Chi-square Distribution Research Articles

Empirical likelihood with twice censored data

. We consider the problem of the construction of confidence intervals for some functionals of a distribution of interest. These functionals include the moments of the distribution, the survival function, and the mean residual life, among others. The empirical likelihood method has been extensively used to construct such confidence intervals in the cases of complete and right censored data. In this article, we deal with the case of twice censored data, where right and left censoring are simultaneously involved. In this context, we propose an empirical likelihood ratio function which we use to construct confidence intervals for some functionals of the distribution of interest. The advantage of our method lies in the fact that our proposed empirical likelihood ratio function converges to a chi-square distribution without being multiplied by any scale parameters, which is the case for many empirical likelihood versions that exist in the setting of censored data. Therefore, we use our empirical likelihood ratio function to construct the desired confidence intervals without involving any further estimation of the scale parameters. This allows to improve the coverage accuracy of our confidence intervals. Finally, we show the performances of our method through a simulation study and a real data application.

Testing Spherical Symmetry Based on Statistical Representative Points

This paper introduces a novel chisquare test for spherical symmetry, utilizing statistical representative points. The proposed representative-point-based chisquare statistic is shown, through a Monte Carlo study, to considerably improve the power performance compared to the traditional equiprobable chisquare test in many high-dimensional cases. While the test requires relatively large sample sizes to approximate the chisquare distribution, obtaining critical values from existing chisquare tables is simpler compared to many existing tests for spherical symmetry. A real-data application demonstrates the robustness of the proposed method against different choices of representative points. This paper argues that the use of representative points provides a new perspective in high-dimensional goodness-of-fit testing, offering an alternative approach to evaluating spherical symmetry in such contexts. By leveraging the flexibility of choosing the number of representative points, this method ensures more reliable detection of departures from spherical symmetry, especially in high-dimensional datasets. Overall, this research highlights the practical advantages of the proposed approach in statistical analysis, emphasizing its potential as a powerful tool in goodness-of-fit tests within the realm of high-dimensional data.

On a Corrective Decile-based Confidence Interval Estimator of Mean for Normal and Skewed Distributions

This study addresses an issue with the decile t-confidence interval (dt-CI), which fails to achieve the desired coverage probabilities for large samples or skewed distributions (Mokhtar, Yusof & Sapiri, 2024). The article proposes a new corrective decile t-confidence interval (cdt-CI) that resolves these issues by modifying the decile standard deviation. Simulations using normal, chi-squared, log-normal, and gamma distributions show that the cdt-CI outperforms existing methods, particularly for skewed data, in terms of coverage probability and robustness. The real-life data sets analyzed in this study also support the conclusions of the simulation study.

Test for conditional Poissonity in integer-valued conditional autoregressive models

The Poisson distribution is a representative distribution for discrete data, much like the normal distribution is for continuous data. Many time series models for count data have been developed based on the Poisson distribution. However, studies examining the validity of the Poisson assumption in these time series models have been relatively scarce. This study addresses the problem of testing for conditional Poissonity in integer-valued conditional autoregressive models. For this purpose, we introduce a test statistic based on the information matrix of the likelihood function. Under regularity conditions, it is shown that the proposed test has an asymptotic null distribution following a chi-square distribution. Simulation results demonstrate the validity of the proposed test. Additionally, a real data analysis is provided for illustration.

Autonomous residual monitoring of metallurgical digital twins

The importance of digital twin maintenance has recently surfaced through findings from industrial applications. Changes in actual physical systems affect the resemblance between digital and physical twins, which can be seen in the continuously changing variation in model residuals. In this study, a method that autonomously updates itself is proposed for monitoring multivariate residuals. It is independent of the models used and monitors normalised residuals based on the squared Mahalanobis distance. The main novelty comes from the normalisation, which is done by using autonomously updated mean and standard deviation values of recent residuals. The method was studied by using an offline simulation model of a grinding circuit in a phosphate concentrator and an online adaptive digital twin model of a flotation circuit in a gold mine. Its performance was compared with conventional squared Mahalanobis distance and principal component analysis methods. The proposed method detected abnormal residual deviations and had low dependence on the characteristics of initial training data, defined by mean and standard deviation. After training with different data sets, the median monitored values of squared Mahalanobis distance remained consistently at values corresponding to 50–57% chi-square distribution probabilities, whereas without autonomous updating, the corresponding values were in the ranges of 3–55% and 39–88% showing inconsistent performance due to the varying distributions of training data sets. The proposed method with transferable and self-configuring properties can advance the online performance monitoring of digital twins.

