Goodness-of-fit tests in mixed models

Mixed models, with both random and fixed effects, are most often estimated on the assumption that the random effects are normally distributed. In this paper we propose several formal tests of the hypothesis that the random effects and/or errors are normally distributed. Most of the proposed methods can be extended to generalized linear models where tests for non-normal distributions are of interest. Our tests are nonparametric in the sense that they are designed to detect virtually any alternative to normality. In case of rejection of the null hypothesis, the nonparametric estimation method that is used to construct a test provides an estimator of the alternative distribution.     

Assessment of variance components in nonlinear mixed-effects elliptical models

The issue of assessing variance components is essential in deciding on the inclusion of random effects in the context of mixed models. In this work we discuss this problem by supposing nonlinear elliptical models for correlated data by using the score-type test proposed in Silvapulle and Silvapulle (1995). Being asymptotically equivalent to the likelihood ratio test and only requiring the estimation under the null hypothesis, this test provides a fairly easy computable alternative for assessing one-sided hypotheses in the context of the marginal model. Taking into account the possible non-normal distribution, we assume that the joint distribution of the response variable and the random effects lies in the elliptical class, which includes light-tailed and heavy-tailed distributions such as Student-t, power exponential, logistic, generalized Student-t, generalized logistic, contaminated normal, and the normal itself, among others. We compare the sensitivity of the score-type test under normal, Student-t and power exponential models for the kinetics data set discussed in Vonesh and Carter (1992) and fitted using the model presented in Russo et al. (2009). Also, a simulation study is performed to analyze the consequences of the kurtosis misspecification.     

Linear profiles monitoring in the presence of nonnormal random errors

Profile monitoring is a technique to test the stability of the relationship between a response variable and explanatory variables over time. The most relevant linear profile monitoring methods have been constructed using the normality assumption. However, the normality assumption could be violated in many quality control applications. In this study, we consider a situation in which the random errors in a linear profile model follow a skew‐normal distribution. The skew‐normal distribution is a generalized version of the normal distribution. A new Shewhart‐type chart and exponentially weighted moving average (EWMA) chart, named the Shewhart^R and EWMA^R charts, respectively, are constructed based on residuals to monitor the parameters of linear profile model. The simulation results show that the multivariate EWMA chart is sensitive to the normality assumption and that the proposed Shewhart^R and EWMA^R charts have good ability to detect big and small‐to‐moderate process shifts, respectively. An example using photo mask techniques in semiconductor manufacturing is provided to illustrate the applications of the Shewhart^R and EWMA^R charts.     

A goodness-of-fit testing approach for normality based on the posterior predictive distribution

In this paper, we propose several new goodness-of-fit tests for normality based on the distance between the observed sample and the predictive sample drawn from the posterior predictive distribution. Note that the predictive sample is stochastic for a set of given sample observations, the distance being consequently random. To circumvent the randomness, we choose the conditional expectation and qth quantile as the test statistics. Two statistics are related to the well-known Shapiro–Francia test, and their asymptotic distributions are derived. The simulation study shows that the new tests are able to better discriminate between the normal distribution and heavy-tailed distributions or mixed normal distributions. Against those alternatives, the new tests are more powerful than existing tests including the Anderson–Darling test and the Shapiro–Wilk test, which are two of the best tests of normality in the literature.     

A central limit theorem for sums of functions of residuals in a high-dimensional regression model with an application to variance homoscedasticity test

We establish a joint central limit theorem for sums of squares and the fourth powers of residuals in a high-dimensional regression model. We then apply this CLT to detect the existence of heteroscedasticity for linear regression models without assuming randomness of covariates when the sample size n tends to infinity and the number of covariates p may be fixed or tend to infinity.     

