Asymptotic Chi-square Distribution Research Articles

Covariance structure tests for multivariate t-distribution

AbstractWe derive an equation system for finding Maximum Likelihood Estimators (MLEs) for the parameters of a p-dimensional t-distribution with $$\nu $$ ν degrees of freedom, $$t_{p,\nu }$$ t p , ν , and use the MLEs for testing covariance structures for the $$t_{p,\nu }$$ t p , ν -distributed population. The likelihood ratio test (LRT), Rao score test (RST) and Wald test (WT) statistics are derived under the general null-hypothesis $$\textrm{H}_0:\varvec{\Sigma }=\varvec{\Sigma }_0$$ H 0 : Σ = Σ 0 , using a matrix derivative technique. Here the $$p\times p$$ p × p -matrix $$\varvec{\Sigma }$$ Σ is a dispersion/scale parameter. Convergence to the asymptotic chi-square distribution under the null hypothesis is examined in extensive simulation experiments. Also the convergence to the chi-square distribution is studied empirically in the situation when the MLEs of a $$t_{p,\nu }$$ t p , ν -distribution are changed to the corresponding estimators for a normal population. Type I errors and the power of the tests are also examined by simulation. In the simulation study the RST behaved more adequately than all remaining statistics in the situation when the dimensionality p was growing.

Identifying Effective Connectivity between Stochastic Neurons with Variable-Length Memory Using a Transfer Entropy Rate Estimator.

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:=Xt:t∈Z and Y:=Yt: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.

Empirical likelihood in a partially linear single-index model with censored response data

An empirical likelihood (EL) approach for a partial linear single-index model with censored response data is studied. A bias-corrected EL ratio is proposed, and the asymptotic chi-squared distribution of this ratio is obtained. The result can be directly used to construct the confidence regions of the regression parameters. The estimators of regression parameters and link function are constructed, and their asymptotic distributions are obtained. Also, a confidence band of the link function is constructed. The proposed method has two main features: The first feature is that the EL ratio is calibrated directly from within, instead of multiplying an adjustment factor by an EL ratio, which reflects the nature of EL. The second feature is avoiding undersmoothing of nonparametric functions, thus ensuring that the n-consistency of the parameter estimator. As a byproduct, the EL and estimation of a single-index model with censored response data are studied. The performance of the bias-corrected EL is evaluated by the simulation studies. The proposed method is illustrated with an example of a real data analysis.

Testing for coefficient distortion due to outliers with an application to the economic impacts of climate change

Outlying observations can bias regression estimates, requiring the use of outlier-robust estimators. Comparing robust estimates to those obtained using ordinary least squares (OLS) is a common robustness check, however, such comparisons have been mostly informal due to the lack of available tests. Here we introduce a formal test for coefficient distortion due to outliers in regression models. Our proposed test is based on the difference between OLS and robust estimates obtained using a class of Huber-skip M-type estimators (such as Impulse Indicator Saturation or Robustified Least Squares). We show that our distortion test has an asymptotic chi-squared distribution by establishing the asymptotics of the corresponding Huber-skip M-estimators using an empirical process Central Limit Theorem recently developed in the literature. The test is valid for cross-sectional, as well as panel, and stationary or deterministically-trending time series models. To improve finite sample performance and to alleviate concerns on distributional assumptions, we explore several bootstrap testing schemes. We apply our outlier distortion test to estimates of the macro-economic impacts of climate change allowing for adaptation.

Statistical Inference for Partially Linear Varying Coefficient Spatial Autoregressive Panel Data Model

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.

Analysis of conditional randomisation and permutation schemes with application to conditional independence testing

We study properties of two resampling scenarios: Conditional Randomisation and Conditional Permutation schemes, which are relevant for testing conditional independence of discrete random variables X and Y given a random variable Z. Namely, we investigate asymptotic behaviour of estimates of a vector of probabilities in such settings, establish their asymptotic normality and ordering between asymptotic covariance matrices. The results are used to derive asymptotic distributions of the empirical Conditional Mutual Information in those set-ups. Somewhat unexpectedly, the distributions coincide for the two scenarios, despite differences in the asymptotic distributions of the estimates of probabilities. We also prove validity of permutation p-values for the Conditional Permutation scheme. The above results justify consideration of conditional independence tests based on resampled p-values and on the asymptotic chi-square distribution with an adjusted number of degrees of freedom. We show in numerical experiments that when the ratio of the sample size to the number of possible values of the triple exceeds 0.5, the test based on the asymptotic distribution with the adjustment made on a limited number of permutations is a viable alternative to the exact test for both the Conditional Permutation and the Conditional Randomisation scenarios. Moreover, there is no significant difference between the performance of exact tests for Conditional Permutation and Randomisation schemes, the latter requiring knowledge of conditional distribution of X given Z, and the same conclusion is true for both adaptive tests.

