bands. It first shows that there is a simultaneous confidence band associated naturally with a partial F test. This confidence band provides more information than the partial F test and the partial F test can be regarded as a side product of the confidence band. This view point of confidence bands also leads to insights of the major weakness of the partial F tests, that is, a partial F test requires implicitly that the linear regression model holds over the entire range of the covariates in concern. Improved tests are proposed and they are induced by simultaneous confidence bands over restricted regions of the covariates. Power comparisons between the partial F tests and the new tests have been carried out to assess when the new tests are more or less powerful than the partial F tests. Computer programmes have been developed for easy implements of these new confidence band based inferential methods. An illustrative example is provided.     

A study of partial F tests for multiple linear regression models

Partial F tests play a central role in model selections in multiple linear regression models. This paper studies the partial F tests from the view point of simultaneous confidence bands. It first shows that there is a simultaneous confidence band associated naturally with a partial F test. This confidence band provides more information than the partial F test and the partial F test can be regarded as a side product of the confidence band. This view point of confidence bands also leads to insights of the major weakness of the partial F tests, that is, a partial F test requires implicitly that the linear regression model holds over the entire range of the covariates in concern. Improved tests are proposed and they are induced by simultaneous confidence bands over restricted regions of the covariates. Power comparisons between the partial F tests and the new tests have been carried out to assess when the new tests are more or less powerful than the partial F tests. Computer programmes have been developed for easy implements of these new confidence band based inferential methods. An illustrative example is provided.     

Practical methods for computing power in testing the multivariate general linear hypothesis

Four test statistics are commonly used in multivariate general linear hypothesis tests such as MANOVA: Roy's largest root, Wilks' Lambda, Hotelling-Lawley trace and Pillai-Bartlett trace. Closed form, finite series expressions do not exist for the distribution functions in either the null or non-null case (except for special cases). In practice, asymptotic approximations, based on F or chi square distributions, are used for the null case. It is not widely known that similarly accurate and general approximations have been published for the non-null case, for all except the largest root [15,38].In this paper new approximations, based on noncentral F's, are provided for power for all but the largest root. These generalize existing F approximations for the central case. Much less calculation is needed than for earlier approximations and accuracy appears not to suffer in practice. An upper bound F approximation is provided for the largest root power.Power calculation can be used to help choose a test statistic as well as for experimental design. A particularly convenient method for estimating power, using standard linear models programs [20], is generalized to the multivariate case.     

Modeling dichotomous item responses with free-knot splines

Item response theory (IRT) models are a class of generalized mixed effect (GME) models used by psychometricians to describe the response behavior of individuals to a set of categorically scored items. The typical assumptions of IRT are Unidimensionality(U) of the random effect; Conditional (or Local) Independence (CI), the item responses are independent given the random effect; and Monotonicity (M), the probability of a correct response is a non-decreasing function of the random effect. The simple parametric models available in the psychometric literature have proved to be too restrictive in many data sets. Non-parametric regression models are a powerful tool for the estimation of non-linear curves, and have been used in IRT as a flexible way to model the item response function. This paper develops a new method for the non-parametric estimation of item response functions based on reversible-jump Markov Chain Monte Carlo, and demonstrates the practicality of the method by examining two data sets.     

A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models

This article is about testing the equality of several normal means when the variances are unknown and arbitrary, i.e., the set up of the one-way ANOVA. Even though several tests are available in the literature, none of them perform well in terms of Type I error probability under various sample size and parameter combinations. In fact, Type I errors can be highly inflated for some of the commonly used tests; a serious issue that appears to have been overlooked. We propose a parametric bootstrap (PB) approach and compare it with three existing location-scale invariant tests—the Welch test, the James test and the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test is the best among the four tests with respect to Type I error rates. The PB test performs very satisfactorily even for small samples while the Welch test and the GF test exhibit poor Type I error properties when the sample sizes are small and/or the number of means to be compared is moderate to large. The James test performs better than the Welch test and the GF test. It is also noted that the same tests can be used to test the significance of the random effect variance component in a one-way random model under unequal error variances. Such models are widely used to analyze data from inter-laboratory studies. The methods are illustrated using some examples.     

