Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach

We consider goodness-of-fit tests for symmetric stable distributions based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard symmetric stable distribution with the characteristic exponent α estimated from the data. We treat α as an unknown parameter, but for theoretical simplicity we also consider the case that α is fixed. For estimation of parameters and the standardization of data we use the maximum likelihood estimator (MLE). We derive the asymptotic covariance function of the characteristic function process with parameters estimated by MLE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distribution of the test statistic is obtained using complex integration. We find that if the sample size is large the calculated asymptotic critical values of test statistics coincide with the simulated finite sample critical values. Finite sample power of the proposed test is examined. We also present a formula of the asymptotic covariance function of the characteristic function process with parameters estimated by an efficient estimator for general distributions.     

Test on the linear combinations of mean vectors in high-dimensional data

In this study, we propose a procedure for simultaneous testing \(l (l\ge 1)\) linear relations on \(k(k\ge 2)\) high-dimensional mean vectors with heterogeneous covariance matrices, which extends the result derived by Nishiyama et al. (J Stat Plan Inference 143(11):1898–1911, 2013) and does not need the normality assumption. The newly proposed test statistic is motivated by Bai and Saranadasa (Statistica Sinica 6(2):311–329, 1996) and Chen and Qin (Ann Stat 38(2):808–835, 2010). As a special case, our result could be applied to multivariate analysis of variance, that is, testing the equality of k high-dimensional mean vectors.     

Testing of multivariate repeated measures data with block exchangeable covariance structure

A new hypothesis testing of equality of mean vectors in two populations using \(D^2\) statistic for multivariate repeated measures data on q response variables at p sites or time points in a block exchangeable covariance matrix setting is proposed. The minimum sample size needed for our new test is only \(q +1\), unlike \(pq +1\) in Hotelling’s \(T^2\) test. The new test is very efficient in small sample scenario, when the number of observations is not adequate to estimate the \(pq \times pq\) dimensional unknown unstructured variance–covariance matrix. Some simulation studies are performed to compare the power of the new \(D^2\) test and the existing \(BT^2\) test. The performance of the proposed \(D^2\) test is demonstrated with the two medical data sets.     

A moment generating function of a combination of linear rank tests and its asymptotic efficiency

When testing hypotheses in two-sample problems, the Lepage test has often been used to jointly test the location and scale parameters, and has been discussed by many authors over the years. The Lepage test was a combination of the Wilcoxon statistic and the Ansari–Bradley statistic. Various Lepage-type tests were proposed with discussions of an asymptotic relative efficiency (Duran et al., Biometrika 63:173–176, 1976; Goria, Stat Neerl 36:3–13, 1982), a robustness and a power comparison (Neuhäuser, Commun Stat Theory Methods 29:67–78, 2000; Büning, J Appl Stat 29:907–924, 2002) and an adaptive test (Büning and Thadewald, J Stat Comput Sim 65:287–310, 2000). We derive an expression for the moment generating function of a linear combination of two linear rank statistics. As a suggested Lepage-type test, we use a combination of the generalized Wilcoxon statistic and the generalized Mood statistic. Deriving the exact critical value of the statistic can be difficult when the sample sizes are increasing. In this situation, an approximation method to the distribution function of the test statistic can be useful with a higher order moment. We use a moment-based approximation with an adjusted gamma polynomial to evaluate the upper tail probability of a Lepage-type test for a finite sample size. We determine the asymptotic efficiencies of the Lepage and Lepage-type tests for various distributions.     

On the asymptotics of minimum disparity estimation

Inference procedures based on the minimization of divergences are popular statistical tools. Beran (Ann stat 5(3):445–463, 1977) proved consistency and asymptotic normality of the minimum Hellinger distance (MHD) estimator. This method was later extended to the large class of disparities in discrete models by Lindsay (Ann stat 22(2):1081–1114, 1994) who proved existence of a sequence of roots of the estimating equation which is consistent and asymptotically normal. However, the current literature does not provide a general asymptotic result about the minimizer of a generic disparity. In this paper, we prove, under very general conditions, an asymptotic representation of the minimum disparity estimator itself (and not just for a root of the estimating equation), thus generalizing the results of Beran (Ann stat 5(3):445–463, 1977) and Lindsay (Ann stat 22(2):1081–1114, 1994). This leads to a general framework for minimum disparity estimation encompassing both discrete and continuous models.     

