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The length-biased lognormal distribution is a length-biased version of lognormal distribution, which is developed to model the length-biased lifetime data from, for example, biological investigation, medical research, and engineering fields. Owing to the existence of censoring phenomena in lifetime data, we study the change-point-testing problem of length-biased lognormal distribution under random censoring in this paper. A procedure based on the modified information criterion is developed to detect changes in parameters of this distribution. Under the sufficient condition of the Fisher information matrix being positive definite, it is proven that the null asymptotic distribution of the test statistic follows a chi-square distribution. In order to evaluate the uncertainty of change point location estimation, a way of calculating the coverage probabilities and average lengths of confidence sets of change point location based on the profile likelihood and deviation function is proposed. The simulations are conducted, under the scenarios of uniform censoring and exponential censoring, to investigate the validity of the proposed method. And the results indicate that the proposed approach performs better in terms of test power, coverage probabilities, and average lengths of confidence sets compared to the method based on the likelihood ratio test. Subsequently, the proposed approach is applied to the analysis of survival data from heart transplant patients, and the results show that there are differences in the median survival time post-heart transplantation among patients of different ages. [ABSTRACT FROM AUTHOR]

Information theory explains how systems encode and transmit information. This article examines the neuronal system, which processes information via neurons that react to stimuli and transmit electrical signals. Specifically, we focus on transfer entropy to measure the flow of information between sequences and explore its use in determining effective neuronal connectivity. We analyze the causal relationships between two discrete time series, X : = X t : t ∈ Z and Y : = Y t : t ∈ Z , which take values in binary alphabets. When the bivariate process (X , Y) is a jointly stationary ergodic variable-length Markov chain with memory no larger than k, we demonstrate that the null hypothesis of the test—no causal influence—requires a zero transfer entropy rate. The plug-in estimator for this function is identified with the test statistic of the log-likelihood ratios. Since under the null hypothesis, this estimator follows an asymptotic chi-squared distribution, it facilitates the calculation of p-values when applied to empirical data. The efficacy of the hypothesis test is illustrated with data simulated from a neuronal network model, characterized by stochastic neurons with variable-length memory. The test results identify biologically relevant information, validating the underlying theory and highlighting the applicability of the method in understanding effective connectivity between neurons. [ABSTRACT FROM AUTHOR]

Consider k independent random samples from p-dimensional multivariate normal distributions. We are interested in the limiting distribution of the log-likelihood ratio test statistics for testing for the equality of k covariance matrices. It is well known from classical multivariate statistics that the limit is a chi-square distribution when k and p are fixed integers. Jiang and Qi (Scand J Stat 42:988–1009, 2015) and Jiang and Yang (Ann Stat 41(4):2029–2074, 2013) have obtained the central limit theorem for the log-likelihood ratio test statistics when the dimensionality p goes to infinity with the sample sizes. In this paper, we derive the central limit theorem when either p or k goes to infinity. We also propose adjusted test statistics which can be well approximated by chi-squared distributions regardless of values for p and k. Furthermore, we present numerical simulation results to evaluate the performance of our adjusted test statistics and the log-likelihood ratio statistics based on classical chi-square approximation and the normal approximation. [ABSTRACT FROM AUTHOR]

The aim of our paper is to investigate the properties of the classical phase-dispersion minimization (PDM), analysis of variance (AOV), string-length (SL), and Lomb–Scargle (LS) power statistics from a statistician's perspective. We confirm that when the data are perturbations of a constant function, i.e. under the null hypothesis of no period in the data, a scaled version of the PDM statistic follows a beta distribution, the AOV statistic follows an F distribution, and the LS power follows a chi-squared distribution with two degrees of freedom. However, the SL statistic does not have a closed-form distribution. We further verify these theoretical distributions through simulations and demonstrate that the extreme values of these statistics (over a range of trial periods), often used for period estimation and determination of the false alarm probability (FAP), follow different distributions than those derived for a single period. We emphasize that multiple-testing considerations are needed to correctly derive FAP bounds. Though, in fact, multiple-testing controls are built into the FAP bound for these extreme-value statistics, e.g. the FAP bound derived specifically for the maximum LS power statistic over a range of trial periods. Additionally, we find that all of these methods are robust to heteroscedastic noise aimed to mimic the degradation or miscalibration of an instrument over time. Finally, we examine the ability of these statistics to detect a non-constant periodic function via simulating data that mimics a well-detached binary system, and we find that the AOV statistic has the most power to detect the correct period, which agrees with what has been observed in practice. [ABSTRACT FROM AUTHOR]

Diagnostic checking is an important predefined step before using autoregressive models. Although many portmanteau tests were proposed for diagnostic checking, they still struggle with the issue of significant size distortion. In this paper, we develop new diagnostic checking methods based on jackknife empirical likelihood. It is demonstrated that the suggested testing statistics asymptotically have a typical chi-squared distribution. To verify the performance of the finite sample, some simulations are constructed. Additionally, a real example of five agricultural futures is provided to illustrate the merits of our diagnostic checking procedure. [ABSTRACT FROM AUTHOR]

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations. [ABSTRACT FROM AUTHOR]

In this paper, the statistical inference of the partially linear varying coefficient quantile regression model is studied under random missing responses. A two-stage estimation procedure is developed to estimate the parametric and nonparametric components involved in the model. Furthermore, the asymptotic properties of the estimators obtained are established under some mild regularity conditions. In addition, the empirical log-likelihood ratio statistic based on imputation is proposed, and it is proven that this statistic obeys the standard Chi-square distribution; thus, the empirical likelihood confidence interval of the parameter component of the model is constructed. Finally, simulation results show that the proposed estimation method is feasible and effective. [ABSTRACT FROM AUTHOR]