The p value line: a way to choose the tuning constant in tests based on the Huber {\varvec{M}}-estimator

In this paper, we propose a method to choose a test result from several competing ones. The choice is based on the p value under the alternative, i.e., the probability of obtaining an extreme or a more extreme result than the one actually observed, assuming that the alternative hypothesis is true. The function obtained, by moving the alternative, is called the p value line. Because this combines the power of a test and its sensitivity to outliers, it is considered to be the basis for the selection between several classical and/or robust test results. When we change the trimming fraction in the trimmed mean, or the tuning constant when we consider a test based on the Huber \(M\) -estimator, we obtain competing robust test results. For this reason, we can use the p value line to choose the trimming fraction or the tuning constant in an objective way. In this article, we shall focus on the second class of estimators. Because computing the distribution of the test statistic under the alternative could be a very hard problem, a linear approximation of the tail probability functional based on the von Mises expansion can be used to compute the p value line. Applications of the proposal to the one-sample location problem, generalized linear models and generalized additive models are considered in the article. We conclude the article with the random p value line to decide if the observed differences between p value lines are significant or not.     

Safety stock optimisation in two-echelon assembly systems: normal approximation models

This paper tackles the problem of optimising safety stocks in a two-echelon assembly system. It presents and discusses several approximation models for the assembly lead-time under the assumption of normality of the assembly demand and normality of components’ nominal lead times. These approximation models are subsequently used to optimise safety stocks throughout a two-echelon assembly system. They are then tested on a particular two-echelon N-identical component assembly system. The obtained results are compared with the results of a discrete event simulation. Finally, it is shown that lead-times and safety stock results already obtained for a two-echelon distribution system can also be derived without difficulty from those of two-echelon assembly systems.     

Comparison of relative density of two random geometric digraph families in testing spatial clustering

We compare the performance of relative densities of two parameterized random geometric digraph families called proximity catch digraphs (PCDs) in testing bivariate spatial patterns. These PCD families are proportional edge (PE) and central similarity (CS) PCDs and are defined with proximity regions based on relative positions of data points from two classes. The relative densities of these PCDs were previously used as statistics for testing segregation and association patterns against complete spatial randomness. The relative density of a digraph, D, with n vertices (i.e., with order n) represents the ratio of the number of arcs in D to the number of arcs in the complete symmetric digraph of the same order. When scaled properly, the relative density of a PCD is a U-statistic; hence, it has asymptotic normality by the standard central limit theory of U-statistics. The PE- and CS-PCDs are defined with an expansion parameter that determines the size or measure of the associated proximity regions. In this article, we extend the distribution of the relative density of CS-PCDs for expansion parameter being larger than one, and compare finite sample performance of the tests by Monte Carlo simulations and asymptotic performance by Pitman asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each alternative in finite samples and in the limit as the sample size tending to infinity. As a result of our comparisons, we demonstrate that in terms of empirical power (i.e., for finite samples) relative density of CS-PCD has better performance (which occurs for expansion parameter values larger than one) for the segregation alternative, while relative density of PE-PCD has better performance for the association alternative. The methods are also illustrated in a real-life data set from plant ecology.     

On Saint-Venant’s principle in the theory of Cosserat elastic shells

In this paper, we consider the linear theory of Cosserat elastic shells made from an homogeneous and anisotropic material. We study the equilibrium of cylindrical shells loaded by contact forces and couples acting on the end edges. We prove the principle of Saint-Venant in the following formulation: if a system of loads has zero resultant force and moment at each end edge, then the strain energy U_z contained in that portion of the cylindrical shell situated beyond a distance z from the load region has an exponential decay as a function of z.     

Defect structure and the phase diagram of uranium dioxide

Positions of the phase boundaries of pure uranium dioxide UO_{2 ? x} are calculated under the assumption that damage (melting) of the crystal lattice is caused by filling it with defects. The conditions of complete filling of the crystal lattice with individual defects are analyzed. The equilibrium reaction rate constants are determined for the high-temperature forms of interaction of the crystal with oxygen of ambient gaseous medium. The position of the monotectic point is found for x > 0. The melting point parameters are determined for stoichiometric uranium dioxide and uranium dioxide with the maximum dense packing of the defects. A retrograde behavior of the solidus curve in the region x > 0 and anisotropy of properties near the point of congruent melting are predicted. The complex character of the behavior of the boundaries of U₄O₉ and U₃O₈ phases is explained. The reason of discrepancy between the results of two different measurements of solidus temperature in the region x < 0 is explained qualitatively, and the possibility of the existence of solid uranium dioxide up to 3400 K under certain conditions is predicted.     

