Adaptive wavelet series estimation in separable nonparametric regression models

It is well-known that multivariate curve estimation suffers from the curse of dimensionality. However, reasonable estimators are possible, even in several dimensions, under appropriate restrictions on the complexity of the curve. In the present paper we explore how much appropriate wavelet estimators can exploit a typical restriction on the curve such as additivity. We first propose an adaptive and simultaneous estimation procedure for all additive components in additive regression models and discuss rate of convergence results and data-dependent truncation rules for wavelet series estimators. To speed up computation we then introduce a wavelet version of functional ANOVA algorithm for additive regression models and propose a regularization algorithm which guarantees an adaptive solution to the multivariate estimation problem. Some simulations indicate that wavelets methods complement nicely the existing methodology for nonparametric multivariate curve estimation.     

Asymptotically Distribution-Free Goodness-of-Fit Testing: A Unifying View

We outline a general paradigm for constructing asymptotically distribution-free (ADF) goodness-of-fit tests, which can be regarded as a generalization of Khmaladze (1993 Khmaladze , E. V. ( 1993 ). Goodness of fit problem and scanning innovation martingales . Ann. Statist. 21 ( 2 ): 798 – 829 .[Crossref], [Web of Science ®] , [Google Scholar]). This is achieved by a nonorthogonal projection of a class of functions onto the ortho-complement of the extended tangent space (ETS) associated with the null hypothesis. In parallel with the work of Bickel et al. (2006 Bickel , P. J. , Ritov , Y. , Stoker , T. M. ( 2006 ). Tailor-made tests for goodness of fit to semiparametric hypotheses . Ann. Statist. 34 ( 2 ): 721 – 741 . [Google Scholar]), we obtain transformed empirical processes (TEP) which are the building blocks for constructing omnibus tests such as the usual Kolmogorov–Smirnov type tests and Crámer–von Mise type tests, as well as Portmanteau tests and directional tests. The critical values can be tabulated due to the ADF property. All the tests are capable of detecting local (Pitman) alternative at the root-n scale. We shall illustrate the framework in several examples, mostly in regression model specification testing.     

Frailty Models and Rank Tests

Connections are established between the theories of weighted logrank tests and of frailty models. These connections arise because omission of a balanced covariate from a proportional hazards model generally leads to a model with non-proportional hazards, for which the simple logrank test is no longer optimal. The optimal weighting function and the asymptotic relative efficiencies of the simple logrank test and of the optimally weighted logrank test relative to the adjusted test that would be used if the covariate values were known, are expressible in terms of the Laplace transform of the hazard ratio for the distribution of the omitted covariate. For example if this hazard ratio has a gamma distribution, the optimal test is a member of the G class introduced by Harrington and Fleming (1982). We also consider positive stable, inverse Gaussian, displaced Poisson and two-point frailty distribution. Results are obtained for parametric and nonparametric tests and are extended to include random censoring. We show that the loss of efficiency from omitting a covariate is generally more important than the additional loss due to misspecification of the resulting non-proportional hazards model as a proportional hazards model. However two-point frailty distributions can provide exceptions to this rule. Censoring generally increases the efficiency of the simple logrank test to the adjusted logrank test.     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

The Estimation and Testing of the Cointegration Order Based on the Frequency Domain

ABSTRACT

This article proposes a method to estimate the degree of cointegration in bivariate series and suggests a test statistic for testing noncointegration based on the determinant of the spectral density matrix for the frequencies close to zero. In the study, series are assumed to be I(d), 0 < d ? 1, with parameter d supposed to be known. In this context, the order of integration of the error series is I(d ? b), b ∈ [0, d]. Besides, the determinant of the spectral density matrix for the dth difference series is a power function of b. The proposed estimator for b is obtained here performing a regression of logged determinant on a set of logged Fourier frequencies. Under the null hypothesis of noncointegration, the expressions for the bias and variance of the estimator were derived and its consistency property was also obtained. The asymptotic normality of the estimator, under Gaussian and non-Gaussian innovations, was also established. A Monte Carlo study was performed and showed that the suggested test possesses correct size and good power for moderate sample sizes, when compared with other proposals in the literature. An advantage of the method proposed here, over the standard methods, is that it allows to know the order of integration of the error series without estimating a regression equation. An application was conducted to exemplify the method in a real context.     

