Non-normal Type Research Articles

Earthquake Focal Mechanisms of Non-Normal Type in the South Baikal Basin

Earthquake Focal Mechanisms of Non-Normal Type in the South Baikal Basin

Existence of Analytic Solutions for Some Classes of Singular Integral Equations of Non-normal Type with Convolution Kernel

Existence of Analytic Solutions for Some Classes of Singular Integral Equations of Non-normal Type with Convolution Kernel

Performance of Coefficient Alpha and Its Alternatives: Effects of Different Types of Non-Normality.

We examined the performance of coefficient alpha and its potential competitors (ordinal alpha, omega total, Revelle's omega total [omega RT], omega hierarchical [omega h], greatest lower bound [GLB], and coefficient H) with continuous and discrete data having different types of non-normality. Results showed the estimation bias was acceptable for continuous data with varying degrees of non-normality when the scales were strong (high loadings). This bias, however, became quite large with moderate strength scales and increased with increasing non-normality. For Likert-type scales, other than omega h, most indices were acceptable with non-normal data having at least four points, and more points were better. For different exponential distributed data, omega RT and GLB were robust, whereas the bias of other indices for binomial-beta distribution was generally large. An examination of an authentic large-scale international survey suggested that its items were at worst moderately non-normal; hence, non-normality was not a big concern. We recommend (a) the demand for continuous and normally distributed data for alpha may not be necessary for less severely non-normal data; (b) for severely non-normal data, we should have at least four scale points, and more points are better; and (c) there is no single golden standard for all data types, other issues such as scale loading, model structure, or scale length are also important.

Examining the Robustness of the Graded Response and 2-Parameter Logistic Models to Violations of Construct Normality.

When fitting unidimensional item response theory (IRT) models, the population distribution of the latent trait (θ) is often assumed to be normally distributed. However, some psychological theories would suggest a nonnormal θ. For example, some clinical traits (e.g., alcoholism, depression) are believed to follow a positively skewed distribution where the construct is low for most people, medium for some, and high for few. Failure to account for nonnormality may compromise the validity of inferences and conclusions. Although corrections have been developed to account for nonnormality, these methods can be computationally intensive and have not yet been widely adopted. Previous research has recommended implementing nonnormality corrections when θ is not "approximately normal." This research focused on examining how far θ can deviate from normal before the normality assumption becomes untenable. Specifically, our goal was to identify the type(s) and degree(s) of nonnormality that result in unacceptable parameter recovery for the graded response model (GRM) and 2-parameter logistic model (2PLM).

A NEW EFFICIENT METHOD FOR TWO CLASSES OF SINGULAR CONVOLUTION INTEGRAL EQUATIONS OF NON-NORMAL TYPE WITH CAUCHY KERNEL

In this article, our task is to study the existence and Noethericity of solution for two classes of singular convolution integral equations with Cauchy kernels in the non-normal type case. To obtain the conditions of solvability for such equations, we establish regularity theory of solvability. By means of the theory of Fourier analysis, we will transform the equations into boundary value problems for holomorphic functions. The holomorphic solutions and conditions of solvability are obtained by using the method of complex analysis in class {0}. Moreover, we also discuss the asymptotic property of solution near nodes. Therefore, our work generalizes and improves the theories of integral equations and the classical boundary value problems for holomorphic functions.

An efficient method for singular integral equations of non-normal type with two convolution kernels

This article deals with the solvability and explicit solutions of one class of singular integral equations with two convolution kernels in the non-normal type case. Via using techniques of complex analysis, such equations are transformed into Riemann–Hilbert boundary value problems (R-HPs) with the discontinuous property. We establish a regularity theory and existence of solutions, and obtain the general solutions and the conditions of Noether solvability. Especially, we investigate the asymptotic behaviours of solutions at nodes. Thus, this paper generalizes the theories of integral equations and Riemann–Hilbert boundary value problems.

The solvability and explicit solutions of singular integral–differential equations of non-normal type via Riemann–Hilbert problem

In this article, we study non-normal type singular integral–differential equations involving convolution kernel and Cauchy kernel in class {0}, and we prove the existence of solutions and the Noether conditions. By using theory of Fourier analysis, the equations under consideration are transformed into boundary value problems for analytic functions (i.e., Riemann–Hilbert problems) with discontinuous coefficients. For such problems, we propose one method different from classical one, and we obtain the analytic solutions and the conditions of Noether solvability. At nodal points, all cases as regards the regularity and the asymptotic behaviors of the solution are discussed in detail. In particular, we generalize the case of normal type discussed in the literature (Li, 2017 [7,8], 2020, 2019) to the case of non-normal type.

EXISTENCE OF SOLUTIONS FOR DUAL SINGULAR INTEGRAL EQUATIONS WITH CONVOLUTION KERNELS IN CASE OF NON-NORMAL TYPE

This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.

