Non-central Χ2 Distribution Research Articles

Robust Bayesian target vector optimization for multi-stage manufacturing processes

This research focuses on optimizing multi-stage manufacturing processes using Bayesian optimization (BO) with a robust Expected Improvement (EI) acquisition function. The aim is to optimize towards pre-selected target vectors, not to just minimize or maximize a function. To achieve this, we minimize the Euclidean distance between the actual and target output vectors, which requires transforming the Gaussian surrogate model posterior distribution into a non-central χ2(NCχ2) distribution. Furthermore, the distance measure additionally uses aleatoric uncertainty estimates of the actual output vectors to achieve robustness. We use a cascaded method that also considers the optimization results of intermediate stages, whereby optimization results are propagated from the last stage towards the first stage in each optimization iteration. By considering intermediate process outputs and aleatoric effects, our approach provides a robust optimization method for multi-stage manufacturing processes. To validate our method and to evaluate its properties, we use two artificial use cases. Moreover, we evaluate our approach in an industrial multi-stage forging process for the manufacturing of a nickel basis superalloy turbine disk, where involved stages are represented by corresponding 2D finite element method (FEM) DEFORM simulations. Evaluation suggests that our approach is superior in optimizing multi-stage manufacturing processes by considering all stage outcomes, robust distance measures, and the use of appropriate uncertainty distributions.

Reprint: Statistical inference for linear mediation models with high-dimensional mediators and application to studying stock reaction to COVID-19 pandemic

Mediation analysis draws increasing attention in many research areas such as economics, finance and social sciences. In this paper, we propose new statistical inference procedures for high dimensional mediation models, in which both the outcome model and the mediator model are linear with high dimensional mediators. Traditional procedures for mediation analysis cannot be used to make statistical inference for high dimensional linear mediation models due to high-dimensionality of the mediators. We propose an estimation procedure for the indirect effects of the models via a partially penalized least squares method, and further establish its theoretical properties. We further develop a partially penalized Wald test on the indirect effects, and prove that the proposed test has a χ2 limiting null distribution. We also propose an F-type test for direct effects and show that the proposed test asymptotically follows a χ2-distribution under null hypothesis and a noncentral χ2-distribution under local alternatives. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed tests and compare their performance with existing ones. We further apply the newly proposed statistical inference procedures to study stock reaction to COVID-19 pandemic via an empirical analysis of studying the mediation effects of financial metrics that bridge company’s sector and stock return.

Approximating Inverse Cumulative Distribution Functions to Produce Approximate Random Variables

For random variables produced through the inverse transform method, approximate random variables are introduced, which are produced using approximations to a distribution’s inverse cumulative distribution function. These approximations are designed to be computationally inexpensive and much cheaper than library functions, which are exact to within machine precision and, thus, highly suitable for use in Monte Carlo simulations. The approximation errors they introduce can then be eliminated through use of the multilevel Monte Carlo method. Two approximations are presented for the Gaussian distribution: a piecewise constant on equally spaced intervals and a piecewise linear using geometrically decaying intervals. The errors of the approximations are bounded and the convergence demonstrated, and the computational savings are measured for C and C++ implementations. Implementations tailored for Intel and Arm hardware are inspected alongside hardware agnostic implementations built using OpenMP. The savings are incorporated into a nested multilevel Monte Carlo framework with the Euler-Maruyama scheme to exploit the speedups without losing accuracy, offering speed ups by a factor of 5–7. These ideas are empirically extended to the Milstein scheme and the non-central χ 2 distribution for the Cox-Ingersoll-Ross process, offering speedups of a factor of 250 or more.

Statistical inference for linear mediation models with high-dimensional mediators and application to studying stock reaction to COVID-19 pandemic

Mediation analysis draws increasing attention in many research areas such as economics, finance and social sciences. In this paper, we propose new statistical inference procedures for high dimensional mediation models, in which both the outcome model and the mediator model are linear with high dimensional mediators. Traditional procedures for mediation analysis cannot be used to make statistical inference for high dimensional linear mediation models due to high-dimensionality of the mediators. We propose an estimation procedure for the indirect effects of the models via a partially penalized least squares method, and further establish its theoretical properties. We further develop a partially penalized Wald test on the indirect effects, and prove that the proposed test has a χ2 limiting null distribution. We also propose an F-type test for direct effects and show that the proposed test asymptotically follows a χ2-distribution under null hypothesis and a noncentral χ2-distribution under local alternatives. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed tests and compare their performance with existing ones. We further apply the newly proposed statistical inference procedures to study stock reaction to COVID-19 pandemic via an empirical analysis of studying the mediation effects of financial metrics that bridge company’s sector and stock return.

