Improving the reliability of bootstrap tests with the fast double bootstrap

Two procedures are proposed for estimating the rejection probabilities (RPs) of bootstrap tests in Monte Carlo experiments without actually computing a bootstrap test for each replication. These procedures are only about twice as expensive (per replication) as estimating RPs for asymptotic tests. Then a new procedure is proposed for computing bootstrap P values that will often be more accurate than ordinary ones. This “fast double bootstrap” (FDB) is closely related to the double bootstrap, but it is far less computationally demanding. Simulation results for three different cases suggest that the FDB can be very useful in practice.     

Bootstrapping the likelihood ratio cointegration test in error correction models with unknown lag order

The finite-sample size and power properties of bootstrapped likelihood ratio system cointegration tests are investigated via Monte Carlo simulations when the true lag order of the data generating process is unknown. Recursive bootstrap schemes are employed which differ in the way in which the lag order is chosen. The order is estimated by minimizing different information criteria and by combining the corresponding order estimates. It is found that, in comparison to the standard asymptotic likelihood ratio test based on an estimated lag order, bootstrapping can lead to improvements in small samples even when the true lag order is unknown, while the power loss is moderate.     

The Size and Power of Bootstrap Tests for Spatial Dependence in a Linear Regression Model

In this paper, we define the spatial bootstrap test as a residual-based bootstrap method for hypothesis testing of spatial dependence in a linear regression model. Based on Moran’s I statistic, the empirical size and power of bootstrap and asymptotic tests for spatial dependence are evaluated and compared. Under classical normality assumption of the model, the performance of the spatial bootstrap test is equivalent to that of the asymptotic test in terms of size and power. For more realistic heterogeneous non-normal distributional models, the applicability of asymptotic normal tests is questionable. Instead, spatial bootstrap tests have shown superiority in smaller size distortion and higher power when compared to asymptotic counterparts, especially for cases with a small sample and dense spatial contiguity. Our Monte Carlo experiments indicate that the spatial bootstrap test is an effective alternative to the theoretical asymptotic approach when the classical distributional assumption is violated.     

Empirical likelihood based diagnostics for heteroscedasticity in partial linear models

In this paper, we propose a diagnostic technique for checking heteroscedasticity based on empirical likelihood for the partial linear models. We construct an empirical likelihood ratio test for heteroscedasticity. Also, under mild conditions, a nonparametric version of Wilk’s theorem is derived, which says that our proposed test has an asymptotic chi-square distribution. Simulation results reveal that the finite sample performance of our proposed test is satisfactory in both size and power. An empirical likelihood bootstrap simulation is also conducted to overcome the size distortion in small sample sizes.     

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.     

Finite-Sample Adjustments for Homogeneity and Symmetry Tests in Systems of Demand Equations: A Monte Carlo Evaluation

This paper gives simulation results comparing the finite-sample performance of three commonly used homogeneity and symmetry asymptotic tests, and some size-corrected tests that can be used when the sample size is small. The results suggest that such finite-sample corrections can be effective in bringing the empirical sizes of the tests closer to their nominal levels. They also suggest that the likelihood ratio test is, in general, more reliable than the Wald and Lagrange multiplier tests. Finally, it is shown that size-corrections of homogeneity tests tend to introduce reductions in power, which can be very large when bootstrap corrections are used. An application to a well known data set is also presented.     

The Exact Maximum Likelihood-Based Test for Fractional Cointegration: Critical Values,Power and Size

The exact maximum likelihood (EML) procedure can be used as a residual-based test of the hypothesis of no cointegration against the alternative of fractional cointegration. Since the corresponding asymptotic properties have not yet been established, this paper provides simulated critical values, power and size relating to the EML-based test for fractional cointegration. Monte Carlo simulations indicate that the simulated density of the EML-based test is shifted to the left compared to the standard normal distribution and exhibits a strong excess of kurtosis in the absence of autoregressive components in the regression residuals. The power and size comparison indicates that the EML-based test is more powerful than other fractional cointegration tests (Lo, Lobato-Robinson and Geweke and Porter-Hudak) in small and medium sample sizes. Moreover, by simulating integrated time series with AR(1), and respectively MA(1), disturbances, it is shown that, whatever the sample size, the EML-based test exhibits the lowest size distortions for positive AR(1) and negative MA(1) coefficients, respectively.JEL classifications: C15; C22     

Constant-partially accelerated life tests for Burr type-XII distribution with progressive type-II censoring

Based on progressively type-II censored samples, constant-partially accelerated life tests (PALTs) when the lifetime of items under use condition follow the two-parameter Burr type-XII (Burr(c,k)) distribution are considered. The likelihood equations of the involved parameters are derived and then reduced to a single nonlinear equation to be solved numerically to obtain the maximum likelihood estimates (MLEs) of the parameters. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLEs are derived. Approximate confidence intervals (CIs) for the parameters, based on normal approximation to the asymptotic distribution of MLEs, studentized-t and percentile bootstrap CIs are derived. A Monte Carlo simulation study is carried out to investigate the precision of the MLEs and to compare the performance of the CIs considered. Finally, two examples presented to illustrate our results are followed by conclusions.     