Several major theorems pertaining to necessary and sufficient conditions under which quadratic forms in normal random vectors are chi-square distributed (both central and non central) are revisited and illuminated. Some of these results date back to the early 1950's and are scattered among journals that are not all easily accessible. Moreover, some of these presentations are very terse, reflecting the tendency to reduce the amount of technical typesetting. Most of these results are not available in their entirety in current graduate level statistics texts. Accordingly, the purpose here is to collect these in one place and to offer complete detailed and clear proofs with some new perspectives. The general case in which the variance matrix of the normal random vector can be positive semidefinite is emphasized, since those results will subsume the positive definite case with appropriate reductions of conditions. [ABSTRACT FROM AUTHOR]

Partially linear models find extensive application in biometrics, econometrics, social sciences, and various other fields due to their versatility in accommodating both parametric and nonparametric elements. This study aims to establish statistical inference for the parametric component effects within these models, employing a nonparametric empirical likelihood approach. The proposed method involves a projection step to eliminate the nuisance nonparametric component and utilizes an empirical-likelihood-based technique, along with the Bartlett correction, to enhance the coverage probability of the confidence interval for the parameter of interest. This method demonstrates robustness in handling normally and non-normally distributed errors. The proposed empirical likelihood ratio statistic converges to a limiting chi-square distribution under certain regulations. Simulation studies demonstrate that this method provides better inference in terms of coverage probabilities compared to the conventional normal-approximation-based method. The proposed method is illustrated by analyzing the Boston housing data from a real study. [ABSTRACT FROM AUTHOR]

In many psychometric applications, the relationship between the mean of an outcome and a quantitative covariate is too complex to be described by simple parametric functions; instead, flexible nonlinear relationships can be incorporated using penalized splines. Penalized splines can be conveniently represented as a linear mixed effects model (LMM), where the coefficients of the spline basis functions are random effects. The LMM representation of penalized splines makes the extension to multivariate outcomes relatively straightforward. In the LMM, no effect of the quantitative covariate on the outcome corresponds to the null hypothesis that a fixed effect and a variance component are both zero. Under the null, the usual asymptotic chi-square distribution of the likelihood ratio test for the variance component does not hold. Therefore, we propose three permutation tests for the likelihood ratio test statistic: one based on permuting the quantitative covariate, the other two based on permuting residuals. We compare via simulation the Type I error rate and power of the three permutation tests obtained from joint models for multiple outcomes, as well as a commonly used parametric test. The tests are illustrated using data from a stimulant use disorder psychosocial clinical trial. [ABSTRACT FROM AUTHOR]

Two-sample hypothesis testing is a classical problem in statistics. Given i.i.d. samples X1, :::, Xm ~ P and Y1, :::, Yn ~ Q, the two-sample testing is to test whether P is significantly different from Q or not. Among the various statistical tests, the maximum mean discrepancy (MMD) has been proven to have excellent performance compared to traditional methods. In application of MMD, we are confronted with two challenges: (I) For large data set, MMD is computationally demanding with computational cost O((m + n)²); (II) The limiting distribution under the null hypothesis usually contains infinitely many unknown parameters. In this paper, we propose two testing methods that solve (I) and (II) respectively. To address (I), the data are independently and evenly divided into k groups, calculate the MMD on each group and then combine them by empirical likelihood to yield a test statistic Tk: It converges in distribution to the chi-square distribution with degree of freedom one (X1²) under the null hypothesis as k goes to infinity. To address (II), we propose to compute the two-sample jackknife pseudo-values on each group and combine them by empirical likelihood to get a new test statistic Rnm: Under the null hypothesis, Rnm converges in distribution to X1² if k = o(n): The simulation and real data application indicates that the proposed tests performs well. [ABSTRACT FROM AUTHOR]

Empirical likelihood enables a nonparametric, likelihood-driven style of inference without restrictive assumptions routinely made in parametric models. We develop a framework for applying empirical likelihood to the analysis of experimental designs, addressing issues that arise from blocking and multiple hypothesis testing. In addition to popular designs such as balanced incomplete block designs, our approach allows for highly unbalanced, incomplete block designs. We derive an asymptotic multivariate chi-square distribution for a set of empirical likelihood test statistics and propose two single-step multiple testing procedures: asymptotic Monte Carlo and nonparametric bootstrap. Both procedures asymptotically control the generalised family-wise error rate and efficiently construct simultaneous confidence intervals for comparisons of interest without explicitly considering the underlying covariance structure. A simulation study demonstrates that the performance of the procedures is robust to violations of standard assumptions of linear mixed models. We also present an application to experiments on a pesticide. [ABSTRACT FROM AUTHOR]

This paper studies the estimation and inference of a partially linear varying coefficient spatial autoregressive panel data model with fixed effects. By means of the basis function approximations and the instrumental variable methods, we propose a two-stage least squares estimation procedure to estimate the unknown parametric and nonparametric components, and meanwhile study the asymptotic properties of the proposed estimators. Together with an empirical log-likelihood ratio function for the regression parameters, which follows an asymptotic chi-square distribution under some regularity conditions, we can further construct accurate confidence regions for the unknown parameters. Simulation studies show that the finite sample performance of the proposed methods are satisfactory in a wide range of settings. Lastly, when applied to the public capital data, our proposed model can also better reflect the changing characteristics of the US economy compared to the parametric panel data models. [ABSTRACT FROM AUTHOR]

