Fast and robust bootstrap

In this paper we review recent developments on a bootstrap method for robust estimators which is computationally faster and more resistant to outliers than the classical bootstrap. This fast and robust bootstrap method is, under reasonable regularity conditions, asymptotically consistent. We describe the method in general and then consider its application to perform inference based on robust estimators for the linear regression and multivariate location-scatter models. In particular, we study confidence and prediction intervals and tests of hypotheses for linear regression models, inference for location-scatter parameters and principal components, and classification error estimation for discriminant analysis.     

On the Power of Bootstrapped Specification Tests

Abstract

Decisions based on econometric model estimates may not have the expected effect if the model is misspecified. Thus, specification tests should precede any analysis. Bierens' specification test is consistent and has optimality properties against some local alternatives. A shortcoming is that the test statistic is not distribution free, even asymptotically. This makes the test unfeasible. There have been many suggestions to circumvent this problem, including the use of upper bounds for the critical values. However, these suggestions lead to tests that lose power and optimality against local alternatives. In this paper we show that bootstrap methods allow us to recover power and optimality of Bierens' original test. Bootstrap also provides reliable p-values, which have a central role in Fisher's theory of hypothesis testing. The paper also includes a discussion of the properties of the bootstrap Nonlinear Least Squares Estimator under local alternatives.     

Robust parameter estimation for the Ornstein–Uhlenbeck process

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein–Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et?al. (Stat Methods Appl 19:333–354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator.     

Approximate bounded influence estimation for longitudinal data with outliers and measurement errors

Mixed effects models or random effects models are popular for the analysis of longitudinal data. In practice, longitudinal data are often complex since there may be outliers in both the response and the covariates and there may be measurement errors. The likelihood method is a common approach for these problems but it can be computationally very intensive and sometimes may even be computationally infeasible. In this article, we consider approximate robust methods for nonlinear mixed effects models to simultaneously address outliers and measurement errors. The approximate methods are computationally very efficient. We show the consistency and asymptotic normality of the approximate estimates. The methods can also be extended to missing data problems. An example is used to illustrate the methods and a simulation is conducted to evaluate the methods.     

Maximum studentized score tests for the detection of outliers in time series regression models

Efficient score tests exist among others, for testing the presence of additive and/or innovative outliers that are the result of the shifted mean of the error process under the regression model. A sample influence function of autocorrelation-based diagnostic technique also exists for the detection of outliers that are the result of the shifted autocorrelations. The later diagnostic technique is however not useful if the outlying observation does not affect the autocorrelation structure but is generated due to an inflation in the variance of the error process under the regression model. In this paper, we develop a unified maximum studentized type test which is applicable for testing the additive and innovative outliers as well as variance shifted outliers that may or may not affect the autocorrelation structure of the outlier free time series observations. Since the computation of the p-values for the maximum studentized type test is not easy in general, we propose a Satterthwaite type approximation based on suitable doubly non-central F-distributions for finding such p-values [F.E. Satterthwaite, An approximate distribution of estimates of variance components, Biometrics 2 (1946), pp. 110–114]. The approximations are evaluated through a simulation study, for example, for the detection of additive and innovative outliers as well as variance shifted outliers that do not affect the autocorrelation structure of the outlier free time series observations. Some simulation results on model misspecification effects on outlier detection are also provided.     

Applications and asymptotic power of marginal-free tests of stochastic vectorial independence

Fully nonparametric tests for the independence between random vectors are studied in this paper. The test statistics are functionals of an empirical process defined as the difference between the joint empirical copula and the product of the empirical copulas associated to the vectors that are suspected to be independent. The validity of a weighted bootstrap procedure is established, which allows for a quick computation of p-values. A special attention is given to the asymptotic behavior of the tests under contiguous sequences of distributions. Finally, a characteristic of the copulas in the Archimedean class in terms of independence of vectors is exploited in order to propose a new goodness-of-fit procedure.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

On robust forecasting in dynamic vector time series models

In this article, robust estimation and prediction in multivariate autoregressive models with exogenous variables (VARX) are considered. The conditional least squares (CLS) estimators are known to be non-robust when outliers occur. To obtain robust estimators, the method introduced in Duchesne [2005. Robust and powerful serial correlation tests with new robust estimates in ARX models. J. Time Ser. Anal. 26, 49–81] and Bou Hamad and Duchesne [2005. On robust diagnostics at individual lags using RA-ARX estimators. In: Duchesne, P., Rémillard, B. (Eds.), Statistical Modeling and Analysis for Complex Data Problems. Springer, New York] is generalized for VARX models. The asymptotic distribution of the new estimators is studied and from this is obtained in particular the asymptotic covariance matrix of the robust estimators. Classical conditional prediction intervals normally rely on estimators such as the usual non-robust CLS estimators. In the presence of outliers, such as additive outliers, these classical predictions can be severely biased. More generally, the occurrence of outliers may invalidate the usual conditional prediction intervals. Consequently, the new robust methodology is used to develop robust conditional prediction intervals which take into account parameter estimation uncertainty. In a simulation study, we investigate the finite sample properties of the robust prediction intervals under several scenarios for the occurrence of the outliers, and the new intervals are compared to non-robust intervals based on classical CLS estimators.     