Maximum likelihood estimation of ARFIMA models with a Box-Cox transformation

In this paper we study the interaction between the estimation of the fractional differencing parameter d of ARFIMA models and the common practice of instantaneous transformation of the observed time series. At this aim, we first discuss the effect of a nonlinear transformation of the data on the identification of the process and on the estimate of d. Thus, we propose a joint estimation of the Box-Cox parameter and d by means of a modified normalized version of the Whittle likelihood. Then, the variance and covariance matrix of the parameters estimates is obtained. Finally, a Monte Carlo study is performed in order to check the behaviour of the proposed estimators in finite samples.The paper is the result of a joint research of the two authors. As far as it concerns this version of the work, A. DElia wrote Sects. 2, 3, 4, while D. Piccolo wrote Sects. 1, 5, 6.     

Nonparametric Benchmark Dose Estimation with Continuous Dose‐Response Data

We propose a new method for risk‐analytic benchmark dose (BMD) estimation in a dose‐response setting when the responses are measured on a continuous scale. For each dose level d, the observation X(d) is assumed to follow a normal distribution: . No specific parametric form is imposed upon the mean μ(d), however. Instead, nonparametric maximum likelihood estimates of μ(d) and σ are obtained under a monotonicity constraint on μ(d). For purposes of quantitative risk assessment, a ‘hybrid’ form of risk function is defined for any dose d as R(d) = P[X(d) < c], where c > 0 is a constant independent of d. The BMD is then determined by inverting the additional risk functionR_A(d) = R(d) ? R(0) at some specified value of benchmark response. Asymptotic theory for the point estimators is derived, and a finite‐sample study is conducted, using both real and simulated data. When a large number of doses are available, we propose an adaptive grouping method for estimating the BMD, which is shown to have optimal mean integrated squared error under appropriate designs.     

The Fisher Transform of the Pearson Product Moment Correlation Coefficient and Its Square: Cumulants,Moments, and Applications

This paper extends results on the distribution of the Fisher transform of the correlation coefficient (Fisher, 1921 Fisher , R. A. ( 1921 ). On the “probable error” of a coefficient of correlation deduced from a small sample . Metron 1 : 1 – 32 . [Google Scholar]). Approaches to obtain exact moments of the Fisher transform for both null and non-null correlations are presented. We extend the classic series expansion formulae of Hotelling (1953 Hotelling , H. ( 1953 ). New light on the correlation coefficient and its transforms . Journal of the Royal Statistical Society, Ser. B 15 : 192 – 232 . [Google Scholar]) for the moments of the Fisher transform. These results are considered in the context of quadratic functions of the Fisher transform. Some applications of these results are discussed in the context of correlational hypothesis tests and confidence intervals, and a Monte Carlo experiment is used to demonstrate how application of these results impact the small sample performance of select tests on correlations.     

Testing sphericity and intraclass covariance structures under a growth curve model in high dimension

In this article, we consider the problem of testing (a) sphericity and (b) intraclass covariance structure under a growth curve model. The maximum likelihood estimator (MLE) for the mean in a growth curve model is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. The MLE for the covariance matrix is based on the MLE for the mean, which can be very poor for p close to N. For both structures (a) and (b), we modify the MLE for the mean to an unweighted estimator and based on this estimator we propose a new estimator for the covariance matrix. This new estimator leads to new tests for (a) and (b). We also propose two other tests for each structure, which are just based on the sample covariance matrix.

To compare the performance of all four tests we compute for each structure (a) and (b) the attained significance level and the empirical power. We show that one of the tests based on the sample covariance matrix is better than the likelihood ratio test based on the MLE.     

Multiple unit root tests under uncertainty over the initial condition: some powerful modifications

We modify the union-of-rejection unit root test of Harvey et?al. ??Unit Root Testing in Practice: Dealing with Uncertainty over the Trend and Initial Condition?? (Harvey, Econom Theory 25:587?C636, 2009). This test rejects if either of two different unit root tests rejects but controls the inherent multiple testing issue by suitably modifying the critical values to ensure the desired null rejection probability. We evaluate the new tests?? power relative to existing ones?? and to the Gaussian asymptotic power envelope. An empirical application illustrates the usefulness of the new statistics.     