Background: Pro–b-type natriuretic peptide (Pro-BNP) is an inflammatory marker that indicates cardiac damage and inflammation. The elevation of this marker in COVID-19 patients can be used as a predictive factor in the prognosis of these patients. Method: Our cross-sectional study investigated the evaluation of cardiac diagnostic test findings based on pro-BNP levels in pregnant COVID-19 patients in Sayyad Shirazi Hospital, Gorgan, Iran, in 2020–2022. A hundred and ten pregnant patients diagnosed with COVID-19 infection were evaluated for cardiac diagnostic tests (electrocardiogram (ECG) and echocardiography (Echo)) and pro-BNP levels. Data were analyzed using SPSS 25 software. Chi-square and Student's t-test will be used to test and compare the relationship between variables and compare them. A P-value less than 0.05 is considered statistically significant. The chi-square test was used to compare the ratio of qualitative variables among the groups if the presuppositions of chi-square distribution were established. Otherwise, Fisher's exact test was used. Result: The mean age of participants were 31.06 ± 5.533 years and 49.1% of patients had pro-BNP levels above the cut-off value for predicting an adverse outcome of COVID-19. The mean ± standard deviation of pro-BNP levels in the low group was 46.125 ± 17.523 pg/mL and in the high group was 878.814 ± 1038.060 pg/mL. This study revealed that patients with higher pro-BNP plasma levels had a significant relation between, myocardial infarction (MI), pericardial effusion (PE), urgent Caesarean section (C/S), and mortality. In addition, no significant relation between gravid, trimester, vaccination, arrhythmia, heart block, and valves diseases with high pro-BNP levels was found. Conclusion: The current research showed that pro-BNP levels can be used as a diagnostic and valuable prognostic tool in pregnant women to diagnose cardiac complications by using ECG and Echo. [ABSTRACT FROM AUTHOR]

The Guangdong Lightning Mapping Array (GDLMA), as the first LMA in China, was deployed in Guangzhou, Guangdong Province, China, in November 2018 by the Chinese Academy of Meteorological Sciences and New Mexico Institute of Mining and Technology. An evaluation was conducted using Monte Carlo and an aircraft track. The average timing uncertainty of GDLMA is 35 ns based on the distributions of reduced chi‐square values. Based on the aircraft track, the average horizontal error is 13 m and the average vertical error is 41 m at an altitude of 4–5 km over the network, consistent with the Monte Carlo results. Location errors outside the network exhibit noticeable directionality. The ability to characterize lightning channels varies with different location errors. In locations that are far from the network center, only the basic structure of lightning flash can be presented, while closer to the network, the flash channel structure can be mapped well. Compared with Low‐to‐Mid Frequency E‐field Detection Array (MLFEDA), they were generally similar in overall structure, and some lightning flash characteristics such as flash duration and coverage area exhibited consistency. However, GDLMA demonstrated better flash channel structure characterization capability, while MLFEDA performed better in processes such as leader/return strokes. In addition, based on the comparison of spatial positions of one‐on‐one discharge events, we found that very high frequency sources were more located ahead of low frequency sources in the direction of lightning channel development. Key Points: The detection performance of Guangdong Lightning Mapping Array has been evaluated using Monte Carlo and an aircraft track, and both methods showed good consistencyLocation errors over the network are within 18 m horizontally and 50 m vertically, when outside the network they show directionalityVery high frequency (VHF) and low frequency lightning location results are similar, but VHF sources are more located ahead along the channel [ABSTRACT FROM AUTHOR]

This work examined the sensitivity of randomization test and F‐test in completely randomized design, randomized complete block design and Latin square design in order to ascertain which is more robust in terms of power (sensitivity) of the test and type I error rate in cases with normal error, skewed error (Chi‐square and Exponential distributions), by varying the sample sizes. Using Monte Carlo simulation, response observations were generated for normal error, skewed error at various sample sizes for each of the experimental designs. The results showed that for the normal error, the power and type I error rate of the F‐test were slightly smaller and greater respectively in each of the designs than that of the randomization test. Furthermore, for the skewed error, the randomization test was more sensitive than the F‐test, and also, the type I error was smaller in each of the designs. [ABSTRACT FROM AUTHOR]

We propose a new class of goodness‐of‐fit tests for the logistic distribution based on a characterization related to the density approach in the context of Stein's method. This characterization‐based test is a first of its kind for the logistic distribution. The asymptotic null distribution of the test statistic is derived and it is shown that the test is consistent against fixed alternatives. The finite sample power performance of the newly proposed class of tests is compared to various existing tests by means of a Monte Carlo study. It is found that this new class of tests are especially powerful when the alternative distributions are heavy tailed, like Student's t and Cauchy, or for skew alternatives such as the log‐normal, gamma and chi‐square distributions. [ABSTRACT FROM AUTHOR]

In this paper, we study functional ordinary least squares estimator and its properties in testing the hypothesis of a constant zero mean function or an unknown constant nonzero mean function. We exploit the recent work by Cho, Phillips, and Seo (Int Econ Rev 170:391–456, 2022) and show that the associated Wald test statistics have standard chi-square limiting null distributions, standard noncentral chi-square distributions for local alternatives converging to zero at a n rate, and are consistent against global alternatives. These properties permit computationally convenient tests of hypotheses involving nuisance parameters. In particular, we develop new alternatives to tests for regression misspecification using the neural network model, that involves nuisance parameters identified only under the alternative. Our Monte Carlo simulations affirm the theory of the current study. Finally, we apply our methodology to the probit models for voter turnout that are estimated by Wolfinger and Rosenstone (Who votes? Yale University Press, New Haven, 1980), Nagler (Am Political Sci Rev 85:1393–1405, 1991) and test whether the models are correctly specified or not. [ABSTRACT FROM AUTHOR]