Estimation of and testing for random effects in dynamic panel data models

In this article, estimation of moments up to the fourth order of random effects and errors is first investigated for dynamic panel data models. Using the QR decomposition of a matrix, the moments of random individual effects and errors are estimated without affecting each other so that the estimation procedure is simple to implement, and the asymptotic behavior of estimation is derived. On the basis of these estimations, we construct a test for the existence of individual effects. This test is asymptotically normally distributed under the null hypothesis without any distributional assumptions on the individual effects and errors other than moments. A power study shows that our test is able to detect local alternatives that are distinct from the null at a parametric rate. Monte Carlo simulations are carried out for illustration.     

Robust and adaptive tests for the two-sample location problem

In the two-sample location problem the application of thet-test depends on very restrictive assumptions such as normality and equal variances of the two random variablesX andY. If the assumptions of thet-test are not satisfied it is more appropriate to apply a robust version of thet-test, like the Welch test or the trimmedt-test, or a nonparametric test, like the Wilcoxon. But usually we have no information about the underlying distribution of the data. Therefore, an adaptive test should be applied. Some of these tests are discussed and compared with each other and with the classicalt-test under different models of nonnormality and for unequal variances. It is shown that an adaptive test behaves well over a broad class of distribution functions.     

On quantitative trait locus mapping with an interference phenomenon

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a gene with quantitative effect on a trait) on the interval [0, T] representing a chromosome. The observation is the trait and the composition of the genome at some locations called “markers”. We focus on the interference phenomenon, i.e. a recombination event inhibits the formation of another recombination event nearby. We give the asymptotic distribution of the LRT process under the null hypothesis that there is no QTL on [0, T] and under local alternatives with a QTL at \(t^{\star}\) on [0, T]. We show that the LRT process is asymptotically the square of a “linear interpolated and normalized process”. We prove that under the null hypothesis, the distribution of the maximum of the LRT process is the same for a model with or without interference. However, the powers of detection are totally different between the two models.     

Linear models for non-Gaussian processes and applications to linear random vibration

Linear models are finite sums of specified deterministic, continuous functions of time with random coefficients. It is shown that linear models provide (i) accurate approximations for real-valued non-Gaussian processes with continuous samples defined on bounded time intervals, (ii) simple solutions for linear random vibration problems with non-Gaussian input, and (iii) efficient techniques for selecting optimal designs from collections of proposed alternatives. Theoretical arguments and numerical examples are presented to establish properties of linear models, illustrate the construction of linear models, solve linear random vibration with non-Gaussian input, and propose an approach for optimal design of linear dynamic systems. It is shown that the proposed linear model provides an efficient tool for analyzing linear systems in non-Gaussian environment.     

Tests for exponentiality against theM andLM-Classes of life distributions

This paper studies tests for exponentiality against the nonparametric classesM andLM of life distributions introduced by Klar and Müller (2003). The test statistics are integrals of the difference between the empirical moment generating function of given data and the moment generating function of a fitted exponential distribution. We derive the limit distributions of the test statistics in case of a general underlying distribution and the local approximate Bahadur efficiency of the procedures against several parametric families of alternatives to exponentiality. The finite sample behavior of the tests is examined by means of a simulation study. Finally, the tests under discussion are applied to two data sets, and we discuss the applicability of the tests under random censorship.     