Homogeneity tests for several Poisson populations

In this paper we compare the size distortions and powers for Pearson’s χ²-statistic, likelihood ratio statistic LR, score statistic SC and two statistics, which we call UT and VT here, proposed by [Potthoff, R.F., Whittinghill, M., 1966. Testing for homogeneity: II. The Poisson distribution. Biometrika 53, 183–190] for testing the equality of the rates of K Poisson processes. Asymptotic tests and parametric bootstrap tests are considered. It is found that the asymptotic UT test is too conservative to be recommended, while the other four asymptotic tests perform similarly and their powers are close to those of their parametric bootstrap counterparts when the observed counts are large enough. When the observed counts are not large, Monte Carlo simulation suggested that the asymptotic test using SC, LR and UT statistics are discouraged; none of the parametric bootstrap tests with the five statistics considered here is uniformly best or worst, and the asymptotic tests using Pearson’s χ² and VT always have similar powers to their bootstrap counterparts. Thus, the asymptotic Pearson’s χ² and VT tests have an advantage over all five parametric bootstrap tests in terms of their computational simplicity and convenience, and over the other four asymptotic tests in terms of their powers and size distortions.     

A frequency domain test for detecting nonstationary time series

We propose a frequency domain generalized likelihood ratio test for testing nonstationarity in time series. The test is constructed in the frequency domain by comparing the goodness of fit in the log-periodogram regression under the varying coefficient fractionally exponential models. Under such a locally stationary specification, the proposed test is capable of detecting dynamic changes of short-range and long-range dependences in a regression framework. The asymptotic distribution of the proposed test statistic is known under the null stationarity hypothesis, and its finite sample distribution can be approximated by bootstrap. Numerical results show that the proposed test has good power against a wide range of locally stationary alternatives.     

ANOVA extensions for mixed discrete and continuous data

This paper is concerned with ANOVA-like tests in the context of mixed discrete and continuous data. The likelihood ratio approach is used to obtain a location test in the mixed data setting after specifying a general location model for the joint distribution of the mixed discrete and continuous variables. The approach allows the problem to be treated from a multivariate perspective to simultaneously test both the discrete and continuous parameters of the model, thus avoiding the problem of multiple significance testing. Moreover, associations among variables are accounted for, resulting in improved power performance of the test. Unlike existing distance-based alternatives which rely on asymptotic theory, the likelihood ratio test is exact. In addition, it can be viewed as an extension to the mixed data setting of the classical multivariate ANOVA. We compare its performance against those of currently available tests via Monte Carlo simulations. Two real-data examples are presented to illustrate the methodology.     

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.     

Bootstrapping divergence statistics for testing homogeneity in multinomial populations

We consider the problem of testing the equality of ν$\nu$ (ν≥2$\nu \ge 2$) multinomial populations, taking as test statistic a sample version of an f-dissimilarity between the populations, obtained by the replacement of the unknown parameters in the expression of the f-dissimilarity among the theoretical populations, by their maximum likelihood estimators. The null distribution of this test statistic is usually approximated by its limit, the asymptotic null distribution. Here we study another way to approximate it, the bootstrap. We show that the bootstrap yields a consistent distribution estimator. We also study by simulation the finite sample performance of the bootstrap distribution and compare it with the asymptotic approximation. From the simulations it can be concluded that it is worth calculating the bootstrap estimator, because it is more accurate than the approximation yielded by the asymptotic null distribution which, in addition, cannot always be exactly computed.     

Bootstrap hypothesis testing for some common statistical problems: A critical evaluation of size and power properties

The construction of bootstrap hypothesis tests can differ from that of bootstrap confidence intervals because of the need to generate the bootstrap distribution of test statistics under a specific null hypothesis. Similarly, bootstrap power calculations rely on resampling being carried out under specific alternatives. We describe and develop null and alternative resampling schemes for common scenarios, constructing bootstrap tests for the correlation coefficient, variance, and regression/ANOVA models. Bootstrap power calculations for these scenarios are described. In some cases, null-resampling bootstrap tests are equivalent to tests based on appropriately constructed bootstrap confidence intervals. In other cases, particularly those for which simple percentile-method bootstrap intervals are in routine use such as the correlation coefficient, null-resampling tests differ from interval-based tests. We critically assess the performance of bootstrap tests, examining size and power properties of the tests numerically using both real and simulated data. Where they differ from tests based on bootstrap confidence intervals, null-resampling tests have reasonable size properties, outperforming tests based on bootstrapping without regard to the null hypothesis. The bootstrap tests also have reasonable power properties.     