Approximating the Mode of the Non-Central Chi-Squared Distribution

In this paper we consider the probability density function (pdf) of the non-central χ2 distribution with arbitrary number of degrees of freedom and non-centrality. For this function we find the approximate location of the maximum and discuss related edge cases of 1 and 2 degrees of freedom. We also use this expression to demonstrate the improved performance of the C++ Boost’s implementation of the non-central χ2 and extend the domain of its applicability.

Higher Order AMMI (HO-AMMI) analysis: A novel stability model to study genotype-location interactions

Additive main effects and multiplicative interaction (AMMI) model is most widely used to analyze genotype x environment interactions (GEI) wherein interaction effects of location is masked by year effect. Hence, presently available models are not able to estimate interaction effects of genotype x location (GLI) and genotype x year (GYI) separately. Moreover, genotype ranking differs as number of years of evaluation vary making selection of genotype for target location difficult. In the present study, a novel stability model i.e., Higher-order-AMMI (HO-AMMI) analysis which can calculate GLI without the confounding effect of GYI and GLYI has been proposed. GEI of AMMI model and all 2-way interactions of HO-AMMI model follow χ2 distribution, whereas 3-way interaction (GLYI) of HO-AMMI follow noncentral χ2 distribution. With increase in number of years of evaluation contribution of GLI towards total variation increased whereas in AMMI model contribution of GEI towards total variation decreased. Variation explained by multiplicative components is higher in HO-AMMI compared to AMMI model. Genotypes were ranked using GL, GY and GL+GY+GLY interactions of HO-AMMI and GEI of AMMI for stability and yield and compared their ranks with field ranking. Correlation and linear regression analysis have indicated high association of GLI (HO-AMMI) with field ranking with high R2 values. Further, HO-AMMI model was able to remove the confounding effect of GYI and GLYI on GLI for accurate identification of genotype for target location irrespective of number of years of evaluation. Hence, HO-AMMI model can be used under multi-environment trials(MET) for selecting genotypes efficiently.

Confidence Intervals of Fit Indexes by Inverting a Bootstrap Test

Fit indexes are an important tool in the evaluation of model fit in structural equation modeling (SEM). Currently, the newest confidence interval (CI) for fit indexes proposed by Zhang and Savalei (2016) is based on the quantiles of a bootstrap sampling distribution at a single level of misspecification. This method, despite a great improvement over naive and model-based bootstrap methods, still suffers from unsatisfactory coverage. In this work, we propose a new method of constructing bootstrap CIs for various fit indexes. This method directly inverts a bootstrap test and produces a CI that involves levels of misspecification that would not be rejected in a bootstrap test. Similar in rationale to a parametric CI of root mean square error of approximation (RMSEA) based on a noncentral χ2 distribution and a profile-likelihood CI of model parameters, this approach is shown to have better performance than the approach of Zhang and Savalei (2016), with more accurate coverage and more efficient widths.

On the statistical power of Baarda’s outlier test and some alternative

Abstract Baarda’s outlier test is one of the best established theories in geodetic practice. The optimal test statistic of the local model test for a single outlier is known as the normalized residual. Also other model disturbances can be detected and identified with this test. It enjoys the property of being a uniformly most powerful invariant (UMPI) test, but is not a uniformly most powerful (UMP) test. In this contribution we will prove that in the class of test statistics following a common central or non-central χ2 distribution, Baarda’s solution is also uniformly most powerful, i.e. UMPχ2 for short. It turns out that UMPχ2 is identical to UMPI, such that this proof can be seen as another proof of the UMPI property of the test. We demonstrate by an example that by means of the Monte Carlo method it is even possible to construct test statistics, which are regionally more powerful than Baarda’s solution. They follow a so-called generalized χ2 distribution. Due to high computational costs we do not yet propose this as a ”new outlier detection method”, but only as a proof that it is in principle possible to outperform the statistical power of Baarda’s test.

Empirical Bayes block shrinkage of wavelet coefficients via the noncentral χ2 distribution

Empirical Bayes approaches to the shrinkage of empirical wavelet coefficients have generated considerable interest in recent years. Much of the work to date has focussed on shrinkage of individual wavelet coefficients in isolation. In this paper we propose an empirical Bayes approach to simultaneous shrinkage of wavelet coefficients in a block, based on the block sum of squares. Our approach exploits a useful identity satisfied by the noncentral Χ 2 density and provides some tractable Bayesian block shrinkage procedures. Our numerical results indicate that the new procedures perform very well.

