Asymptotic refinements of bootstrap tests in a linear regression model ; A CHM bootstrap using the first four moments of the residuals

We consider linear regression models and we suppose that disturbances are either Gaussian or non Gaussian. Then, by using Edgeworth expansions, we compute the exact errors in the rejection probability (ERPs) for all one-restriction tests (asymptotic and bootstrap) which can occur in these linear models. More precisely, we show that the ERP is the same for the asymptotic test as for the classical parametric bootstrap test it is based on as soon as the third cumulant is nonnul. On the other side, the non parametric bootstrap performs almost always better than the parametric bootstrap. There are two exceptions. The first occurs when the third and fourth cumulants are null, in this case parametric and non parametric bootstrap provide exactly the same ERPs, the second occurs when we perform a t-test or its associated bootstrap (parametric or not) in the models y =μ+u and y=ax+u where the disturbances have nonnull kurtosis coefficient and a skewness coefficient equal to zero. In that case, the ERPs of any test (asymptotic or bootstrap) we perform are of the same order.Finally, we provide a new parametric bootstrap using the first four moments of the distribution of the residuals which is as accurate as a non parametric bootstrap which uses these first four moments implicitly. We will introduce it as the parametric bootstrap considering higher moments (CHM), and thus, we will speak about the CHM parametric bootstrapNon parametric bootstrap, Parametric Bootstrap, Cumulants, Skewness, kurtosis.

Higher-order Improvements of the Parametric Bootstrap for Markov Processes

This paper provides bounds on the errors in coverage probabilities of maximum likelihood-based, percentile-t, parametric bootstrap confidence intervals for Markov time series processes. These bounds show that the parametric bootstrap for Markov time series provides higher-order improvements (over confidence intervals based on first order asymptotics) that are comparable to those obtained by the parametric and nonparametric bootstrap for iid data and are better than those obtained by the block bootstrap for time series. Additional results are given for Wald-based confidence regions. The paper also shows that k-step parametric bootstrap confidence intervals achieve the same higher-order improvements as the standard parametric bootstrap for Markov processes. The k-step bootstrap confidence intervals are computationally attractive. They circumvent the need to compute a nonlinear optimization for each simulated bootstrap sample. The latter is necessary to implement the standard parametric bootstrap when the maximum likelihood estimator solves a nonlinear optimization problem.Asymptotics, Edgeworth expansion, Gauss-Newton, k-step bootstrap, maximum likelihood estimator, Newton-Raphson, parametric bootstrap, t statistic

A comparison of block and semi-parametric bootstrap methods for variance estimation in spatial statistics

Efron (1979) introduced the bootstrap method for independent data but it cannot be easily applied to spatial data because of their dependency. For spatial data that are correlated in terms of their locations in the underlying space the moving block bootstrap method is usually used to estimate the precision measures of the estimators. The precision of the moving block bootstrap estimators is related to the block size which is difficult to select. In the moving block bootstrap method also the variance estimator is underestimated. In this paper, first the semi-parametric bootstrap is used to estimate the precision measures of estimators in spatial data analysis. In the semi-parametric bootstrap method, we use the estimation of the spatial correlation structure. Then, we compare the semi-parametric bootstrap with a moving block bootstrap for variance estimation of estimators in a simulation study. Finally, we use the semi-parametric bootstrap to analyze the coal-ash data

AIDS VERSUS THE ROTTERDAM DEMAND SYSTEM: A COX TEST WITH PARAMETRIC BOOTSTRAP

A Cox test with parametric bootstrap is developed to select between the linearized version of the First-Difference Almost Ideal Demand System (FDAIDS) and the Rotterdam model. A Cox test with parametric bootstrap has been shown to be more powerful than encompassing tests like those used in past research. The bootstrap approach is used with U.S. meat demand (beef, pork, chicken, fish) and compared to results obtained with an encompassing test. The Cox test with parametric bootstrap consistently indicates the Rotterdam model is preferred to the FDAIDS, while the encompassing test sometimes fails to reject FDAIDS.Research Methods/ Statistical Methods,