A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models

This article is about testing the equality of several normal means when the variances are unknown and arbitrary, i.e., the set up of the one-way ANOVA. Even though several tests are available in the literature, none of them perform well in terms of Type I error probability under various sample size and parameter combinations. In fact, Type I errors can be highly inflated for some of the commonly used tests; a serious issue that appears to have been overlooked. We propose a parametric bootstrap (PB) approach and compare it with three existing location-scale invariant tests—the Welch test, the James test and the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test is the best among the four tests with respect to Type I error rates. The PB test performs very satisfactorily even for small samples while the Welch test and the GF test exhibit poor Type I error properties when the sample sizes are small and/or the number of means to be compared is moderate to large. The James test performs better than the Welch test and the GF test. It is also noted that the same tests can be used to test the significance of the random effect variance component in a one-way random model under unequal error variances. Such models are widely used to analyze data from inter-laboratory studies. The methods are illustrated using some examples.     

A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution

A variety of methods of modelling overdispersed count data are compared. The methods are classified into three main categories. The first category are ad hoc methods (i.e. pseudo-likelihood, (extended) quasi-likelihood, double exponential family distributions). The second category are discretized continuous distributions and the third category are observational level random effects models (i.e. mixture models comprising explicit and non-explicit continuous mixture models and finite mixture models). The main focus of the paper is a family of mixed Poisson distributions defined so that its mean μ is an explicit parameter of the distribution. This allows easier interpretation when μ is modelled using explanatory variables and provides a more orthogonal parameterization to ease model fitting. Specific three parameter distributions considered are the Sichel and Delaporte distributions. A new four parameter distribution, the Poisson-shifted generalized inverse Gaussian distribution is introduced, which includes the Sichel and Delaporte distributions as a special and a limiting case respectively. A general formula for the derivative of the likelihood with respect to μ, applicable to the whole family of mixed Poisson distributions considered, is given. Within the framework introduced here all parameters of the distributions are modelled as parametric and/or nonparametric (smooth) functions of explanatory variables. This provides a very flexible way of modelling count data. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models.     

Estimation of the mean squared error of predictors of small area linear parameters under a logistic mixed model

An accuracy measure (mean squared error, MSE) is necessary when small area estimators of linear parameters are provided. Even in the case when such estimators arise from the assumption of relatively simple models for the variable of interest, as linear mixed models, the analytic form of the MSE is not suitable to be calculated explicitly. Some good and widely used approximations are available for those models. For generalized linear mixed models, a rough approximation can be obtained by a linearization of the model and application of Prasad-Rao approximation for linear mixed models. Resampling methods, although computationally demanding, represent a conceptually simple alternative. Under a logistic mixed linear model for the characteristic of interest, the Prasad-Rao-type formula is compared with a bootstrap estimator obtained by a wild bootstrap designed for estimating under finite populations. A simulation study is developed in order to study the performance of both methods for estimating a small area proportion.     

Estimating reliability in proportional odds ratio models

In this paper, we are mainly interested in inference on the reliability coefficient, R=P(X<Y), in proportional odds ratio models based on the new family of tilted survival functions introduced by Marshall and Olkin [Marshall, A.W., Olkin, I., 1997. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84 (3), 641-652]. We also present some results on stochastic comparison between the survival distribution functions. Asymptotic and various bootstrap confidence intervals of R are investigated. The performance of asymptotic and bootstrap confidence intervals is studied through a simulation. A numerical example based on real-life data is presented to illustrate the implementation of the proposed procedure.     

Testing for Impulsive Behavior: A Bootstrap Approach

Brown, C. L., and Zoubir, A. M., Testing for Impulsive Behavior: A Bootstrap Approach, Digital Signal Processing11 (2001) 120–132Increasingly, systems are being designed to account for impulsive behavior that may be present in signals, with one of the widely used statistical models used being the α stable distribution. Two techniques are presented that test for the level of impulsive behavior: testing the parameter α directly and a characteristic function (cf) based technique. The parametric bootstrap is used in both cases to estimate the distribution of the test statistics and in the setting of critical values. Simulation results show both tests maintain their level and achieve high rejection rates under alternatives.     