Efficient Bootstrap Tests for the Goodness of Fit in Covariance Structure Analysis

Asymptotic chi-square tests, such as the normal theory likelihood ratio test, are often used to evaluate the goodness-of-fit of a covariance structure analysis model. Another approach is to use the bootstrap test, which is known to have the desired asymptotic level if model restrictions are taken into account in designing a resampling algorithm. The bootstrap test is. however, computationally very tedious and the problem of nonconvergence and improper solutions often arise in bootstrap resampling. In this paper, we propose a bootstrap test which is based on an approximation, by a quadratic form, to the minimum value of a discrepancy function calculated from each bootstrap sample. Hence, the proposed bootstrap test is efficient in the sense of the amount of computing needed and is free from the problem of nonconvergence and improper solutions with resampling. A Monte Carlo experiment is conducted to compare the performance of the proposed method with that of asymptotic chi-square tests for each combination of three distributions and four sample sizes.

     

On the number of Eisenstein polynomials of bounded height

We obtain a more precise version of an asymptotic formula of A. Dubickas for the number of monic Eisenstein polynomials of fixed degree d and of height at most H, as $H\rightarrow \infty $ . In particular, we give an explicit bound for the error term. We also obtain an asymptotic formula for arbitrary Eisenstein polynomials of height at most H.     

Asymptotic tensile crack-tip stress fields in elastic-perfectly plastic crystals

For a crack in an elastic-perfectly plastic crystal, there exist more than one asymptotic stress field around the crack tip. The condition governing which field occurs cannot be determined by the asymptotic fields, but depends on external loading conditions. For a plane-strain tensile crack in the (0 1 0) plane and growing in the [1 0 1] direction in a face-centered-cubic (fcc) crystal, there exist three asymptotic stress fields around the crack tip, including a previously established four-sector field and two new families of three-sector fields. The four-sector field gives a large mean stress ahead of the crack, 6.12, where is the critical resolved shear stress for slip systems {1 1 1}1 1 0. The first family of three-sector fields is parameterized by a single parameter p ranging from 0 to 1. In the limit p=0, the field degenerates to uniform tension in the direction parallel to the crack surface. In the other limit p=1, the corresponding field gives the maximum mean stress, 4.90 , in the family. The other family of three-sector fields also has two limits: one corresponds to uniform compression parallel to the crack, and the other provides the maximum mean stress in the family, 2.45 much less than the four-sector field. The stress distributions obtained by the finite element method confirm not only the four-sector field, but also two families of fields.     

Testing the adequacy of semiparametric transformation models

We consider a semiparametric model whereby the response variable following a transformation can be expressed by means of a regression model. In this model, the form of the transformation is specified analytically (up to an unknown transformation parameter), while the regression function is completely unknown. We develop testing procedures for the null hypothesis that this semiparametric model adequately describes the data at hand. In doing so, the test statistic is formulated on the basis of Fourier-type conditional expectations, an idea first put forward by Bierens (J Econom 20:105–134, 1982). The asymptotic distribution of the test statistic is obtained under the null as well as under alternative hypotheses. Since the limit null distribution is nonstandard, a bootstrap version is utilized in order to actually carry out the test procedure. Monte Carlo results are included that illustrate the finite-sample properties of the new method.     

The boundary-layer effect on the crack tip field in mechanism-based strain gradient plasticity

The theory of mechanism-based strain gradient (MSG) plasticity involves two material length parameters, namely the intrinsic material length land the mesoscale cell size l , which are on the order of a few microns and 0.1 m, respectively. Prior studies suggest that l has essentially no effect on the macroscopic quantities, but it may affect the local stress distribution. We demonstrate in this paper that there is a boundary layer effect associated with l in MSG plasticity, and the thickness of the boundary layer is on the order of   l ² big/l. By neglecting this boundary layer effect, a stress-dominated asymptotic field around a crack tip in MSG plasticity is obtained. This asymptotic field is valid at a distance to the crack tip between l and l(i.e., from 0.1 m to a few microns). The stress in this asymptotic field has an approximate singularity of r ^–2/3, which is more singular than not only the HRR field in classical plasticity but also the classical elastic Kfield (r ^–1/2). The stress level in this asymptotic field is two to three times higher than the HRR field, which provides an alternative mechanism for cleavage fracture in ductile materials observed in experiments.     

Flows with variable viscosity: an asymptotic model

Introducing a variable dynamic viseosity coefficient in the Navier equations for an incompressible fluid of small viscosity (Re1, where Re is the classical Reynolds number), we exhibit a three-layer asymptotic model: ideal fluid layer, boundary layer and lower viscous layer. Surprisingly we find that the interaction between the boundary layer and the lower viscous layer is realized only starting with the second-order approximation. We give the full mathematical formulation of the corresponding boundary-layer problem, for the second approximation, with the new boundary conditions obtained by matching from the first-order lower viscous layer, of which the thickness is O(1/Re).As an application of this three-layer asymptotic model we solve completely the classical Blasius problem. In this case the expression for the skin friction coefficient shows that the classical Blasius value is multiplied by a positive term, directly linked to the variability of the dynamic viscosity coefficient.     