Powerful Test of Heterogeneity in Two-Sample Summary-Data Mendelian Randomization.

The success of a Mendelian randomization (MR) study critically depends on the validity of the assumptions underlying MR. We focus on detecting heterogeneity (also known as horizontal pleiotropy) in two-sample summary-data MR. A popular approach is to apply Cochran's statistic method, developed for meta-analysis. However, Cochran's statistic, including its modifications, is known to lack power when its degrees of freedom are large. Furthermore, there is no theoretical justification for the claimed null distribution of the minimum of the modified Cochran's statistic with exact weighting ( ), although it seems to perform well in simulation studies. The principle of our proposed method is straightforward: if a set of variables are valid instruments, then any linear combination of these variables is still a valid instrument. Specifically, this principle holds when these linear combinations are formed using eigenvectors derived from a variance matrix. Each linear combination follows a known normal distribution from which a value can be calculated. We use the minimum value for these eigenvector-based linear combinations as the test statistic. Additionally, we explore a modification of the modified Cochran's statistic by replacing the weighting matrix with a truncated singular value decomposition. Extensive simulation studies reveal that the proposed methods outperform Cochran's statistic, including those with modified weights, and MR-PRESSO, another popular method for detecting heterogeneity, in cases where the number of instruments is not large or the Wald ratios take two values. We also demonstrate these methods using empirical examples. Furthermore, we show that does not follow, but is dominated by, the claimed null chi-square distribution. The proposed methods are implemented in an R package iGasso. Dimension reduction techniques are useful for generating powerful tests of heterogeneity in MR.

Asymptotic Confidence Intervals for MRL and TVaR of the Inverse Weibull Distribution with Type II Censored Data

We consider interval estimation for the parameters, the Mean Residual Life (MRL) and the Tail Value at Risk (TVaR) of the inverse Weibull distribution. We constructed the confidence intervals based on the asymptotic normality of the maximum likelihood estimator in addition to intervals based on the asymptotic Chi-Square distribution of the likelihood ratio (LR) statistic under Type II Censoring. The performance of intervals is investigated and compared using the lower, upper, and total error rates, in addition to expected lengths using simulation. We found that LR intervals showed better overall performance and are more accurate than confidence intervals based on the asymptotic normality of maximum likelihood estimator.

APMG: 3D Molecule Generation Driven by Atomic Chemical Properties.

Recently, mask-fill-based 3D Molecular Generation (MG) methods have become very popular in virtual drug design. However, the existing MG methods ignore the chemical properties of atoms and contain inappropriate atomic position training data, which limits their generation capability. To mitigate the above issues, this paper presents a novel mask-fill-based 3D molecule generation model driven by atomic chemical properties (APMG). Specifically, we construct a new attention-MPNN-based encoder and introduce the electronic information into atom representations to enrich chemical properties. Also, a multi-functional classifier is designed to predict the electronic information of each generated atom, guiding the type prediction of elements and bonds. By design, the proposed method uses the chemical properties of atoms and their correlations for high-quality molecule generation. Second, to optimize the atomic position training data, we propose a novel atomic training position generation approach using the Chi-Square distribution. We evaluate our APMG method on the CrossDocked dataset and visualize the docking states of the pockets and generated molecules. The obtained results demonstrate the superiority and merits of APMG over the state-of-the-art approaches. The dataset and codes will be available on the project homepage: https://github.com/JU-HuaY/APMG.

VAE-SIMCA — Data-driven method for building one class classifiers with variational autoencoders

The paper proposes a new method for building one class classifiers based on variational autoencoders (VAE). The classification decision is built on a linear combination of two squared distances: computed for the original and the reconstructed image as well as for the representation of the original image inside the latent space formed by VAE. Because both distances are well approximated by scaled chi-square distribution, the decision boundary is computed using the theoretical quantile function for this distribution and the predefined probability for Type I error, ⍺. Thereby the boundary does not require any specific optimization and is solely based on the model outcomes computed for the training set.The original idea of the proposed method is inherited from another OCC approach, Data Driven Soft Independent Method for Class Analogies, where singular value decomposition is employed for building the latent space. In this paper we show how this idea can be adopted to be used with VAE for detection of anomalies on images. The paper describes the theoretical background, introduces the main outcomes as well as tools for visual exploration of the classification results, and shows how the method works on several simulated and real datasets.