Bootstrap hypothesis testing for some common statistical problems: A critical evaluation of size and power properties

The construction of bootstrap hypothesis tests can differ from that of bootstrap confidence intervals because of the need to generate the bootstrap distribution of test statistics under a specific null hypothesis. Similarly, bootstrap power calculations rely on resampling being carried out under specific alternatives. We describe and develop null and alternative resampling schemes for common scenarios, constructing bootstrap tests for the correlation coefficient, variance, and regression/ANOVA models. Bootstrap power calculations for these scenarios are described. In some cases, null-resampling bootstrap tests are equivalent to tests based on appropriately constructed bootstrap confidence intervals. In other cases, particularly those for which simple percentile-method bootstrap intervals are in routine use such as the correlation coefficient, null-resampling tests differ from interval-based tests. We critically assess the performance of bootstrap tests, examining size and power properties of the tests numerically using both real and simulated data. Where they differ from tests based on bootstrap confidence intervals, null-resampling tests have reasonable size properties, outperforming tests based on bootstrapping without regard to the null hypothesis. The bootstrap tests also have reasonable power properties.     

Size and power of tests for a zero random effect variance or polynomial regression in additive and linear mixed models

Several tests for a zero random effect variance in linear mixed models are compared. This testing problem is non-regular because the tested parameter is on the boundary of the parameter space. Size and power of the different tests are investigated in an extensive simulation study that covers a variety of important settings. These include testing for polynomial regression versus a general smooth alternative using penalized splines. Among the test procedures considered, three are based on the restricted likelihood ratio test statistic (RLRT), while six are different extensions of the linear model F-test to the linear mixed model. Four of the tests with unknown null distributions are based on a parametric bootstrap, the other tests rely on approximate or asymptotic distributions. The parametric bootstrap-based tests all have a similar performance. Tests based on approximate F-distributions are usually the least powerful among the tests under consideration. The chi-square mixture approximation for the RLRT is confirmed to be conservative, with corresponding loss in power. A recently developed approximation to the distribution of the RLRT is identified as a rapid, powerful and reliable alternative to computationally intensive parametric bootstrap procedures. This novel method extends the exact distribution available for models with one random effect to models with several random effects.     

平滑转移向量误差修正模型的非线性调节bootstrap检验

对原假设为线性不含平滑转移均衡趋势关系的两类平滑转移向量误差修正模型提出了非线性调节检验;提出了检验这类非线性调节关系的SupWald检验、相应的渐近分布和残差bootstrap方法模拟p值. 模拟实验中证实了有限样本下SupWald检验统计量的良好效用, 并给出了其适用范围, 进而应用该方法检验出几组美国国库券收益率间存在明显的平滑转移非线性调节.     

Small sample improvements in the threshold cointegration test using residual-based moving block bootstrap

A residual-based moving block bootstrap procedure for testing the null hypothesis of linear cointegration versus cointegration with threshold effects is proposed. When the regressors and errors of the models are serially and contemporaneously correlated, our test compares favourably with the Sup LM test proposed by Gonzalo and Pitarakis. Indeed, shortcomings of the former motivated the development of our test. The small sample performance of the bootstrap test is investigated by Monte Carlo simulations, and the results show that the test performs better than the Sup LM test.     

Size and power properties of some tests in the Birnbaum-Saunders regression model

The Birnbaum-Saunders distribution has been used quite effectively to model times to failure for materials subject to fatigue and for modeling lifetime data. In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the non-null distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the Birnbaum-Saunders regression model. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the shape parameter. Monte Carlo simulation is presented in order to compare the finite-sample performance of these tests. We also present two empirical applications.     

Improved likelihood inference in Birnbaum-Saunders regressions

The Birnbaum-Saunders regression model is commonly used in reliability studies. We address the issue of performing inference in this class of models when the number of observations is small. Our simulation results suggest that the likelihood ratio test tends to be liberal when the sample size is small. We obtain a correction factor which reduces the size distortion of the test. Also, we consider a parametric bootstrap scheme to obtain improved critical values and improved p-values for the likelihood ratio test. The numerical results show that the modified tests are more reliable in finite samples than the usual likelihood ratio test. We also present an empirical application.