Asymptotic Correlations Between Rotated Solutions in Factor Analysis

The asymptotic correlations between differently rotated solutions in exploratory factor analysis are derived. The solutions are orthogonally or obliquely rotated for unstandardized or standardized manifest variables. To obtain the asymptotic correlations between different solutions, the covariance models for manifest variables have been constructed so that two sets of solutions are involved in a single covariance structure. The asymptotic correlations can be used for the statistical test of the differences of rotated solutions. The correlation matrix between the rotated factors of the first solution and those of the second is also introduced in the models with appropriate restrictions to identify the models. The asymptotic standard errors of the estimates of the correlations between the factors in different solutions are simultaneously provided. A numerical example is presented with simulated values.

     

Fast Computing for Distance Covariance

Distance covariance and distance correlation have been widely adopted in measuring dependence of a pair of random variables or random vectors. If the computation of distance covariance and distance correlation is implemented directly accordingly to its definition then its computational complexity is O(n²), which is a disadvantage compared to other faster methods. In this article we show that the computation of distance covariance and distance correlation of real-valued random variables can be implemented by an O(nlog?n) algorithm and this is comparable to other computationally efficient algorithms. The new formula we derive for an unbiased estimator for squared distance covariance turns out to be a U-statistic. This fact implies some nice asymptotic properties that were derived before via more complex methods. We apply the fast computing algorithm to some synthetic data. Our work will make distance correlation applicable to a much wider class of problems. A supplementary file to this article, available online, includes a Matlab and C-based software that realizes the proposed algorithm.     

On the Bounds of the Bilinear Complexity of Multiplication in Some Finite Fields

From the existence of a tower of algebraic function fields, we improve upper bounds on the bilinear complexity of multiplication in all the extensions of the finite fields and where p is a prime 5. In particular, we improve asymptotic upper bounds on this complexity for prime finite fields of characteristic p>5.     

A heuristic reference recursive recipe for adaptively tuning the Kalman filter statistics part-1: formulation and simulation studies

Since the innovation of the ubiquitous Kalman filter more than five decades back it is well known that to obtain the best possible estimates the tuning of its statistics \({\mathbf{X}}_{\mathbf{0}}\), \({\mathbf{P}}_{\mathbf{0}}\), \(\Theta \), R and Q namely initial state and covariance, unknown parameters, and the measurement and state noise covariances is very crucial. The manual and other approaches have not matured to a routine approach applicable for any general problem. The present reference recursive recipe (RRR) utilizes the prior, posterior, and smoothed state estimates as well as their covariances to balance the state and measurement equations and thus form generalized cost functions. The filter covariance at the end of each pass is heuristically scaled up by the number of data points and further trimmed to provide the \({\mathbf{P}}_{\mathbf{0}}\) for subsequent passes. The importance of \({\mathbf{P}}_{\mathbf{0}}\) as the probability matching prior between the frequentist approach via optimization and the Bayesian approach of the Kalman filter is stressed. A simultaneous and proper choice for Q and R based on the filter sample statistics and other covariances leads to a stable filter operation after a few iterations. A typical simulation study of a spring, mass and damper system with a weak nonlinear spring constant by RRR shows it to be better than earlier techniques. Part-2 of the paper further consolidates the present approach based on an analysis of real flight test data.     

Finite-size effects in surface tension: Thermodynamics and the Gaussian interface model

It has been suggested by Kayser that finite-size effects associated with capillary waves might play a significant role in some surface tension measurements; for capillary rise between plates a distance D apart, an effect varying as 1/D and apparently observable in measurements, was proposed. In reconsidering this problem, one must analyze the thermodynamics of finite-size corrections to surface tension. In particular, one sees that capillary rise between plates does not measure the interfacial free energy density but, rather, a derivative of the interfacial free energy with respect to a system dimension. The quantity needed to draw definite conclusions, the finite-size residual free energy, can be calculated within the harmonic or Gaussian capillary wave model in d spatial dimensions with the aid of Poisson summation techniques and should yield the correct leading asymptotic behavior. For d=3 and experimentally relevant parameter values, the results are independent of the short-wavelength cutoff needed in the model and can be checked against the theory of conformai covariance at two-dimensional critical points. It is found that the finite-size effects in capillary-rise measurements of surface tension vary as 1/D ² (with a universal coefficient) but are too small to be seen in current experiments.Invited paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.