Research on bridge safety early warning method based on strain energy theory and health monitoring data

Bridges are technology-intensive and heavily invested permanent infrastructure. After completion and opening to traffic, bridge structures are easy to be affected by factors such as traffic load and atmospheric environment. Therefore, it is necessary to do safety warning and evaluation of bridges, especially the abnormal behavior in the early stages of bridge operation. In this research paper, a large-span continuous rigid frame bridge installed with the health monitoring system (HMS), of which a large amount of health monitoring data are collected by the HMS, is used as an example, and then a bridge safety early warning method is proposed when the bridge is during early operating period. First of all, the research finding that the internal stress of the prototype bridge obeys normal distribution through statistical analysis is used; next, we deduce that the strain energy inside the prototype bridge is subject to the Non-central Chi-square distribution combined with the strain energy and statistical theories; in the end, the key probability density distribution function of strain energy and its parameters are derived by using the key stress distribution function of the high performance concrete C50 strength grade used in the prototype bridge. The method recommended in this paper is conducive to the formulation of bridge preventive maintenance strategies.

ON IMPROVED GAUSSIAN CORRELATION INEQUALITIES FOR SYMMETRICAL -RECTANGLES EXTENDED TO CERTAIN MULTIVARIATE GAMMA DISTRIBUTIONS AND SOME FURTHER PROBABILITY INEQUALITIES

The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint cumulative distribution function (cdf), which is MTP2 (multivariate totally positive of order 2). Inequalities of the given type hold at least for all MTP2-cdfs on or with everywhere positive smooth densities. In particular, at least some infinitely divisible multivariate chi-square distributions (gamma distributions in the sense of Krishnamoorthy and Parthasarathy) with any positive real “degree of freedom” are shown to be MTP2. Moreover, further numerically calculable probability inequalities for a broader class of multivariate gamma distributions are derived. A different improvement for inequalities of the GCI-type, and of a similar type with three instead of two groups of components with more special correlation structures is also obtained. The main idea behind these inequalities is to find for a given correlation matrix with positive correlations, a further correlation matrix with smaller correlations whose inverse is an M-matrix and where the corresponding multivariate gamma distribution function is numerically available.

Dynamic model of arriving passengers based on dual value drive

In order to optimize the service process of integrated transportation hubs, the dynamic characteristics of the gathering behavior of arriving passengers were analyzed, and the transformation mechanism of the gathering behavior of arriving passengers in the time dimension of inverse gamma distribution, gamma distribution and chi-square distribution was studied. It was revealed that the gathering behavior was jointly affected by the flight operation time and the utility of the transfer mode, and was converted into the time value and utility value of the arriving passengers respectively. On this basis, a dual-value-driven dynamic model of arriving passengers was established based on the three distributions, and the output was the parameter-adjustable distribution of arriving passengers. The results show that the simulation output with adjustable parameters is consistent with the real distribution, which provides a method and basis for the accurate prediction of the distribution of arriving passengers at hub airports.

An efficient non-binary QC-LDPC coded modulation scheme for long-haul optical systems

Abstract Coded modulation (CM), which combines channel codes with high-dimensional modulation schemes, is an attractive technique to improve the spectral efficiency in high-speed optical transmission systems. Non-binary low-density parity check (NB LDPC) codes can provide larger coding gains with smaller codeword lengths compared to their binary counterpart, in addition to reducing the latency of the system. The majority of works based on NB LDPC CM focus on bit-interleaved CM (BICM) and the widely used additive white Gaussian noise (AWGN) channel model. However, the Gaussian channel model is not a realistic model for long-haul intensity-modulated direct-detection (IM/DD) optical systems using erbium-doped fiber amplifiers. In this paper, it is mathematically proved that noise in the received signals after photodetection can be more accurately represented using chi-squared distribution. The efficiency of CM depends on the mapping between coded symbols and the constellation signal points of the modulation scheme. Hence, a symbol interleaved CM with subcarrier modulation for NB LDPC codes based on the proposed channel model is presented in this paper to improve the bit-error rate (BER) performance. The simulation results show that the proposed CM for NB LDPC codes with M-ary quadrature amplitude modulation schemes provides better performance than BICM. The analysis of the error performance of the proposed system gives a coding gain of 1 dB at a BER of 10−7 compared to that of the AWGN channel model. The performance improvement of the system is also evaluated in terms of spectral efficiency and link distance.