Permutation Tests for Assessing Potential Non-Linear Associations between Treatment Use and Multivariate Clinical Outcomes

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.

Empirical likelihood for the analysis of experimental designs

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.

Designing to detect heteroscedasticity in a regression model

Abstract We consider the problem of designing experiments to detect the presence of a specified heteroscedastity in Gaussian regression models. We study the relationship of the Ds- and KL-criteria with the noncentrality parameter of the asymptotic chi-squared distribution of a likelihood-based test, for local alternatives. We found that, when the heteroscedastity depends on one parameter, the two criteria coincide asymptotically and that the D1-criterion is proportional to the noncentrality parameter. Differently, when it depends on several parameters, the KL-optimum design converges to the design that maximizes the noncentrality parameter. Our theoretical findings are confirmed through a simulation study.

Gaussian universal likelihood ratio testing

Summary 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.

Selecting age structure in integrated population models

Integrated population modelling is widely used in ecology when data at the individual level are combined with independent time series measuring population abundance. However there is no formal assessment of how to select the best integrated model. Here we focus on the important case of determining the age-structure for annual survival probabilities of wild animals, involving comparing state–space models with different numbers of states. The work is motivated by real data sets, and evaluated by simulation. We reject the naïve use of AIC, and advocate the use of likelihood-ratio tests, based on combined data. We demonstrate using simulation that typical asymptotic chi-square distributions of likelihood-ratio test statistics to compare integrated models apply when the corresponding state–space models have the same state variables. In addition, for linear state–space models with matching initial conditions the correct chi-square distributions may also hold when models apparently have different state–spaces. The results for comparing integrated models also have relevance for state–space modelling alone. A senescence case study is provided which incorporates a step-up approach and illustrates the use of the recommendations of the paper in practice.

Analysis of medians under two-way model with and without interaction for Birnbaum–Saunders distributed response

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.

Confidence regions near singular information and boundary points with applications to mixed models

We propose confidence regions with asymptotically correct uniform coverage probability of parameters whose Fisher information matrix can be singular at important points of the parameter set. Our work is motivated by the need for reliable inference on scale parameters close or equal to zero in mixed models, which is obtained as a special case. The confidence regions are constructed by inverting a continuous extension of the score test statistic standardized by expected information, which we show exists at points of singular information under regularity conditions. Similar results have previously only been obtained for scalar parameters, under conditions stronger than ours, and applications to mixed models have not been considered. In simulations our confidence regions have near-nominal coverage with as few as $n = 20$ independent observations, regardless of how close to the boundary the true parameter is. It is a corollary of our main results that the proposed test statistic has an asymptotic chi-square distribution with degrees of freedom equal to the number of tested parameters, even if they are on the boundary of the parameter set.

Empirical likelihood analysis for accelerated failure time model using length-biased data

Length-biased data arise when the probability of observing a subject in a sample is proportional to its corresponding value. The statistical tools that should be applied to analyse such sets of data differ from those used for random samples. Ignoring this issue in an investigation results in a biased inference. In this paper, we propose an empirical likelihood-based method to draw inference on covariate effects in an accelerated failure time model while the observations are subject to length-bias. The asymptotic distribution of the empirical log-likelihood ratio statistic is derived to be a weighted sum of independent chi-square distributions. Hence, we then extend the results by an exploration into the adjusted empirical likelihood. We derive an asymptotic standard chi-square distribution for the adjusted log-likelihood statistic. The limiting distributions are applied to obtain confidence regions for the regression parameters. A simulation study is carried out to evaluate and compare the performance of the proposed procedures with an existing method based on the normal approximation approach. Finally, the procedures are illustrated by modelling the regression parameter and estimating confidence intervals for a set of real data on widths of shrubs.

Human UPF3A and UPF3B enable fault-tolerant activation of nonsense-mediated mRNA decay.

The paralogous human proteins UPF3A and UPF3B are involved in recognizing mRNAs targeted by nonsense‐mediated mRNA decay (NMD). UPF3B has been demonstrated to support NMD, presumably by bridging an exon junction complex (EJC) to the NMD factor UPF2. The role of UPF3A has been described either as a weak NMD activator or an NMD inhibitor. Here, we present a comprehensive functional analysis of UPF3A and UPF3B in human cells using combinatory experimental approaches. Overexpression or knockout of UPF3A as well as knockout of UPF3B did not substantially change global NMD activity. In contrast, the co‐depletion of UPF3A and UPF3B resulted in a marked NMD inhibition and a transcriptome‐wide upregulation of NMD substrates, demonstrating a functional redundancy between both NMD factors. In rescue experiments, UPF2 or EJC binding‐deficient UPF3B largely retained NMD activity. However, combinations of different mutants, including deletion of the middle domain, showed additive or synergistic effects and therefore failed to maintain NMD. Collectively, UPF3A and UPF3B emerge as fault‐tolerant, functionally redundant NMD activators in human cells.