A score test under a semiparametric finite mixture model

We propose a score statistic to test the mixing proportion under a semiparametric finite mixture model introduced by Anderson [Anderson, J.A., 1979, Multivariate logistic compounds. Biometrika, 66, 17–26]. The proposed score test is based on the semiparametric profile log-likelihood function under a three-sample semiparametric model. The proposed score statistic has an asymptotic χ2 distribution under the null hypothesis and an asymptotic noncentral χ2 distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to the Wald test and the empirical likelihood ratio test. We present some results on simulation and on the analysis of one data set.

Evaluating the Noncentral Chi-Square Distribution for the Cox-Ingersoll-Ross Process

The conditional distribution of the short rate in the Cox-Ingersoll-Ross process can be expressed in terms of the noncentral χ2-distribution. The three standard methods for evaluating this function are by its representation in terms of a series of gamma functions, by analytic approximation, and by its asymptotic expansion. We perform numerical tests of these methods over parameter ranges typical for the Cox-Ingersoll-Ross process. We find that the gamma series representation is accurate over a wide range of parameters but has a runtime that increases proportional to the square root of the noncentrality parameter. Analytic approximations and the asymptotic expansion run quickly but have an accuracy that varies significantly over parameter space. We develop a fourth method for evaluating the upper and lower tails of the noncentral χ2-distribution based on a Bessel function series representation. We find that the Bessel method is accurate over a wide range of parameters and has a runtime that is insensitive to increases in the noncentrality parameter. We show that by using all four methods it is possible to efficiently evaluate the noncentral χ2-distribution to a relative precision of six significant figures.

Computation of Baarda?s lower bound of the non-centrality parameter

Baarda’s reliability measures for outliers, as well as sensitivity and separability measures for deformations, are functions of the lower bound of the non-centrality parameter (LBNP). This parameter, which is taken from Baarda’s well-known nomograms, is actually a non-centrality parameter of the cumulative distribution function (CDF) of the non-central χ2-distribution yielding a complementary probability of the desired power of the test, i.e. probability of Type II error. It is investigated how the LBNP can be computed for desired probabilities (power of the test and significance level) and known degrees of freedom. Two recursive algorithms, namely bisection and the Newton algorithm, were applied to compute the LBNP after the definition of a stable and accurate algorithm for the computation of the corresponding CDF. Despite the fact that the recursive algorithms ensure some desired accuracy, it is presented numerically that the Newton algorithm has a faster convergence to the solution than the bisection algorithm.

Calculating Power for the Comparison of Dependent κ-Coefficients

Summary In the psychosocial and medical sciences, some studies are designed to assess the agreement between different raters and/or different instruments. Often the same sample will be used to compare the agreement between two or more assessment methods for simplicity and to take advantage of the positive correlation of the ratings. Although sample size calculations have become an important element in the design of research projects, such methods for agreement studies are scarce. We adapt the generalized estimating equations approach for modelling dependent κ-statistics to estimate the sample size that is required for dependent agreement studies. We calculate the power based on a Wald test for the equality of two dependent κ-statistics. The Wald test statistic has a non-central χ2-distribution with non-centrality parameter that can be estimated with minimal assumptions. The method proposed is useful for agreement studies with two raters and two instruments, and is easily extendable to multiple raters and multiple instruments. Furthermore, the method proposed allows for rater bias. Power calculations for binary ratings under various scenarios are presented. Analyses of two biomedical studies are used for illustration.

Evaluation of the extent of nonlinearity in reportable range studies.

Abstract Objectives.—To extend the polynomial method for evaluating linearity in 2 ways. First, we developed a screen to ascertain whether the data were precise enough to permit a reliable evaluation of linearity and therefore eliminate findings of linearity due to low statistical power. Second, we assessed whether the degree of nonlinearity detected by the polynomial method was clinically relevant using a statistically rigorous method. Methods.—Because we assessed linearity relative to a clinically determined level of importance instead of the default value of zero, we used sampling theory based on the noncentral χ2 distribution. Using statistical power calculations, we incorporated a screen for imprecision that guarantees that the probability of correctly identifying nonlinear methods is at least 80%. Results.—With the described methods, we achieved a sensitivity of at least 80% and a specificity of at least 95%. When the data were too imprecise to achieve a sensitivity of 80%, no determination of linea...

Evaluation of the extent of nonlinearity in reportable range studies.

To extend the polynomial method for evaluating linearity in 2 ways. First, we developed a screen to ascertain whether the data were precise enough to permit a reliable evaluation of linearity and therefore eliminate findings of linearity due to low statistical power. Second, we assessed whether the degree of nonlinearity detected by the polynomial method was clinically relevant using a statistically rigorous method. Because we assessed linearity relative to a clinically determined level of importance instead of the default value of zero, we used sampling theory based on the noncentral chi(2) distribution. Using statistical power calculations, we incorporated a screen for imprecision that guarantees that the probability of correctly identifying nonlinear methods is at least 80%. With the described methods, we achieved a sensitivity of at least 80% and a specificity of at least 95%. When the data were too imprecise to achieve a sensitivity of 80%, no determination of linearity was made. This procedure mimics the practice in manual inspection of flagging data that appear imprecise by visual inspection and halting the evaluation. Formal statistical tests for precision and amount of nonlinearity are advantageous because they allow us to quantify and limit classification errors. By formalizing these various aspects of linearity assessment, we maintain some of the complex features of manual methods while making the linearity assessment feasible to apply to a high volume of assessments and removing the between-analyst variability.