The random projection method in goodness of fit for functional data

The possibility of considering random projections to identify probability distributions belonging to parametric families is explored. The results are based on considerations involving invariance properties of the family of distributions as well as on the random way of choosing the projections. In particular, it is shown that if a one-dimensional (suitably) randomly chosen projection is Gaussian, then the distribution is Gaussian. In order to show the applicability of the methodology some goodness-of-fit tests based on these ideas are designed. These tests are computationally feasible through the bootstrap setup, even in the functional framework. Simulations providing power comparisons of these projections-based tests with other available tests of normality, as well as to test the Black-Scholes model for a stochastic process are presented.     

Using the bootstrap for finite sample confidence intervals of the log periodogram regression

Log periodogram regression is widely applied in empirical applications to estimate the memory parameter, d, of long memory time series. This estimator is consistent for d<1 and pivotal asymptotically normal for d<3/4. However, the asymptotic distribution is a poor approximation of the (unknown) finite sample distribution if the sample size is small. Finite sample improvements in the construction of confidence intervals can be achieved by different nonparametric bootstrap procedures based on the residuals of log periodogram regression. In addition to the basic residual bootstrap, the local and block bootstraps seem adequate for replicating the structure that may arise in the errors of the regression when the series shows weak dependence in addition to long memory. The performances of different bias correcting bootstrap techniques and a bias reduced log periodogram regression are also analyzed with a view to adjusting the bias caused by that structure. Finally, an application to the Nelson and Plosser US macroeconomic data is included.     

An empirical study of a test for polynomial relationships in randomly right censored regression models

In this paper, a test statistic is constructed to test polynomial relationships in randomly right censored regression models based on the local polynomial smoothing technique. Two bootstrap procedures, namely the residual-based bootstrap and the naive bootstrap procedures, are suggested to derive the p-value of the test. Some simulations are conducted to empirically assess the performance of the two bootstrap procedures. The results demonstrate that the residual-based bootstrap performs much better than the naive bootstrap and the test method with the residual-based bootstrap to derive the p-value works satisfactorily. Although the limiting distribution of the test statistic and the consistency of the bootstrap approximations remain to be investigated, simulation results indicate that the proposed test method may be of some practical use. As a real example, the proposed test is applied to the Stanford heart transplant data.     

k-Sample tests based on the likelihood ratio

General k-sample tests are developed, including the classical Kolmogorov-Smirnov, Cramér-von Mises and Anderson-Darling k-sample tests, as well as new powerful omnibus tests based on the likelihood ratio. Conventional tests are sensitive to location difference among distributions, but are dull to detect the variation in shape. The new tests are sensitive to both location and shape. In fact, if the distributions of k-sampled populations are different in location only, the new tests are as powerful as the old ones. Otherwise, they are much more powerful.     

Goodness-of-fit tests based on empirical characteristic functions

A class of goodness-of-fit tests based on the empirical characteristic function is studied. They can be applied to continuous as well as to discrete or mixed data with any arbitrary fixed dimension. The tests are consistent against any fixed alternative for suitable choices of the weight function involved in the definition of the test statistic. The bootstrap can be employed to estimate consistently the null distribution of the test statistic. The goodness of the bootstrap approximation and the power of some tests in this class for finite sample sizes are investigated by simulation.     

Exact optimal inference in regression models under heteroskedasticity and non-normality of unknown form

Simple point-optimal sign-based tests are developed for inference on linear and nonlinear regression models with non-Gaussian heteroskedastic errors. The tests are exact, distribution-free, robust to heteroskedasticity of unknown form, and may be inverted to build confidence regions for the parameters of the regression function. Since point-optimal sign tests depend on the alternative hypothesis considered, an adaptive approach based on a split-sample technique is proposed in order to choose an alternative that brings power close to the power envelope. The performance of the proposed quasi-point-optimal sign tests with respect to size and power is assessed in a Monte Carlo study. The power of quasi-point-optimal sign tests is typically close to the power envelope, when approximately 10% of the sample is used to estimate the alternative and the remaining sample to compute the test statistic. Further, the proposed procedures perform much better than common least-squares-based tests which are supposed to be robust against heteroskedasticity.