AIDS VERSUS ROTTERDAM: A COX NONNESTED TEST WITH PARAMETRIC BOOTSTRAP

A Cox nonnested test with parametric bootstrap is developed to select between the linearized version of the First Difference Almost Ideal Demand System (FDAIDS) and the Rotterdam model. The Cox test with parametric bootstrap is expected to be more powerful than the various orthodox tests used in past research. The new approach is then used for U. S. meat demand (beef, pork, and chicken) and compared to results obtained with an orthodox test. The orthodox test gives inconsistent results. In contrast, under the same varied conditions, the Cox test with parametric bootstrap consistently indicates that the Rotterdam model is preferred to the FDAIDS.Demand and Price Analysis,

Bootstrap tests for the error distribution in linear and nonparametric regression models

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and nonparametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution-free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness-of-fit testing of assumptions regarding the error distribution in linear and nonparametric regression models. --goodness-of-fit,residual process,parametric bootstrap,linear model,analysis of variance,M-estimation,nonparametric regression

Beyond first-order asymptotics for Cox regression

To go beyond standard first-order asymptotics for Cox regression, we develop parametric bootstrap and second-order methods. In general, computation of $P$-values beyond first order requires more model specification than is required for the likelihood function. It is problematic to specify a censoring mechanism to be taken very seriously in detail, and it appears that conditioning on censoring is not a viable alternative to that. We circumvent this matter by employing a reference censoring model, matching the extent and timing of observed censoring. Our primary proposal is a parametric bootstrap method utilizing this reference censoring model to simulate inferential repetitions of the experiment. It is shown that the most important part of improvement on first-order methods - that pertaining to fitting nuisance parameters - is insensitive to the assumed censoring model. This is supported by numerical comparisons of our proposal to parametric bootstrap methods based on usual random censoring models, which are far more unattractive to implement. As an alternative to our primary proposal, we provide a second-order method requiring less computing effort while providing more insight into the nature of improvement on first-order methods. However, the parametric bootstrap method is more transparent, and hence is our primary proposal. Indications are that first-order partial likelihood methods are usually adequate in practice, so we are not advocating routine use of the proposed methods. It is however useful to see how best to check on first-order approximations, or improve on them, when this is expressly desired.Comment: Published at http://dx.doi.org/10.3150/13-BEJ572 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A parametric bootstrap for heavytailed distributions

It is known that Efron's resampling bootstrap of the mean of random variables with common distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting distribution of the bootstrap mean is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be quite unreliable in other than very large samples. A parametric bootstrap is derived by considering the distribution of the bootstrap P value instead of that of the bootstrap statistic. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed.bootstrap inconsistency, stable distribution, domain of attraction, infinite variance

A Parametric Bootstrap Test for Cycles

The paper proposes a simple test for the hypothesis of strong cycles and as a by-product a test for weak dependence for linear processes. We show that the limit distribution of the test is the maximum of a (semi)Gaussian process G(t), t ? [0; 1]. Because the covariance structure of G(t) is a complicated function of t and model dependent, to obtain the critical values (if possible) of maxt?[0;1] G(t) may be difficult. For this reason we propose a bootstrap scheme in the frequency domain to circumvent the problem of obtaining (asymptotically) valid critical values. The proposed bootstrap can be regarded as an alternative procedure to existing bootstrap methods in the time domain such as the residual-based bootstrap. Finally, we illustrate the performance of the bootstrap test by a small Monte Carlo experiment and an empirical example.Cyclical data, strong and weak dependence, spectral density functions, Whittle estimator, bootstrap algorithms

Testing an autoregressive structure in binary time series models

This paper introduces a Lagrange Multiplier (LM) test for testing an autoregressive structure in a binary time series model proposed by Kauppi and Saikkonen (2008). Simulation results indicate that the two versions of the proposed LM test have reasonable size and power properties when the sample size is large. A parametric bootstrap method is suggested to obtain approximately correct sizes also in small samples. The use of the test is illustrated by an application to recession forecasting models using monthly U.S. data.LM test, Binary response, Dynamic probit model, Parametric bootstrap, Recession forecasting