The reliability of the uncertainty characterization, also known as uncertainty realism, is of the uttermost importance for Space Situational Awareness (SSA) services. Among the many sources of uncertainty in the space environment, the most relevant one is the inherent uncertainty of the dynamic models, which is generally not considered in the batch least-squares Orbit Determination (OD) processes in operational scenarios. A classical approach to account for these sources of uncertainty is the theory of consider parameters. In this approach, a set of uncertain parameters are included in the underlying dynamical model, in such a way that the model uncertainty is represented by the variances of these parameters. However, realistic variances of these consider parameters are not known a priori. This work introduces a methodology to infer the variance of consider parameters based on the observed distribution of the Mahalanobis distance of the orbital differences between predicted and estimated orbits, which theoretically should follow a chi-square distribution under Gaussian assumptions. Empirical Distribution Function statistics such as the Cramer-von-Mises and the Kolmogorov–Smirnov distances are used to determine optimum consider parameter variances. The methodology is presented in this paper and validated in a series of simulated scenarios emulating the complexity of operational applications. [ABSTRACT FROM AUTHOR]

The Birnbaum–Saunders (BS) distribution, well-known as the fatigue-life distribution, has been used in numerous disciplines ranging from engineering to medical sciences. In this article, we develop a test for analysis of medians for BS distributed response to assess the impact of two interacting factors on the median, where no test is presently available. The proposed integrated likelihood ratio test (ILRT) eliminates the nuisance shape parameters by integrating them out. The second-order accurate asymptotic chi-square distribution of ILRT is derived. An in-depth simulation study strongly supports its excellent performance even under small group sizes. Furthermore, ILRT developed under the one-way model is found uniformly superior over its peers, is straightway extendable under general multiway setup, and has potential to be extended to other non-normal response variables. Its genuine need in industry, where non-normal responses are commonly encountered, is highlighted through analysis of three real data sets: ILRT strongly picked out the deposition time as influential factor in epitaxial layer experiment, revealed significant impact of spools on fiber life for the failure times of Kevlar 49 fiber data, and gave more accurate parameter estimates in delivery time data experiment, as assessed by various model adequacy tools, where its competitors failed to deliver desired results. [ABSTRACT FROM AUTHOR]

Magnetic resonance imaging (MRI) has become a widely used medical imaging method. Affected by imaging mechanism, magnetic field inhomogeneity and other factors, MRI images are often interfered by non-Gaussian noise such as Rician noise and non-central chi-square distribution noise. However, the existing MRI denoising methods cannot effectively remove different kinds of noise, and the image is prone to blur and detail loss, even artifacts. Thus, this paper proposes an adaptive denoising algorithm for MRI based on Nonlocal Structural Similarity and Low-Rank Sparse Representation (NSS-LRSR). Different from the existing methods, it is a new paradigm to adaptively filter non-Gaussian noise of MRI and it has a good effect on both spatially stable and spatially varying Rician or non-central chi-square distribution noise. The forward variance stable transformation is used to correct the deviation caused by non-Gaussian noise, and then the non-local information is regrouped. And considering the sparse of similar image blocks, we use improved weighted kernel norm minimization to represent the non-local image blocks based on estimation of noise's standard deviation; thereby the processed image block are aggregated and outputted. Experimental results show that compared with the currently popular algorithms, the proposed NSS-LRSR achieves better results in PSNR and SSIM quantitative indexes. [ABSTRACT FROM AUTHOR]

We consider the process capability index (PCI), a widely used quality‐related statistic used to assess the quality of products and performance of monitored processes in various industries. It is widely known that the conventional PCIs perform well when the quality process being monitored has a normal distribution. Unfortunately, using the indices to evaluate a non‐normally distributed process often leads to inaccurate results. In this article, we consider a new PCI, Cpc$C_{pc}$, that can be used in both normal and non‐normal scenarios. The objective of this article is threefold: (i) We provide a corrected form of the confidence interval for Cpc$C_{pc}$. (ii) We compare the performance of three nonparametric bootstrap confidence intervals (BCIs) for Cpc$C_{pc}$. Specifically, the standard bootstrap, percentile bootstrap, and bias‐corrected percentile bootstrap. Under various distributional assumptions such as the normal, chi‐square, Student t, Laplace, and two‐parameter exponential distributions, the estimated coverage probabilities and average width of the confidence intervals and BCIs for Cpc$C_{pc}$ are compared. (iii) The power of the respective bootstrap approaches is evaluated by using the equivalence relation between confidence interval construction and two‐sided hypothesis testing. We also provide the receiver operating characteristic curves to evaluate their performance. Finally, as an illustrative example, an actual data set related to groove dimensions (in inches) measured from a manufacturing process of ignition keys is re‐analyzed to illustrate the utility of the proposed methods. [ABSTRACT FROM AUTHOR]

In recent years, two-sample testing problems with one sample from a mixture distribution have been studied in the literature. Some of these studies ignore an auxiliary information that the exposure or treatment involved in the two-sample problems may have a positive (or negative) effect on the response variable if the effect exists. The positive (or negative) effect implies an inequality constraint. In this paper, we establish the homogeneity tests for the two-sample problem with a mixture structure by considering the inequality constraint, and their null limiting distributions are shown to be a mixture of chi-square distributions. Simulation studies and real data analysis show that the proposed tests have better performance than the existing methods. [ABSTRACT FROM AUTHOR]

We study spatial dynamic panel data models with both spatial lags and spatial errors. The empirical likelihood (EL) ratio statistics are constructed for the parameters of the models. It is shown that the limiting distributions of the EL ratio statistics are chi-square distributions, which are used to construct confidence regions for the parameters of the models. A simulation study is conducted to compare the performances of the EL based and the normal approximation (NA) based confidence regions. Simulation results show that the EL can be computationally easier than the NA method to implement in practice. [ABSTRACT FROM AUTHOR]

M. Loeve wrote that "the fundamental limit theorems of Probability theory may be classified into two groups. One group deals with the problem of limit laws of sequences of some of random variables, the other deals with the problem of limits of random variables, in the sense of almost sure convergence, of such sequences. The list of topics is extensive, and it includes classical models of sums of both independent and various types of dependent random variables, probabilities of large deviations, functional limit theorems, and limit theorems for random processes, in high-dimensional spaces, for spectra of random matrices and random graphs, and more. It is assumed that the covariance structure is one of three covariance structures: (1) an independent covariance structure with the same variance, (2) an independent covariance structure with different variances, and (3) a uniform covariance structure. [Extracted from the article]