THE SEQUENTIAL BOOTSTRAP: A COMPARISON WITH REGULAR BOOTSTRAP

Based on Bradley Efron's observation that individual resamples in the regular bootstrap have support on approximately 63% of the original observations, C. R. Rao, P. K. Pathak and V. I. Koltchinskii [1] Rao, C. R., Pathak, P. K. and Koltchinskii, V. I. 1997. Bootstrap by Sequential Resampling. Journal of Statistical Planning and Inference, 64: 257–281. [Crossref], [Web of Science ®] , [Google Scholar]have proposed a sequential resampling scheme. This sequential bootstrap stabilizes the information content of each resample by fixing the number of unique observations and letting N, the number of observatons in each resample, vary. The Rao-Pathak-Koltchinskii paper establishes the asymptotic correctness (consistency) of the sequential bootstrap. The main object of our investigation is to study the empirical properties of the Rao-Pathak-Koltchinskii sequential bootstrap as compared to the regular bootstrap. In all our settings, sequential bootstrap performs as well or better than regular bootstrap. In the particular case where we estimate standard errors of sample medians, we find that sequential bootstrap outperforms regular bootstrap by reducing variability in the final bootstrap estimates.     

Testing for sub-models of the skew <Emphasis Type="Italic">t</Emphasis>-distribution

The skew t-distribution includes both the skew normal and the normal distributions as special cases. Inference for the skew t-model becomes problematic in these cases because the expected information matrix is singular and the parameter corresponding to the degrees of freedom takes a value at the boundary of its parameter space. In particular, the distributions of the likelihood ratio statistics for testing the null hypotheses of skew normality and normality are not asymptotically \(\chi ^2\). The asymptotic distributions of the likelihood ratio statistics are considered by applying the results of Self and Liang (J Am Stat Assoc 82:605–610, 1987) for boundary-parameter inference in terms of reparameterizations designed to remove the singularity of the information matrix. The Self–Liang asymptotic distributions are mixtures, and it is shown that their accuracy can be improved substantially by correcting the mixing probabilities. Furthermore, although the asymptotic distributions are non-standard, versions of Bartlett correction are developed that afford additional accuracy. Bootstrap procedures for estimating the mixing probabilities and the Bartlett adjustment factors are shown to produce excellent approximations, even for small sample sizes.     

P-Value Precision and Reproducibility

P-values are useful statistical measures of evidence against a null hypothesis. In contrast to other statistical estimates, however, their sample-to-sample variability is usually not considered or estimated, and therefore not fully appreciated. Via a systematic study of log-scale p-value standard errors, bootstrap prediction bounds, and reproducibility probabilities for future replicate p-values, we show that p-values exhibit surprisingly large variability in typical data situations. In addition to providing context to discussions about the failure of statistical results to replicate, our findings shed light on the relative value of exact p-values vis-a-vis approximate p-values, and indicate that the use of *, **, and *** to denote levels 0.05, 0.01, and 0.001 of statistical significance in subject-matter journals is about the right level of precision for reporting p-values when judged by widely accepted rules for rounding statistical estimates.     

Model-free feature screening for ultrahigh dimensional censored regression

In this paper we design a sure independent ranking and screening procedure for censored regression (cSIRS, for short) with ultrahigh dimensional covariates. The inverse probability weighted cSIRS procedure is model-free in the sense that it does not specify a parametric or semiparametric regression function between the response variable and the covariates. Thus, it is robust to model mis-specification. This model-free property is very appealing in ultrahigh dimensional data analysis, particularly when there is lack of information for the underlying regression structure. The cSIRS procedure is also robust in the presence of outliers or extreme values as it merely uses the rank of the censored response variable. We establish both the sure screening and the ranking consistency properties for the cSIRS procedure when the number of covariates p satisfies \(p=o\{\exp (an)\}\), where a is a positive constant and n is the available sample size. The advantages of cSIRS over existing competitors are demonstrated through comprehensive simulations and an application to the diffuse large-B-cell lymphoma data set.     

Robust nonparametric estimation with missing data

In this paper, under a nonparametric regression model, we introduce two families of robust procedures to estimate the regression function when missing data occur in the response. The first proposal is based on a local M$M$-functional applied to the conditional distribution function estimate adapted to the presence of missing data. The second proposal imputes the missing responses using the local M$M$-smoother based on the observed sample and then estimates the regression function with the completed sample. We show that the robust procedures considered are consistent and asymptotically normally distributed. A robust procedure to select the smoothing parameter is also discussed.     

Online Control Charts for Process Averages Based on Repeated Median Filters

Two types of estimates of process level, namely repeated median estimates (Siegel, 1982 Siegel , A. F. ( 1982 ). Robust regression using repeated medians . Biometrika 69 : 242 – 244 .[Crossref], [Web of Science ®] , [Google Scholar]) and full online estimates (Gather et al., 2006 Gather , U. , Schettlinger , K. , Fried , R. ( 2006 ). Online signal extraction by robust linear regression . Computational Statistics 21 : 33 – 51 .[Crossref], [Web of Science ®] , [Google Scholar]) based on repeated median filters, are used to develop control charts. The distributional properties of the estimates are studied using simulation and these are found to closely follow normal distribution. The repeated median being robust against outliers with asymptotically 50% breakdown value and having small standard deviation is found to be useful as a basis for monitoring process averages. The control charts using repeated median estimates have been recommended for general use.     

Symmetric regression quantile and its application to robust estimation for the nonlinear regression model

Populational conditional quantiles in terms of percentage α are useful as indices for identifying outliers. We propose a class of symmetric quantiles for estimating unknown nonlinear regression conditional quantiles. In large samples, symmetric quantiles are more efficient than regression quantiles considered by Koenker and Bassett (Econometrica 46 (1978) 33) for small or large values of α, when the underlying distribution is symmetric, in the sense that they have smaller asymptotic variances. Symmetric quantiles play a useful role in identifying outliers. In estimating nonlinear regression parameters by symmetric trimmed means constructed by symmetric quantiles, we show that their asymptotic variances can be very close to (or can even attain) the Cramer–Rao lower bound under symmetric heavy-tailed error distributions, whereas the usual robust and nonrobust estimators cannot.