Numerical performance of block thresholded wavelet estimators

Usually, methods for thresholding wavelet estimators are implemented term by term, with empirical coefficients included or excluded depending on whether their absolute values exceed a level that reflects plausible moderate deviations of the noise. We argue that performance may be improved by pooling coefficients into groups and thresholding them together. This procedure exploits the information that coefficients convey about the sizes of their neighbours. In the present paper we show that in the context of moderate to low signal-to-noise ratios, this block thresholding approach does indeed improve performance, by allowing greater adaptivity and reducing mean squared error. Block thresholded estimators are less biased than term-by-term thresholded ones, and so react more rapidly to sudden changes in the frequency of the underlying signal. They also suffer less from spurious aberrations of Gibbs type, produced by excessive bias. On the other hand, they are more susceptible to spurious features produced by noise, and are more sensitive to selection of the truncation parameter.     

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behavior under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behavior of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. We also provide a numerical comparison with other competing tests.     

On minimaxity of the normal precision matrix estimator of Krishnamoorthy and Gupta

We consider the orthogonally invariant estimation problem of the inverse of the scale matrix of Wishart distribution using Stein's loss (entropy loss). In this problem Krishnamoorthy and Gupta [2] Krishnamoorthy, K. and Gupta, A. K. (1989). Improved minimax estimation of a normal precision matrix. Canad. J. Statist., 17: 91–102. [Crossref], [Web of Science ®] , [Google Scholar] proposed an estimator and showed its good performance in a Monte Carlo simulation. They conjectured their estimator is minimax. Perron [3] Perron, F. (1997). On a conjecture of Krishnamoorthy and Gupta. J. Multivariate Anal., 62: 110–120.  [Google Scholar] proved its minimaxity for p?=?2. In this paper we prove it for p?=?3 by using a new method.     

On singular elliptical models

The probability density function (pdf) ofsingular elliptical distributions is represented as an integralseries of singular normal distributions. Explicit formulas for the pdf and the cdf of the generalized Chi-square distribution are derived under singular elliptical assumptions extending the result of Díaz-García [(2002). Singular elliptical distribution: density and applications. Commun. Stat.—Theory Methods 31:665–681]. Applications are given of the proposed result for singular mixedmodels.     

On elliptical multilevel models

Multilevel models have been widely applied to analyze data sets which present some hierarchical structure. In this paper we propose a generalization of the normal multilevel models, named elliptical multilevel models. This proposal suggests the use of distributions in the elliptical class, thus involving all symmetric continuous distributions, including the normal distribution as a particular case. Elliptical distributions may have lighter or heavier tails than the normal ones. In the case of normal error models with the presence of outlying observations, heavy-tailed error models may be applied to accommodate such observations. In particular, we discuss some aspects of the elliptical multilevel models, such as maximum likelihood estimation and residual analysis to assess features related to the fitting and the model assumptions. Finally, two motivating examples analyzed under normal multilevel models are reanalyzed under Student-t and power exponential multilevel models. Comparisons with the normal multilevel model are performed by using residual analysis.     

On reciprocal symmetry

On the positive half line, there are two natural, and complementary, analogues of the single notion of symmetry of distributions on the real line. One is the R-symmetry recently proposed and investigated by Mudholkar and Wang [2007. IG-symmetry and R-symmetry: interrelations and applications to the inverse Gaussian theory. J. Statist. Plann. Inference 137, 3655–3671]; the other is the ‘log-symmetry’ investigated here. Log-symmetry can be thought of either in terms of a random variable having the same distribution as its reciprocal or as ordinary symmetry of the distribution of the logged random variable. Various properties, analogies, comparisons and consequences are investigated.     

On the estimation of the mean vector of a multivariate normal distribution under symmetry

The problem of estimation of the mean vector of a multivariate normal distribution with unknown covariance matrix, under uncertain prior information (UPI) that the component mean vectors are equal, is considered. The shrinkage preliminary test maximum likelihood estimator (SPTMLE) for the parameter vector is proposed. The risk and covariance matrix of the proposed estimato are derived and parameter range in which SPTMLE dominates the usual preliminary test maximum likelihood estimator (PTMLE) is investigated. It is shown that the proposed estimator provides a wider range than the usual premilinary test estimator in which it dominates the classical estimator. Further, the SPTMLE has more appropriate size for the preliminary test than the PTMLE.     

Inference in affine shape theory under elliptical models

This paper studies the elliptical statistical affine shape theory under certain particular conditions on the evenness or oddness of the number of landmarks. In such a case, the related distributions are polynomials, and the inference is easily performed; as an example, a landmark data is studied, and the performance of the polynomial density versus the usual series density is compared.     

On local influence for elliptical linear models

The local influence method plays an important role in regression diagnostics and sensitivity analysis. To implement it, we need the Delta matrix for the underlying scheme of perturbations, in addition to the observed information matrix under the postulated model. Galea, Paula and Bolfarine (1997) has recently given the observed information matrix and the Delta matrix for a scheme of scale perturbations and has assessed of local influence for elliptical linear regression models. In the present paper, we consider the same elliptical linear regression models. We study the schemes of scale, predictor and response perturbations, and obtain their corresponding Delta matrices, respectively. To illustrate the methodology for assessment of local influence for these schemes and the implementation of the obtained results, we give an example.     