Based on empirical likelihood method, we investigate statistical inference in partially linear single-index quantile regression with high dimensional linear and single-index parameters when the observations are missing at random, which allows the response or covariates or response and covariates simultaneously missing. In particular, applying B-spline approximation to the unknown link function, we establish asymptotic normality of bias-corrected empirical likelihood ratio function and maximum empirical likelihood estimators of the parameters. Variable selection is considered by using the SCAD penalty. Meanwhile, we propose a penalized empirical likelihood ratio statistic to test hypothesis, and prove its asymptotically chi-square distribution under the null hypothesis. Also, simulation study and a real data analysis are conducted to evaluate the performance of the proposed methods. • Empirical likelihood for partially linear single-index quantile regression model. • High dimensional statistical inference with observations missing at random. • Hypothesis testing based on penalized empirical likelihood ratio statistic. [ABSTRACT FROM AUTHOR]

A great deal of economic problems are related to detecting the stability of time series data, where the main interest is in the unit root test. In this paper, we consider the unit root testing problem with errors being long-memory processes with the GARCH structure. A new test statistic is developed by using the random weighted bootstrap method. It turns out that the proposed statistic has a chi-squared distribution asymptotically regardless of the process being stationary or nonstationary, and with or without an intercept term. The simulation results show that the statistic has a desired finite sample performance in terms of both size and power. A real data application is also given relying on the inflation rate data of 17 countries. [ABSTRACT FROM AUTHOR]

In this paper, we study the empirical likelihood method for special self-exciting threshold autoregressive models. We assume that the parameters of models depend on some positive integer n, the threshold effect diminishes to zero as n increases, and the errors have heavy-tailed distributions. After replacing the indicator function in the model with a smooth function, we obtain a self-weighted and smoothed least absolute deviation estimator for the threshold parameter and the asymptotic normality of the estimator is proved. Then, the confidence intervals for the threshold parameter can be constructed by the normal approximation method. In order to improve the coverage of confidence intervals, we further propose a profile empirical likelihood ratio, and prove that this statistic has the asymptotically standard chi-squared distribution. Therefore, the confidence interval of the threshold parameter can also be constructed by the empirical likelihood method. Simulations and empirical results demonstrate that the confidence interval constructed by empirical likelihood method is superior to that constructed by normal approximation method. [ABSTRACT FROM AUTHOR]

We demonstrate that the most probable state of a conserved system with a limited number of entities or molecules is the state where non-Gaussian and non-chi-square distributions govern. We have conducted a thought experiment using a specific setup. We have verified the mathematical derivation of the most probable state accurately predicts the results obtained by computer simulations. The derived distributions approach the Gaussian and the chi-square distributions as the number of entities approaches infinity. [ABSTRACT FROM AUTHOR]

Football is a popular sport, but little is known about the youth-to-senior transition rates in elite players, particularly in large and successful countries. This study aims to investigate the youth-to-senior transition rate in the Italian national football team, both prospectively and retrospectively, and to explore if Relative Age Effects (RAEs) affect this transition. Data from 885 players selected in youth and senior Italian national teams between 2000 and 2021 were included in the study. For each player, the birthdate and the number of selection in Under 16, 17, 19, 21 and senior team was considered. The transition rate was determined by the number of youth players competing in the Senior National team (and vice versa), whilst birth quarter (Q) distributions with a chi-square goodness-of-fit test. Prospectively, the transition rate increased as age increased (i.e., from ~20% in U16 to ~50% in U19). Retrospectively, less than 10–20% of youth players were subsequently selected for the senior team. Data revealed a skewed birth date distribution in all age groups, and the RAEs magnitude decreased when age increased (i.e., ORs for Q1 vs Q4 was ~ 9 in U16 and ~ 1.7 in senior teams). Nevertheless, the RAE magnitude was smaller for successfully transitioned players. In conclusion, most players in the senior team were not previously selected for youth teams suggesting that junior international experience may not be a prerequisite for later success. Moreover, while the birthdate strongly influences the selection of youth national teams, its impact is less evident in the youth-to-senior transition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the application of penalized empirical likelihood to the high-dimensional generalized linear models with longitudinal data. Under regular conditions, it is shown that the penalized empirical likelihood has the oracle property. That is, with probability converging to one, the penalized empirical likelihood identifies the true model and estimates the nonzero coefficients as efficiently as if the sparsity of the true model was known in advance. Also, we find the asymptotic distribution of the penalized empirical likelihood ratio test statistic is the chi-square distribution. Some simulations and a real data analysis are conducted to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we first proposed a statistical test for the detrended multiple moving average cross-correlation coefficient DMMC (s). The DMMC (s) mainly was used to analyze the correlation between the dependent variable y and other n independent variables x i . We proved that DMMC (s) approximately obeys the chi-square distribution. We studied the statistical properties of the DMMC (s) between normally distributed random sequences and power-law ARFIMA long memory random sequences. Furthermore, we discussed the influence of the cross-correlation among the target variable and independent variables on DMMC (s). Finally, we further study the application of DMMC (s) to China's stock markets and China carbon emission trading markets to investigate multiple cross-correlation. The empirical results show that there is a strong multiple correlation between China's Shanghai, Shenzhen and Hong Kong stock markets, while the correlation between China's carbon markets is not significant. This paper provides new ideas and theoretical support for exploring the correlation between multiple variables, which has implications for investors and policymakers. [ABSTRACT FROM AUTHOR]