Non-normal type singular integral-differential equations by Riemann-Hilbert approach

This article deals with one class of singular integral-differential equations of non-normal type with convolution and Cauchy principal value integral in class {0}. By using Fourier transform, this classes of equations are transformed into a Riemann boundary value problem with nodal point, and we prove the existence of solutions and the Noether theory for the equations. For such equations, we propose one method different from classical one, and we obtain the general solutions and the conditions of solvability. All cases about the behaviors of the solution are considered at nodal points. Therefore, the theory of integral equations and the classical Riemann boundary value problems is extended further.

Contributions to Estimation of Polychoric Correlations

ABSTRACTThis research concerns the estimation of polychoric correlations in the context of fitting structural equation models to observed ordinal variables by multistage estimation. The first main contribution of this research is to propose and evaluate a Monte Carlo estimator for the asymptotic covariance matrix (ACM) of the polychoric correlation estimates. In multistage estimation, the ACM plays a prominent role, as overall test statistics, derived fit indices, and parameter standard errors all depend on this quantity. The ACM, however, must itself be estimated. Established approaches to estimating the ACM use a sample-based version, which can yield poor estimates with small samples. A simulation study demonstrates that the proposed Monte Carlo estimator can be more efficient than its sample-based counterpart. This leads to better calibration for established test statistics, in particular with small samples. The second main contribution of this research is a further exploration of the consequences of violating the normality assumption for the underlying response variables. We show the consequences depend on the type of nonnormality, and the number and location of thresholds. The simulation study also demonstrates that overall test statistics have little power to detect the studied forms of nonnormality, regardless of the ACM estimator.

Doubly Periodic Riemann Boundary Value Problem of Non-Normal Type for Analytic Functions on Two Parallel Curves

In this paper, we present and study a kind of Riemann boundary value problem of non-normal type for analytic functions on two parallel curves. Making use of the method of complex functions, we give the method for solving this kind of doubly periodic Riemann boundary value problem of non-normal type and obtain the explicit expressions of solutions and the solvable conditions for it.

Correlation of Anterior Corneal Higher-order Aberrations With Age

The aim of this study was to investigate the distribution of higher-order aberrations (HOAs) of the anterior cornea of eyes with cataract in a senior Chinese population and their correlations with age. Four hundred ten Chinese patients (male/female: 196/214; age: 67.9 ± 10.0 years) with cataract were categorized into 5 groups according to their age. The HOAs in the central 6-mm zone of the cornea were measured with an iTrace wavefront analyzer. The correlations between age and total HOA, coma aberration, trefoil aberration, and spherical aberration were assessed using Spearman analysis. The differences in total HOA, coma aberration, trefoil aberration, spherical aberration among various age groups were analyzed using the Kruskal-Wallis H test. The corneal HOA exhibited a nonnormal distribution. A weak correlation was found between age and total HOA, coma aberration (Z3), trefoil aberration (Z3) (P < 0.001 for all). No correlation was found between age and spherical aberration (Z4). Differences in the root mean square of the total HOA, Z3, and Z3 were found among various age groups (P < 0.05). No difference was found in the root mean square of Z4, spherical diopter, corneal curvature, corneal astigmatism, corneal effective refractive power, corneal asphericity Q value and corneal central power across the groups. The distribution of anterior corneal HOAs of eyes with cataract in a senior Chinese population is of the nonnormal type. The total HOA, Z3, and Z3 are correlated to age, unlike Z4 of the anterior cornea.

Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line

Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12]. In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.

Rank Test of Location Optimal for Hyperbolic Secant Distribution

There are at least two reasons for a symmetric, unimodal, diffuse tailed hyperbolic secant distribution to be interesting in real-life applications. It displays one of the common types of non normality in natural data and is closely related to the logistic and Cauchy distributions that often arise in practice. To test the difference in location between two hyperbolic secant distributions, we develop a simple linear rank test with trigonometric scores. We investigate the small-sample and asymptotic properties of the test statistic and provide tables of the exact null distribution for small sample sizes. We compare the test to the Wilcoxon two-sample test and show that, although the asymptotic powers of the tests are comparable, the present test has certain practical advantages over the Wilcoxon test.

The finite-sample performance of robust unit root tests

This paper investigates the relative small sample performance of several robust unit root tests by means of a simulation study. It is confirmed that the traditional least-squares based Dickey-Fuller test has substantially lower power than several robust alternatives if the error distribution is fat-tailed while its power gain is small at the normal model. Particularly good results are achieved by a quasi-maximum likelihood test. However, all robust tests under consideration exhibit severe size distortions if the disturbances follow a skewed distribution. Moreover, under additive outliers, robust tests fail to produce stable sizes and good power properties. Consequently, the value of using robust unit root tests depends heavily of the type of nonnormality at hand.