Score test for marks in Hawkes processes

AbstractA score statistic for detecting the impact of marks in a linear Hawkes self-exciting point process is proposed, with its asymptotic properties, finite sample performance, power properties using simulation and application to real data presented. A major advantage of the proposed inference procedure is that the Hawkes process can be fitted under the null hypothesis that marks do not impact the intensity process. Hence, for a given record of a point process, the intensity process is estimated once only and then assessed against any number of potential marks without refitting the joint likelihood each time. Marks can be multivariate and serially dependent. The score function for any given set of marks is easily constructed as the covariance of functions of future intensities fits the unmarked process with functions of the marks under assessment. The asymptotic distribution of the score statistic is a Chi-squared distribution, with degrees of freedom equal to the number of parameters required to specify the boost function. Model-based or nonparametric estimation of required features of the mark’s marginal moments and serial dependence can be used. Using sample moments of the marks in the test statistic construction does not impact the size and power properties.

Adaptive use of streams, wells, and tianjing based on riverbanks and drainages in Hakka settlements of Futang Village before access to piped water

ABSTRACT To address the homogenisation of rural water use brought about by the popularity of water appliances, this study attempted to find a solution in the spatial and behavioural correspondence and fitness of environment on water use practice in Hakka settlements, habitats for immigrants. Studies on settlements have focused on landscape features, heritage conservation, and life behaviour. Water use has been analysed based on its relationship and comparison with space composition. Traffic-related studies have detailed planning practices, system functions, and history evolution. Unlike previous studies, this study combines the perspectives of architectural planning and environmental adaptability to examine linkages among riverbanks, drainages, and water use patterns in Futang Village and elucidated 3 of 11 hakka settlements’ adaptive use of streams, wells, and tianjing (courtyards) before piped water is available. Satellite map depictions, field surveys, and chi-square distributions showed that upstream and downstream settlements had distinct yet flexible water use behaviours, depending on their access to clean water bodies and sewage areas, concerns about the distance to water sources to perform household chores, and weather conditions. This demonstrates the practical value of adapting to fluvial landforms for developing water facilities and restoring water use activities.

Distinguishing Time-Varying Factor Models

Time-varying factor models have been widely used to model changing relationships among economic and financial variables. The existing literature usually specifies the time-varying factor loadings as deterministic functions of time or unit root processes. This article proposes two consistent tests to distinguish between these two specifications based on a randomization approach. By setting the null hypothesis as either specification, we show that the proposed test statistics follow an asymptotic Chi-squared distribution under the respective null hypotheses and diverge to infinity in probability under the respective alternatives. Simulation studies reveal that both test statistics perform reasonably well in finite samples. We apply the proposed tests to the U.S. macroeconomic and global macroeconomic and financial datasets. The results suggest that the time-varying factor loadings as deterministic functions of time should be adopted for these two applications.

Test and Measure for Partial Mean Dependence Based on Machine Learning Methods

It is of importance to investigate the significance of a subset of covariates W for the response Y given covariates Z in regression modeling. To this end, we propose a significance test for the partial mean independence problem based on machine learning methods and data splitting. The test statistic converges to the standard Chi-squared distribution under the null hypothesis while it converges to a normal distribution under the fixed alternative hypothesis. Power enhancement and algorithm stability are also discussed. If the null hypothesis is rejected, we propose a partial Generalized Measure of Correlation (pGMC) to measure the partial mean dependence of Y given W after controlling for the nonlinear effect of Z. We present the appealing theoretical properties of the pGMC and establish the asymptotic normality of its estimator with the optimal root-N convergence rate. Furthermore, the valid confidence interval for the pGMC is also derived. As an important special case when there are no conditional covariates Z, we introduce a new test of overall significance of covariates for the response in a model-free setting. Numerical studies and real data analysis are also conducted to compare with existing approaches and to demonstrate the validity and flexibility of our proposed procedures. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