The classical likelihood ratio test based on the asymptotic chi-squared distribution of the log-likelihood is one of the fundamental tools of statistical inference. A recent universal likelihood ratio test approach based on sample splitting provides valid hypothesis tests and confidence sets in any setting for which we can compute the split likelihood ratio statistic, or, more generally, an upper bound on the null maximum likelihood. The universal likelihood ratio test is valid in finite samples and without regularity conditions. This test empowers statisticians to construct tests in settings for which no valid hypothesis test previously existed. For the simple, but fundamental, case of testing the population mean of |$d$| -dimensional Gaussian data with an identity covariance matrix, the classical likelihood ratio test itself applies. Thus, this setting serves as a perfect test bed to compare the classical likelihood ratio test against the universal likelihood ratio test. This work presents the first in-depth exploration of the size, power and relationships between several universal likelihood ratio test variants. We show that a repeated subsampling approach is the best choice in terms of size and power. For large numbers of subsamples, the repeated subsampling set is approximately spherical. We observe reasonable performance even in a high-dimensional setting, where the expected squared radius of the best universal likelihood ratio test's confidence set is approximately 3/2 times the squared radius of the classical likelihood ratio test's spherical confidence set. We illustrate the benefits of the universal likelihood ratio test through testing a nonconvex doughnut-shaped null hypothesis, where a universal inference procedure can have higher power than a standard approach. [ABSTRACT FROM AUTHOR]

Fracture characteristics have a significant effect on the mechanical and hydraulic properties of rock mass. In this paper, before modeling a discrete fracture network, the fracture is simplified as a non-similar ellipse. First, the multifactor coupling stereological relationship of the sampling trace length distribution is established, which proves that the trace length distribution is independent of the fracture occurrence distribution. Using the Volterra integral equation method, the stereological formula of the trace length is inversely solved to obtain the distribution expressions of the major axis and axial ratio of the elliptical fractures. The analytical solutions of the probability density function (PDF) of the characteristic size of the elliptical fractures are derived for cases in which the trace length follows a uniform distribution, fractal distribution, and polynomial distribution. Second, for cases in which the trace length conforms to a negative exponential distribution, gamma distribution, chi-square distribution, and lognormal distribution, the statistical eigenvalues of the major axis and axial ratio of the elliptical fractures are deduced. Finally, a Monte Carlo statistical simulation is performed using the Rock Mass Joint Network Simulation (RJNS3D) toolkit to verify the applicability of the derivation process and the correctness of fitting a polynomial function to the trace length distribution to solve the PDF of the characteristic size of the elliptical fractures. The proposed method can better predict the distribution of the required parameters, such as the major axis and axial ratio of the elliptical fractures, according to the typical two-dimensional data in the sampling windows. This method can be further applied to reconstruct fracture networks in practical engineering tasks and lay a foundation for the analysis of rock mass strength and deformation, representative elementary volume (REV), seepage, and surrounding rock stability. Highlights: A non-similar elliptical model is developed to simulate a fracture network in rock mass. The multifactor coupling stereological relationship of the sampling trace length distribution is established. The correct analytical solutions of the probability density function of the characteristic size of elliptical fractures are derived. Trace sampling in the sampling window is simulated by the Monte Carlo method. [ABSTRACT FROM AUTHOR]

This study aimed to evaluate youth-to-senior transition and the relative age effect in Italian female football national teams. Birthdate data of 774 female players selected for Under 17 (N = 416), 19 (N = 265) and National Senior team (N = 93) were analysed. The youth-to-senior transition rate was determined by the number of youth players competing in the Senior National team (and vice versa), whilst birth quarter (Q) distributions with a chi-square goodness-of-fit test. Only 17.4% of youth players were able to be selected for the Senior National team, whereas 31.2% of the players reached the high-senior level without being selected for youth age groups. Data revealed a skewed birth date distribution in Under 17 and 19 teams (on average, Q1 = 35.6% vs Q4 = 18.5%) but not in the Senior National team. Youth players born in Q1 were two times more likely to be selected than in Q4. In Under 17, goalkeepers, defenders, and midfielders of Q1 players were overrepresented. However, Q4 players recorded higher conversion rates than Q1 (Q1 = 16.4% vs. Q4 = 25.0%). National youth experience may not be a prerequisite for being selected at the senior level. Moreover, this confers a higher probability of playing in the National Senior team than players not selected in youth rosters. [ABSTRACT FROM AUTHOR]

As a typical application of geodesy, the GNSS/INS (Global Navigation Satellite System and Inertial Navigation System) integrated navigation technique was developed and has been applied for decades. For the integrated systems with multiple sensors, data fusion is one of the key problems. As a well-known data fusion algorithm, the Kalman filter can provide optimal estimates with known parameters of the models and noises. In the literature, however, the data fusion algorithm of the GNSS/INS integrated navigation and positioning systems is performed under a certain norm, and performance of the conventional filtering algorithms are improved only under this fixed and limited frame. The mixed norm-based data fusion algorithm is rarely discussed. In this paper, a mixed norm-based data fusion algorithm is proposed, and the hypothesis test statistics are constructed and adopted based on the chi-square distribution. Using the land vehicle data collected through the multi-GNSS and the IMU (Inertial Measurement Unit), the proposed algorithm is tested and compared with the conventional filtering algorithms. Results show that the influences of the outlying measurements and the uncertain noises are weakened with the proposed data fusion algorithm, and the precision of the estimates is further improved. Meanwhile, the proposed algorithm provides an open issue for geodetic applications with mixed norms. [ABSTRACT FROM AUTHOR]

Statistical tables associated with named probability distributions and their families, such as the standard normal, Student's t, and chi-square tables, among others, have been utilized for years and are still widely used today, especially for mathematics and statistics education. While such tables can be found in many statistics textbooks and even on the internet, various shortcomings exist regarding the process of obtaining and distributing tables, as well as the content and appearance of some. In this article, a description is provided of a free, web-based Shiny application created by the author to address these shortcomings and which enables users to generate common statistical tables that are simple and quick to make, customizable, and that users may download and distribute without fear of committing copyright infringement. [ABSTRACT FROM AUTHOR]