Non-Standard Central Bank Loss Functions, Skewed Risks, and Certainty Equivalence

This paper sets out to investigate the role of additive uncertainty under plausible non-standard central bank loss functions over future inflation. Building on a substantial body of evidence in the economic psychology literature, this paper postulates (i) period-by-period loss functions that are non-convex, I.e. displaying diminishing or non-increasing sensitivity to losses, and (ii) non-linear weighing of probabilities, hence departing from the expected utility paradigm. The main conclusion of the study is that if the additive uncertainty is caused by a non-normal distributed additive shock, for instance if the probability distribution of the shock is skewed, then with these departures from the quadratic function the principle of certainty equivalence does not hold anymore. Thus, it appears that with additive uncertainty of the non-normal type the assumption of a quadratic loss function for the central banker may not be as innocuous as it is commonly regarded. JEL Classification: E52, E58

Shear band development predicted by a non-normality theory of plasticity and comparison to crystal plasticity predictions

A phenomenological plasticity model has recently been proposed, which combines a smooth yield surface for an anisotropic solid with a vertex-type plastic flow rule. Both polycrystal plasticity calculations and experiments have shown the type of non-normality of plastic flow, which is represented by this material model. The plasticity model is here implemented in a finite element programme and is used to analyze the plane strain tensile test, thus representing the formation of a neck and the subsequent evolution of shear bands in the neck region. To test the predictions of the phenomenological plasticity model the tensile test problem is also analyzed by polycrystal plasticity, based on the Taylor model for either b.c.c. or f.c.c. crystal structure. It is shown that the phenomenological plasticity model gives a good approximation of the crystal plasticity predictions.

Testing the Robustness of the Likelihood-Ratio Test in a Variance-Component Quantitative-Trait Loci–Mapping Procedure

Detection of linkage to genes for quantitative traits remains a challenging task. Recently, variance components (VC) techniques have emerged as among the more powerful of available methods. As often implemented, such techniques require assumptions about the phenotypic distribution. Usually, multivariate normality is assumed. However, several factors may lead to markedly nonnormal phenotypic data, including (a) the presence of a major gene (not necessarily linked to the markers under study), (b) some types of gene x environment interaction, (c) use of a dichotomous phenotype (i.e., affected vs. unaffected), (d) nonnormality of the population within-genotype (residual) distribution, and (e) selective (extreme) sampling. Using simulation, we have investigated, for sib-pair studies, the robustness of the likelihood-ratio test for a VC quantitative-trait locus-detection procedure to violations of normality that are due to these factors. Results showed (a) that some types of nonnormality, such as leptokurtosis, produced type I error rates in excess of the nominal, or alpha, levels whereas others did not; and (b) that the degree of type I error-rate inflation appears to be directly related to the residual sibling correlation. Potential solutions to this problem are discussed. Investigators contemplating use of this VC procedure are encouraged to provide evidence that their trait data are normally distributed, to employ a procedure that allows for nonnormal data, or to consider implementation of permutation tests.

Consequences of Assumption Violations Revisited: A Quantitative Review of Alternatives to the One-Way Analysis of Variance F Test

The presence of variance heterogeneity and nonnormality in educational and psychological data may frequently invalidate the use of the analysis of variance (ANOVA) F test in one-way independent groups designs. This article offers recommendations to applied researchers on the use of various parametric and nonparametric alternatives to the F test under assumption violation conditions. Meta-analytic techniques were used to summarize the statistical robustness literature on the Type I error properties of the Brown-Forsythe (Brown &amp; Forsythe, 1974 ), James (1951) second-order, Kruskal-Wallis ( Kruskal &amp; Wallis, 1952 ), and Welch (1951) tests. Two variables, based on the theoretical work of Box (1954) , are shown to be highly effective in deciding when a particular alternative procedure should be adopted. Based on the meta-analysis findings, it is recommended that researchers gain a clear understanding of the nature of their data before conducting statistical analyses. Of all of the procedures, the James and Welch tests performed best under violations of the variance homogeneity assumption, although their sensitivity to certain types of nonnormality may preclude their use in all data-analytic situations. Opportunities for further methodological studies of ANOVA alternative procedures are also discussed.

On the Performance of the SPRT for Correlation Coefficient: Normal and Mixtures of Normal Populations

AbstractThe sequential probability ratio test (SPRT) for the correlation coefficient is examined in the normal and non‐normal situations. In the latter case, we evaluate the robustness of the normal procedure when sampling from the mixtures of normal distributions. Two models are introduced for this type of nonnormality. It is shown that in the case of moderate nonnormality, the normal procedure is robust while for extreme nonnormality, the normal procedure is not recommended.