A Wiener Germ approximation of the noncentral chi square distribution and of its quantiles

The cumulative distribution function (cdf) of the noncentral χ2 distribution with positive degrees of freedom ν > 0 and a noncentrality parameter δ2 ≥ 0 is usually expressed as an infinite weighted sum of central χ2 cdf’s. For the purpose of numerical evaluation this infinite sum is being approximated by a finite sum. For large values of the noncentrality parameter, the sum converges slowly. Alternative approximation algorithms have been proposed instead in the literature. A comparison of these is given in Johnson & Kotz (1970). Most of the approximation algorithms have advantages for certain values of the arguments/parameters and perform poorly for other values. We are proposing an approximation algorithm that has a very solid theoretical background and is surprisingly accurate for extremely large set of arguments/parameter values. It is also applied for a reliable approximation of the quantiles of the distribution for large values of noncentrality and degrees of freedom. Although being asymptotic in spirit (with respect to degrees of freedom ν), the algorithm gives quite accurate approximation even down to ν = 1.

Generalization for the distribution of hartley’S and nair’s Fmax statistics

The exact distribution of Hartley’s FmaxH statistics under heterogeneity of variances with or without unequal sample sizes is given by Yanagisawa and Shirakawa (1997). The distribution of Hartley’s FmaxH statistics is further generalized to that when follows a non-central χ2 distribution instead of a central χ2 distribution. The exact distribution of Nair’s FmaxN statistics under heterogeneity of variances with or without unequal sample sizes when follows either a central χ2 or a non-central χ2 distribution is given. These exact distributions provide new statistical tests and the power of the tests under very general conditions, such as a simultaneous statistical test for the Analysis of Variance (ANOVA). For these tests, we provide a Fortran program1 which calculates, to a high accuracy, the upper tail probabilities, probability points and probability densities for FmaxH and FmaxN statistics under very general conditions. Usually the accuracy of the calculation is 13 to 14 digits for double precision arithmetic. The probability density functions under various conditions are shown and an extension statistical table for Nair’s FmaxN statistic is given when the degrees of freedom is small.

Performance Analysis of a Glrt in Late-Time Radar Target Detection - Abstract

This paper addresses the issue of target detection in Gaussian noise for wideband radar systems. Through the use of the parsimonious singularity expansion method (SEM) representation of the late-time transient radar return and Bayesian detection theory, an aspect independent detection method based on a generalized likelihood ratio test (GLRT) is developed. The resulting decision rule has a non-central Χ2 distribution which is used to analyze the performance of the GLRT in terms of the probability of detection (PD) and the probability of false alarm (PF) . Results illustrating the performance of the GLRT for two different thin wire targets are displayed in the form of receiver operating characteristics. Simulation results are generated to verify the analytical receiver operating characteristics.

The equal correlation baseline model for comparative fit assessment in structural equation modeling

In this article I describe and evaluate an alternative baseline model for comparative fit assessment of structural equation models and compare it to the standard “null” baseline model. The new “equal correlation” baseline model constrains all variables to have equal, rather than zero, correlations, whereas all variances are free. The new baseline model reflects the reality of atheoretical background correlation in nonex‐perimental data sets, and it improves the ability of comparative fit indices to distinguish between better and worse target models. It also helps to preserve the statistical link between these indices and the noncentral χ2 distribution. Also, computing the same comparative fit indices using different baseline models will provide more information about model fit than computing multiple comparative fit indices using the same baseline. I also point out some limitations of the proposed baseline model.

On estimation of linkage test power.

Two methods of estimating linkage test power are proposed. The first method is based on the maximum likelihood (ML) estimate of the recombination fraction and is intended for use with a likelihood ratio test (LRT) in the form of a chi 2 or lod score. The power is estimated through a noncentral chi 2 distribution with a specially chosen noncentrality parameter. The second method uses the LRT constructed for a simple alternative hypothesis regarding the recombination fraction value. The approximate distribution of this test and a method of estimating its power is proposed. Using simulated pedigree data, the power estimates for these two methods were shown to be satisfactory. Comparisons among these two methods and the computer simulation approach of Boehnke [1986] are performed.