This article completes our studies on the formal construction of asymptotic approximations for statistics based on a random number of observations. Second order Chebyshev–Edgeworth expansions of asymptotically normally or chi-squared distributed statistics from samples with negative binomial or Pareto-like distributed random sample sizes are obtained. The results can have applications for a wide spectrum of asymptotically normally or chi-square distributed statistics. Random, non-random, and mixed scaling factors for each of the studied statistics produce three different limit distributions. In addition to the expected normal or chi-squared distributions, Student's t-, Laplace, Fisher, gamma, and weighted sums of generalized gamma distributions also occur. [ABSTRACT FROM AUTHOR]

Pearson's chi-squared test is widely used to test the goodness of fit between categorical data and a given discrete distribution function. When the number of sets of the categorical data, say k, is a fixed integer, Pearson's chi-squared test statistic converges in distribution to a chi-squared distribution with k−1 degrees of freedom when the sample size n goes to infinity. In real applications, the number k often changes with n and may be even much larger than n. By using the martingale techniques, we prove that Pearson's chi-squared test statistic converges to the normal under quite general conditions. We also propose a new test statistic which is more powerful than chi-squared test statistic based on our simulation study. A real application to lottery data is provided to illustrate our methodology. [ABSTRACT FROM AUTHOR]

In this paper, we apply the profile least-square method and inverse probability weighted method to define estimation of the error variance in partially linear varying-coefficient model when the covariates are missing at random. At the same time, we construct a jackknife estimator and jackknife empirical likelihood (JEL) statistic of the error variance, respectively. It is proved that the proposed estimators are asymptotical normality and the JEL statistic admits a limiting standard chi-square distribution. A simulation study is conducted to compare the JEL method with the normal approximation approach in terms of coverage probabilities and average interval lengths, and a comparison of the proposed estimators is done based on sample means, biases and mean square errors under different settings. Subsequently, a real data set is analyzed for illustration of the proposed methods. [ABSTRACT FROM AUTHOR]

Fast and reliable dc line protection is one of the key techniques for flexible dc grid with the development of flexible dc transmission technology. However, the serious overcurrent caused by dc faults has a prominent contradiction with the weak overcurrent capability of power electronics. In this article, the high-fidelity dynamic model of the transmission line is first established based on the frequency-dependent characteristic of line parameters. Then, a new dynamic state estimation based protection (DSEBP) scheme is proposed, which uses the redundancy feature of state estimation to improve the dependability of protection. Finally, the actual measurements and the estimated states are compared with the similarity to identify the fault by the chi-square distribution. The proposed method's effectiveness is verified through simulations power systems computer aided design (PSCAD)/electromagnetic transients including DC (EMTDC) and experiments, and the factors affecting the performance of DSEBP are analyzed. Compared with the existing protection methods, the proposed method is more dependable, more sensitive, and available for actual projects. [ABSTRACT FROM AUTHOR]

Diabetes mellitus has become a rapidly growing chronic health problem worldwide. There has been a noticeable increase in diabetes cases in the last two decades. Recent advances in ensemble machine learning methods play an important role in the early detection of diabetes mellitus. These methods are both faster and less costly than traditional methods. This study aims to propose a new super ensemble learning model to enable an early diagnosis of diabetes mellitus. Super learner is a cross-validation-based approach that makes better predictions by combining prediction results of more than one machine learning algorithm. The proposed super learner model was created with four base-learners (logistic regression, decision tree, random forest, gradient boosting) and a meta learner (support vector machines) as a result of a case study. Three different dataset were used to measure the robustness of the proposed model. Chi-square was determined as an optimal feature selection technique from five different techniques, and also hyper-parameter settings were made with GridSearch. Finally, the proposed new super learner model achieved to obtain the best accuracy results in the detection of Diabetes mellitus compared to the base-learners for the early-stage diabetes risk prediction (99.6%), PIMA (92%), and diabetes 130-US hospitals (98%) dataset, respectively. This study revealed that super learner algorithms can be effectively used in the detection of diabetes mellitus. Also, obtaining of the high and convincing statistical scores shows the robustness of the proposed super learner model. [ABSTRACT FROM AUTHOR]

In a clinical trial, the responses to the new treatment may vary among patient subsets with different characteristics in a biomarker. It is often necessary to examine whether there is a cutpoint for the biomarker that divides the patients into two subsets of those with more favourable and less favourable responses. More generally, we approach this problem as a test of homogeneity in the effects of a set of covariates in generalized linear regression models. The unknown cutpoint results in a model with nonidentifiability and a nonsmooth likelihood function to which the ordinary likelihood methods do not apply. We first use a smooth continuous function to approximate the indicator function defining the patient subsets. We then propose a penalized likelihood ratio test to overcome the model irregularities. Under the null hypothesis, we prove that the asymptotic distribution of the proposed test statistic is a mixture of chi‐squared distributions. Our method is based on established asymptotic theory, is simple to use, and works in a general framework that includes logistic, Poisson, and linear regression models. In extensive simulation studies, we find that the proposed test works well in terms of size and power. We further demonstrate the use of the proposed method by applying it to clinical trial data from the Digitalis Investigation Group (DIG) on heart failure. [ABSTRACT FROM AUTHOR]