A Penalized Empirical Likelihood Approach for Estimating Population Sizes under the Negative Binomial Regression Model

In capture–recapture experiments, the presence of overdispersion and heterogeneity necessitates the use of the negative binomial regression model for inferring population sizes. However, within this model, existing methods based on likelihood and ratio regression for estimating the dispersion parameter often face boundary and nonidentifiability issues. These problems can result in nonsensically large point estimates and unbounded upper limits of confidence intervals for the population size. We present a penalized empirical likelihood technique for solving these two problems by imposing a half-normal prior on the population size. Based on the proposed approach, a maximum penalized empirical likelihood estimator with asymptotic normality and a penalized empirical likelihood ratio statistic with asymptotic chi-square distribution are derived. To improve numerical performance, we present an effective expectation-maximization (EM) algorithm. In the M-step, optimization for the model parameters could be achieved by fitting a standard negative binomial regression model via the R basic function glm.nb(). This approach ensures the convergence and reliability of the numerical algorithm. Using simulations, we analyze several synthetic datasets to illustrate three advantages of our methods in finite-sample cases: complete mitigation of the boundary problem, more efficient maximum penalized empirical likelihood estimates, and more precise penalized empirical likelihood ratio interval estimates compared to the estimates obtained without penalty. These advantages are further demonstrated in a case study estimating the abundance of black bears (Ursus americanus) at the U.S. Army’s Fort Drum Military Installation in northern New York.

Statistical Modeling of Within-Laboratory Precision Using a Hierarchical Bayesian Approach.

Reproducibility has been well studied in the field of food analysis; the RSD is said to follow a Horwitz curve with certain exceptions. However, little systematic research has been done on predicting repeatability or intermediate precision. We developed a regression method to estimate within-laboratory SDs using hierarchical Bayesian modeling and analyzing duplicate measurement data obtained from actual laboratory tests. The Hamiltonian Monte Carlo method was employed and implemented using R with Stan. The basic structure of the statistical model was assumed to be a Chi-squared distribution, the fixed effect of the predictor was assumed to be a nonlinear function with a constant term and a concentration-dependent term, and the random effects were assumed to follow a lognormal distribution as a hierarchical prior. By analyzing over 300 instances, we obtained regression results that fit well with the assumed model, except for moisture, which was a method-defined analyte. The developed method applies to a wide variety of analytes measured using general principles, including spectroscopy, GC, and HPLC. Although the estimated precisions were within the Horwitz ratio criteria for repeatability, some cases using high-sensitivity detectors, such as mass spectrometers, showed SDs below that range. We propose utilizing the within-laboratory precision predicted by the model established in this study for internal QC and measurement uncertainty estimation without considering sample matrices. Performing statistical modeling on data from double analysis, which is conducted as a part of internal QCs, will simplify the estimation of the precision that fits each analytical system in a laboratory.

The Effect of Student-Centered Teaching Methods and Techniques on Academic Achievement in Life Sciences Teaching: A Meta-Analysis Study

In this study, it was aimed to determine the general effect of student-centered teaching on students' academic achievement in life science teaching. For this purpose, 25 theses scanned in YÖK Thesis Center, having open access permission and methodologically meeting the criteria for inclusion in meta-analysis constituted the data of this study. From 25 thesis studies, 26 effect values were calculated by meta-analysis method. Cohen's g value was used to determine the effect values. The studies included in the meta-analysis were first tested for publication bias using the Rosenthal method and funnel plot. In addition, Z value, significance of Z value, Q value and chi-square distribution values at 25 degrees of freedom were used to decide the meta-analysis model. It was found that the studies included in the meta-analysis did not show publication bias and were heterogeneously distributed. Accordingly, meta-analysis was conducted according to the random effects model. According to the findings obtained in the study, it was seen that each study included in the analysis independently positively affected students' academic achievement. It was determined that the overall effect size of the studies integrated according to the random effects model was 1.028. From this point of view, it was concluded that student-centered teaching methods and techniques in life science teaching have a large positive effect on students' academic achievement.