Testing for spatial lag and spatial error dependence using double length artificial regressions

This paper obtains the joint and conditional Lagrange multiplier (LM) tests for a spatial lag regression model with spatial auto-regressive error derived in Anselin (Reg Sci Urban Ecom 26:77–104, 1996) using artificial double length regressions (DLR). These DLR tests and their corresponding LM tests are compared using an illustrative example and a Monte Carlo simulation.     

Non parametric regression estimations over Lp risk based on biased data

Using a wavelet basis, Chesneau and Shirazi study the estimation of one-dimensional regression functions in a biased non parametric model over L² risk (see Chesneau, C and Shirazi, E. Non parametric wavelet regression based on biased data, Communication in Statistics – Theory and Methods, 43: 2642–2658, 2014). This article considers d-dimensional regression function estimation over L^p?(1 ? p < ∞) risk. It turns out that our results reduce to the corresponding theorems of Chesneau and Shirazi’s theorems, when d = 1 and p = 2.     

Construction and Nonparametric Testing of Orthogonal Arrays through Algebraic Strata and Inequivalent Permutation Matrices

In this article we use a new methodology, based on algebraic strata, to generate the class of all the orthogonal arrays of given size and strength. From this class we extract all the non isomorphic orthogonal arrays. Then, using all these non isomorphic orthogonal arrays, we suggest a method based on the inequivalent matrices permutations testing procedures Basso et al. (2004 Basso , D. , Evangelaras , H. , Koukouvinos , C. , Salmaso , L. ( 2004 ). Nonparametric testing for main effects on inequivalent designs. Proc. 7th Int. Workshop Model-Oriented Design Anal. Heeze, Netherlands, June 14–18 . [Google Scholar]) in order to obtain separate permutation tests for the effects in unreplicated mixed level fractional factorial designs. In order to validate the proposed method we perform a Monte Carlo simulation study and find out that the permutation tests appear to be a valid solution for testing effects, in particular when the usual normality assumptions cannot be justified.     

Test for the mean matrix in a Growth Curve model for high dimensions

We consider the problem of estimating and testing a general linear hypothesis in a general multivariate linear model, the so-called Growth Curve model, when the p × N observation matrix is normally distributed.

The maximum likelihood estimator (MLE) for the mean is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. We modify the MLE to an unweighted estimator and propose new tests which we compare with the previous likelihood ratio test (LRT) based on the weighted estimator, i.e., the MLE. We show that the performance of these new tests based on the unweighted estimator is better than the LRT based on the MLE.     

Testing drug additivity based on monotherapies

Under the Loewe additivity, constant relative potency between two drugs is a sufficient condition for the two drugs to be additive. Implicit in this condition is that one drug acts like a dilution of the other. Geometrically, it means that the dose‐response curve of one drug is a copy of another that is shifted horizontally by a constant over the log‐dose axis. Such phenomenon is often referred to as parallelism. Thus, testing drug additivity is equivalent to the demonstration of parallelism between two dose‐response curves. Current methods used for testing parallelism are usually based on significance tests for differences between parameters in the dose‐response curves of the monotherapies. A p‐value of less than 0.05 is indicative of non‐parallelism. The p‐value‐based methods, however, may be fundamentally flawed because an increase in either sample size or precision of the assay used to measure drug effect may result in more frequent rejection of parallel lines for a trivial difference. Moreover, similarity (difference) between model parameters does not necessarily translate into the similarity (difference) between the two response curves. As a result, a test may conclude that the model parameters are similar (different), yet there is little assurance on the similarity between the two dose‐response curves. In this paper, we introduce a Bayesian approach to directly test the hypothesis that the two drugs have a constant relative potency. An important utility of our proposed method is in aiding go/no‐go decisions concerning two drug combination studies. It is illustrated with both a simulated example and a real‐life example. Copyright © 2015 John Wiley & Sons, Ltd.