For high-dimensional two-sample Behrens–Fisher problems, several non-scale-invariant and scale-invariant tests have been proposed. Most of them impose strong assumptions on the underlying group covariance matrices so that their test statistics are asymptotically normal. However, in practice, these assumptions may not be satisfied or hardly be checked so that these tests may not be able to maintain the nominal size well in practice. To overcome this difficulty, in this paper, a normal reference scale-invariant test is proposed and studied. It works well by neither imposing strong assumptions on the underlying group covariance matrices nor assuming their equality. It is shown that under some regularity conditions and the null hypothesis, the proposed test and a chi-square-type mixture have the same normal and non-normal limiting distributions. It is then justifiable to approximate the null distribution of the proposed test using that of the chi-square-type mixture. The distribution of the chi-square type mixture can be well approximated by the Welch–Satterthwaite chi-square-approximation with the approximation parameter consistently estimated from the data. The asymptotic power of the proposed test is established. Numerical results demonstrate that the proposed test has much better size control and power than several well-known non-scale-invariant and scale-invariant tests. [ABSTRACT FROM AUTHOR]

A sequential multiple assignment randomized trial (SMART) facilitates the comparison of multiple adaptive treatment strategies (ATSs) simultaneously. Previous studies have established a framework to test the homogeneity of multiple ATSs by a global Wald test through inverse probability weighting. SMARTs are generally lengthier than classical clinical trials due to the sequential nature of treatment randomization in multiple stages. Thus, it would be beneficial to add interim analyses allowing for an early stop if overwhelming efficacy is observed. We introduce group sequential methods to SMARTs to facilitate interim monitoring based on the multivariate chi‐square distribution. Simulation studies demonstrate that the proposed interim monitoring in SMART (IM‐SMART) maintains the desired type I error and power with reduced expected sample size compared to the classical SMART. Finally, we illustrate our method by reanalyzing a SMART assessing the effects of cognitive behavioral and physical therapies in patients with knee osteoarthritis and comorbid subsyndromal depressive symptoms. [ABSTRACT FROM AUTHOR]

This paper focuses mainly on the problem of computing the γ th , γ > 0 , moment of a random variable Y n : = ∑ i = 1 n α i X i in which the α i 's are positive real numbers and the X i 's are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of Y n , especially when not all α i 's are equal. We analytically solve this problem by showing that the γ th moment of Y n can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the γ th moment of Y n when X i is a combination of statistically independent Z i 2 and G i in which the Z i 's are distributed according to normal or Maxwell–Boltzmann distributions and the G i 's are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding γ th conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests. [ABSTRACT FROM AUTHOR]

A new and simple blockwise empirical likelihood moment-based procedure to test if a stationary autoregressive process is Gaussian has been proposed. The proposed test utilizes the skewness and kurtosis moment constraints to develop the test statistic. The test nonparametrically accommodates the dependence in the time series data whilst exhibiting some useful properties of empirical likelihood, such as the Wilks theorem with the test statistic having a chi-square limiting distribution. A Monte Carlo simulation study shows that our proposed test has good control of type I error. The finite sample performance of the proposed test is evaluated and compared to some selected competitor tests for different sample sizes and a variety of alternative applied distributions by means of a Monte Carlo study. The results reveal that our proposed test is on average superior under the log-normal and chi-square alternatives for small to large sample sizes. Some real data studies further revealed the applicability and robustness of our proposed test in practice. [ABSTRACT FROM AUTHOR]

We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, 28(2), 386–400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) (Fan, Zhang and Zhang, Annals of statistics, 29(1),153–193, 2001) and a loss function (LF) (Hong and Lee, Annals of Statistics, 41(3), 1166–1203, 2013)-based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled chi-squared distributions, and both exhibit the Wilks phenomenon, which means the scaling constants and degrees of freedom are independent of nuisance parameters. These tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing. Additionally, the bandwidths that are well suited for model estimation may be useful for testing. We show that in additive models, the LF test is asymptotically more powerful than the GLR test. We use simulations to demonstrate the Wilks phenomenon and the power of these proposed GLR and LF tests, and a real example to illustrate their usefulness. [ABSTRACT FROM AUTHOR]

This study aimed at enhancing the efficiency of Zaman estimators using exponential transformation technique. A new class of estimators was obtained using the concept of Bahl and Tuteja. The bias and mean squared error (MSE) of the new class of suggested estimators was derived up to second degree approximation. The empirical study through simulations was conducted using Normal, exponential, gamma, chi‐square and beta distributions under robust regression methods (Huber‐M, Huber‐MM, LTS (least trimmed squares) and LMS (least median of squares)) and the results revealed that proposed estimators were more efficient. [ABSTRACT FROM AUTHOR]

The accuracy and quality of the landslide susceptibility map depend on the available landslide locations and the sampling strategy for absence data (non-landslide locations). In this study, we propose an objective method to determine the critical value for sampling absence data based on Mahalanobis distances (MD). We demonstrate this method on landslide susceptibility mapping of three subdistricts (Upazilas) of the Rangamati district, Bangladesh, and compare the results with the landslide susceptibility map produced based on the slope-based absence data sampling method. Using the 15 landslide causal factors, including slope, aspect, and plan curvature, we first determine the critical value of 23.69 based on the Chi-square distribution with 14 degrees of freedom. This critical value was then used to determine the sampling space for 261 random absence data. In comparison, we chose another set of the absence data based on a slope threshold of < 3°. The landslide susceptibility maps were then generated using the random forest model. The Receiver Operating Characteristic (ROC) curves and the Kappa index were used for accuracy assessment, while the Seed Cell Area Index (SCAI) was used for consistency assessment. The landslide susceptibility map produced using our proposed method has relatively high model fitting (0.87), prediction (0.85), and Kappa values (0.77). Even though the landslide susceptibility map produced by the slope-based sampling also has relatively high accuracy, the SCAI values suggest lower consistency. Furthermore, slope-based sampling is highly subjective; therefore, we recommend using MD -based absence data sampling for landslide susceptibility mapping. [ABSTRACT FROM AUTHOR]