Testing the skewness of skew-normal distribution by Bayes factors

Likelihood ratio test is based on the ratio of the maximum values of likelihood functions on the whole parameter space and the null space, respectively. However, the likelihood ratio test is sometimes intractable such as the skew-normal distribution because of the irregular likelihood behavior. In this paper, we investigate the testing problem concerning the skew parameter based on Bayes factor, which is just the ratio of the integrals of the likelihoods. Like Wilks phenomenon of the classical likelihood ratio, the asymptotic chi-square distributions of some linear functions of the logarithms of Bayes factors are derived. Then selecting appropriate prior, we can construct tests with asymptotically nominal significance level. At the same time, we give the local power of the tests, which is the same as the power of classical likelihood ratio tests. We also provide an exact test based on the Bayes factor with specific prior, which is competitive with existing tests. Simulation studies show that our tests are effective.

The effect of latent and error non-normality on corrections to the test statistic in structural equation modeling

In structural equation modeling, several corrections to the likelihood-ratio model test statistic have been developed to counter the effects of non-normal data. Previous robustness studies investigating the performance of these corrections typically induced non-normality in the indicator variables. However, non-normality in the indicators can originate from non-normal errors or non-normal latent factors. We conducted a Monte Carlo simulation to analyze the effect of non-normality in factors and errors on six different test statistics based on maximum likelihood estimation by evaluating the effect on empirical rejection rates and derived indices (RMSEA and CFI) for different degrees of non-normality and sample sizes. We considered the uncorrected likelihood-ratio model test statistic and the Satorra–Bentler scaled test statistic with Bartlett correction, as well as the mean and variance adjusted test statistic, a scale-shifted approach, a third moment-adjusted test statistic, and an approach drawing inferences from the relevant asymptotic chi-square mixture distribution. The results indicate that the values of the uncorrected test statistic—compared to values under normality—are associated with a severely inflated type I error rate when latent variables are non-normal, but virtually no differences occur when errors are non-normal. Although no general pattern regarding the source of non-normality for all analyzed measures of fit can be derived, the Satorra–Bentler scaled test statistic with Bartlett correction performed satisfactorily across conditions.

Estimation for a hybrid model of functional and linear measurement errors regression with missing response

In this paper, we consider least squares estimation and empirical likelihood inference for partial functional linear models when the covariates in the non-functional linear component are measured with additive error and the responses are missing at random. Asymptotic properties of the proposed estimators for the parametric and nonparametric components are established. A class of empirical log-likelihood ratio functions of the parametric component and response mean are developed, and the corresponding maximum empirical likelihood estimators are constructed. We also prove that the empirical log-likelihood ratio functions are asymptotic standard chi-squared distribution. The results can be used to construct confidence intervals for the parametric component and response mean. The asymptotic distributions of the corresponding maximum empirical likelihood estimators are also established. Meanwhile, a simulation study is conducted to demonstrate the finite sample performance of the proposed procedure. A real data analysis is also used to illustrate our methods.

Statistical Inference for the Heteroscedastic Partially Linear Varying-Coefficient Errors-in-Variables Model with Missing Censoring Indicators

In this paper, we focus on heteroscedastic partially linear varying-coefficient errors-in-variables models under right-censored data with censoring indicators missing at random. Based on regression calibration, imputation, and inverse probability weighted methods, we define a class of modified profile least square estimators of the parameter and local linear estimators of the coefficient function, which are applied to constructing estimators of the error variance function. In order to improve the estimation accuracy and take into account the heteroscedastic error, reweighted estimators of the parameter and coefficient function are developed. At the same time, we apply the empirical likelihood method to construct confidence regions and maximum empirical likelihood estimators of the parameter. Under appropriate assumptions, the asymptotic normality of the proposed estimators is studied. The strong uniform convergence rate for the estimators of the error variance function is considered. Also, the asymptotic chi-squared distribution of the empirical log-likelihood ratio statistics is proved. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators. Meanwhile, one real data example is provided to illustrate our methods.

Asymptotic distribution of the score test for detecting marks in hawkes processes

The score test is a computationally efficient method for determining whether marks have a significant impact on the intensity of a Hawkes process. This paper provides theoretical justification for use of this test. It is shown that the score statistic has an asymptotic chi-squared distribution under the null hypothesis that marks do not impact the intensity process. For local power, the asymptotic distribution against local alternatives is proved to be non-central chi-squared. A stationary marked Hawkes process is constructed using a thinning method when the marks are observations on a continuous time stationary process and the joint likelihood of event times and marks is developed for this case, substantially extending existing results which only cover independent and identically distributed marks. These asymptotic chi-squared distributions required for the size and local power of the score test extend existing asymptotic results for likelihood estimates of the unmarked Hawkes process model under mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for the exponential decay Hawkes process.