On the E-optimality of complete block designs under a mixed interference model

In experiments in which the response to a treatment can be affected by other treatments, the interference model with neighbor effects is usually used. It is known that circular neighbor balanced designs (CNBDs) are universally optimal under such a model if the neighbor effects are fixed (Druilhet, 1999) or random (4 and 7). However, such designs cannot exist for every combination of design parameters. In the class of block designs with the same number of treatments as experimental units per block, a CNBD cannot exist if the number of blocks, b  , is equal to p(t−1)±1$p(t-1)\pm 1$, where p is a positive integer and t is the number of treatments. Filipiak et al. (2008) gave the structure of the left-neighboring matrix of E-optimal complete block designs with p  =1 under the model with fixed neighbor effects. The purpose of this paper is to generalize E-optimality results for designs with p∈N$p\in \mathbb{N}$ assuming random neighbor effects.     

General saddlepoint approximations to distributions under an elliptical population

General saddlepoint approximations are derived for the distributions of statistics under an elliptical population. The technique is applied to obtain the tail probabilities of latent roots of a sample covariance matrix. It is shown that the method based on normalizing transformations by Tsuchiya and Konishi (1997) is efficient for the sample correlation coefficient in an elliptical sample.     

Statistical theory of shape under elliptical models via QR decomposition

The statistical shape theory via QR decomposition and based on Gaussian and isotropic models is extended in this paper to the families of non-isotropic elliptical distributions. The new shape distributions are easily computable and then the inference procedure can be studied with the resulting exact densities. An application in Biology is studied under two Kotz models, the best distribution (non-Gaussian) is selected by using a modified Bayesian information criterion (BIC)*.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

Optimal Hoeffding-like inequalities under a symmetry assumption

In this paper, we prove a Hoeffding-like inequality for the survival function of a sum of symmetric independent identically distributed random variables, taking values in a segment [?b, b] of the reals. The symmetric case is relevant to the auditing practice and is an important case study for further investigations. The bounds as given by Hoeffding in 1963 cannot be improved upon unless we restrict the class of random variables, for instance, by assuming the law of the random variables to be symmetric with respect to their mean, which we may assume to be zero. The main result in this paper is an improvement of the Hoeffding bound for i.i.d. random variables which are bounded and have a (upper bound for the) variance by further assuming that they have a symmetric law.     

On the block structure of proper block designs

In this paper we consider proper block designs and derive an upper bound for the number of blocks which can have a fixed number of symbols common with a given block of the design. To arrive at the desired bound, a generalization of an integer programming theorem due to Bush (1976) is first obtained. The integer programming theorem is then used to derive the main result of this paper. The bound given here is then compared with a similar bound obtained by Kageyama and Tsuji (1977).     

On additive and block decompositions of WLSEs under a multiple partitioned regression model

While considering the mechanism of weighted least-squares estimators (WLSEs) of regression coefficients in a partitioned linear model, Tian and Takane [On sum decompositions of weighted least-squares estimators under the partitioned linear model, Comm. Statist. Theory Methods 37 (2008), pp. 55–69] gave some identifying conditions for the WLSEs to be the sum of WLSEs under its two small models based on orthogonality of regressors with respect to the given weight matrix. The purpose of this paper is to show how to establish additive and block decompositions of WLSEs under a multiple partitioned linear model and its k small models based on orthogonality of regressors with respect to a given weight matrix.     

Assessing local influence for elliptical linear models under equality constraints

ABSTRACT

In this article, we study the local influence for the elliptical linear regression model under equality constraints. We first obtain the parameter estimators of this model using the penalized log-likelihood function and iterative techniques. Then we obtain the diagnostics under the perturbations of constant variance, responses, and explanatory variables in the spirit of Cook (1986 Cook, R.D. (1986). Assessment of local influence. J. Royal Stat. Soc. Ser. B 48(2):133–169. [Google Scholar]). Finally, a numerical example on the data set of the salinity of water is given to illustrate the theoretical results.     

On influence diagnostic in univariate elliptical linear regression models

We discuss in this paper the assessment of local influence in univariate elliptical linear regression models. This class includes all symmetric continuous distributions, such as normal, Student-t, Pearson VII, exponential power and logistic, among others. We derive the appropriate matrices for assessing the local influence on the parameter estimates and on predictions by considering as influence measures the likelihood displacement and a distance based on the Pearson residual. Two examples with real data are given for illustration.     

FDR control under block dependence

In practical settings such as microarray data analysis, multiple hypotheses with dependence within but not between equal-sized blocks often need to be tested. We consider an adaptive BH procedure to test the hypotheses. Under the condition of positive regression dependence on a subset of the true null hypotheses, the proposed adaptive procedure is shown to control the false discovery rate. The proposed approach is compared to the existing methods in simulation under block dependence and totally uniform pairwise dependence. It is observed that the proposed method performs better than the existing methods in several situations.