A parametric fracture mechanics study of welded joints with toe cracks and lack of penetration

The existing design rules give quite general guidelines to the fatigue assessment of different types of welded joints. The goal of this investigation was to give designers some tools, which would allow more precise assessment of the effect of dimensional variations on the fatigue strength. Therefore the fatigue behaviour of 12 common types of welded joints has been studied parametrically. Two-dimensional (2D) finite element models of the joint were made and evaluated using plane strain linear elastic fracture mechanics (LEFM) calculations. The as-welded condition was assumed with the result that no crack initiation period was considered and stress ranges were fully effective. A maximum tangential stress criterion with the Paris’ crack growth law was used to predict the growth rate and direction of root and toe cracks under mixed mode K_I-K_II conditions. The effects of weld size and joint dimension ratios on the fatigue strength were systematically studied. In addition to tensile loading, bending and combined tension/bending moment loading in both directions are examined for positive and negative mean stress.     

Uniform integrability of the OLS estimators,and the convergence of their moments

The problem of convergence of moments of a sequence of random variables to the moments of its asymptotic distribution is important in many applications. These include the determination of the optimal training sample size in the cross-validation estimation of the generalization error of computer algorithms, and in the construction of graphical methods for studying dependence patterns between two biomarkers. In this paper, we prove the uniform integrability of the ordinary least squares estimators of a linear regression model, under suitable assumptions on the design matrix and the moments of the errors. Further, we prove the convergence of the moments of the estimators to the corresponding moments of their asymptotic distribution, and study the rate of the moment convergence. The canonical central limit theorem corresponds to the simplest linear regression model. We investigate the rate of the moment convergence in canonical central limit theorem proving a sharp improvement of von Bahr’s (Ann Math Stat 36:808–818, 1965) theorem.     

Determination of boundary area and spacing in prismatic structures with applications to dislocation boundaries

An analysis is presented for the determination of boundary area per unit volume (S_V) for structures that can be idealized as two-dimensional prismatic foams. An important application for this analysis is in the determination of S_V for one class of dislocation boundary observed following deformation to high strains. Assuming a complete uniformly random distribution of inclination angles for these dislocation walls S_V=πN_L/2, where N_L is the number of intercepts per unit length along a line perpendicular to the wall traces, in any plane perpendicular to the prism basal plane. Analyses of the inclination angle distribution from both measurements of projected boundary widths and from spacing measurements in two perpendicular sections show that while the assumption of a complete uniformly random distribution is an approximation, the error introduced by making this assumption is small.     

First-Principles Study on the Structural, Electronic, Magnetic and?Optical Properties of UFe2 and PuFe2 Compounds

The structural, electronic and magnetic properties of UFe₂ and PuFe₂ have been calculated in the presence and absence of spin?Corbit interaction using density-functional theory by the WIEN2K package. The total energy calculations indicate that at zero pressure the ferromagnetic phase is the most stable phase. Both the energy band calculation and the density of states curves indicate that spin?Corbit interaction has a considerable effect and cannot be ignored. The magnetic moment calculation within local density approximation (LDA) and generalized gradient approximation (GGA) approaches show that LDA and GGA are not good approaches for this compound. To improve the result, we have calculated the magnetic moment using the GGA+U and LDA+U approaches. The calculation of the magnetic moment as a function of pressure has been investigated.     

A linear-encoding model explains the variability of the target morphology in regeneration

A fundamental assumption of today''s molecular genetics paradigm is that complex morphology emerges from the combined activity of low-level processes involving proteins and nucleic acids. An inherent characteristic of such nonlinear encodings is the difficulty of creating the genetic and epigenetic information that will produce a given self-assembling complex morphology. This ‘inverse problem’ is vital not only for understanding the evolution, development and regeneration of bodyplans, but also for synthetic biology efforts that seek to engineer biological shapes. Importantly, the regenerative mechanisms in deer antlers, planarian worms and fiddler crabs can solve an inverse problem: their target morphology can be altered specifically and stably by injuries in particular locations. Here, we discuss the class of models that use pre-specified morphological goal states and propose the existence of a linear encoding of the target morphology, making the inverse problem easy for these organisms to solve. Indeed, many model organisms such as Drosophila, hydra and Xenopus also develop according to nonlinear encodings producing linear encodings of their final morphologies. We propose the development of testable models of regeneration regulation that combine emergence with a top-down specification of shape by linear encodings of target morphology, driving transformative applications in biomedicine and synthetic bioengineering.