Proposed Test Statistic Research Articles

A High-Dimensional Cramér–von Mises Test

The Cramér–von Mises test provides a useful criterion for assessing goodness of fit in various problems. In this paper, we introduce a novel Cramér–von Mises-type test for testing distributions of high-dimensional continuous data. We establish an asymptotic theory for the proposed test statistics based on quadratic functions in high-dimensional stochastic processes. To estimate the limiting distribution of the test statistic, we propose two practical approaches: a plug-in calibration method and a subsampling method. Theoretical justifications are provided for both techniques. Numerical simulation also confirms the convergence of the proposed methods.

Pairwise likelihood estimation and limited-information goodness-of-fit test statistics for binary factor analysis models under complex survey sampling.

This paper discusses estimation and limited-information goodness-of-fit test statistics in factor models for binary data using pairwise likelihood estimation and sampling weights. The paper extends the applicability of pairwise likelihood estimation for factor models with binary data to accommodate complex sampling designs. Additionally, it introduces two key limited-information test statistics: the Pearson chi-squared test and the Wald test. To enhance computational efficiency, the paper introduces modifications to both test statistics. The performance of the estimation and the proposed test statistics under simple random sampling and unequal probability sampling is evaluated using simulated data.

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.

Diagnostic checking of periodic vector autoregressive time series models with dependent errors

In this article, we study the asymptotic behaviour of the residual autocorrelations for periodic vector autoregressive time series models (PVAR henceforth) with uncorrelated but dependent innovations (i.e., weak PVAR). We then deduce the asymptotic distribution of the Ljung–Box-McLeod modified Portmanteau statistics for weak PVAR models. In Monte Carlo experiments, we illustrate that the proposed test statistics have reasonable finite sample performance. When the innovations exhibit conditional heteroscedasticity or other forms of dependence, it appears that the standard test statistics (under independent and identically distributed innovations) are generally unreliable, overrejecting, or underrejecting severely, while the proposed test statistics offer satisfactory levels. The proposed methodology is employed in the analysis of two river flows.

Testing for Sufficient Follow-Up in Censored Survival Data by Using Extremes.

In survival analysis, it often happens that some individuals, referred to as cured individuals, never experience the event of interest. When analyzing time-to-event data with a cure fraction, it is crucial to check the assumption of "sufficient follow-up," which means that the right extreme of the censoring time distribution is larger than that of the survival time distribution for the noncured individuals. However, the available methods to test this assumption are limited in the literature. In this article, we study the problem of testing whether follow-up is sufficient for light-tailed distributions and develop a simple novel test. The proposed test statistic compares an estimator of the noncure proportion under sufficient follow-up to one without the assumption of sufficient follow-up. A bootstrap procedure is employed to approximate the critical values of the test. We also carry out extensive simulations to evaluate the finite sample performance of the test and illustrate the practical use with applications to leukemia and breast cancer datasets.

A bias-corrected Srivastava-type test for cross-sectional independence

This paper proposes a test for cross-sectional independence with high dimensional panel data. It uses the random matrix theory based approach of Srivastava (2005) in the presence of a large number of cross-sectional units and time series observations. Because the errors are unobservable, the residuals from the regression model for panel data are used. We develop a bias-corrected test after adjusting for the contribution from the regressors. With the aid of the martingale central limit theorem, we prove that the limiting null distribution of the proposed test statistic is normal under mild conditions as cross-sectional dimension and time dimension go to infinity together. We further study the asymptotic relative efficiency of our proposed test with respect to the state-of-art Lagrange multiplier test. An interesting finding is that the newly proposed test can have substantial power gain when the underlying variance magnitudes are not identical across different units.

Conditional symmetry test based on empirical characteristic function

In this paper, we propose a method for testing the conditional symmetry of multivariate random variables, specifically testing whether or not the conditional distribution of one random variable is symmetric around zero given another. Both the conditional symmetry test for observable data and unobserved error terms in parametric models are considered. The proposed test statistics are built on the weighted L 2 -distance between empirical characteristic functions and require no distributional assumption about the data distribution. The asymptotic properties of the test statistics under the null and alternative hypotheses are investigated. Since the limiting null distributions are intractable, a nonparametric Monte Carlo resampling method is used to determine critical values. Simulation results show the good performance of the proposed method in various scenarios. Two real datasets are analysed to illustrate the practical usefulness of the proposed tests.

Reliability test for degradation data based on ranked set sampling

AbstractIn this article, we consider test for the two null hypotheses for and , two widely useful tests in reliability, based on ranked set sampling (RSS). We derive the likelihood ratio test as well as the associated exact and asymptotic results. Considering a fixed significance level and power of the test, we show that the proposed test statistic outperforms the existing test. In small sample cases, the proposed test leads to a much narrower confidence interval for the reliability function . Then, the test statistics obtained from simple random sampling and RSS schemes are compared through which, the efficiency of using RSS is demonstrated. For illustration, we apply the proposed test to a degradation data from the reliability literature. Upon using RSS, the cost of measurement gets reduced and efficiency gets improved, suggesting the importance and use of RSS data in reliability experiments and their design.

A Robust High-Dimensional Test for Two-Sample Comparisons

The Hotelling T2 statistic is used to compare the mean vectors of two independent multivariate Gaussian distributions. Nevertheless, this statistic is highly sensitive to outliers and is not suitable for high-dimensional datasets where the number of variables exceeds the sample size. This study introduces a robust permutation test based on the minimum regularized covariance determinant (MRCD) estimator to address these limitations of the two-sample Hotelling T2 statistic. Simulation studies were performed to evaluate the proposed test’s empirical size, power, and robustness. Additionally, the test was applied to both uncontaminated and contaminated Alzheimer’s Disease datasets. The findings from the simulations and real data examples provide clues that the proposed test can be effectively used with high-dimensional data without being impacted by outliers. Finally, an R function within the “MVTests” package was developed to implement the proposed test statistic on real-world data.

Projection tests for regression coefficients in high-dimensional partial linear models

. To check the significance of the regression coefficients in the linear component of high-dimensional partial linear models, we proposed some projection-based test statistics. These test statistics are connected with U-statistics of order two and they are applicable for diverging dimensions and heteroscedastic model errors. By using the martingale central limit theorem, we show the asymptotic normalities of the proposed test statistics under the null hypothesis and local alternative hypotheses. The performance of test statistics are evaluated by simulation studies. The simulation results show that the proposed test statistics are powerful and have the correct type-I error asymptotically under the null hypothesis.

A class of nonparametric tests for the two-sample problem based on order statistics

In this paper, a new class of distribution-free statistics based on order statistics from two independent samples is introduced to test the equality of two continuous distributions. The null distributions are obtained, and the symmetry property of the proposed test statistics is studied. The exact power functions of the new class of tests under the Lehmann alternative family are derived. Extensions of the proposed class of test statistics to situations with unequal sample sizes are discussed. We also study the power performance of the proposed test under the location-shift and scale-shift alternatives using Monte Carlo simulations. We observe that the power performance of the new class of test procedures is comparable to some commonly used nonparametric tests under various scenarios, which indicates that the new class of test procedures can be a good alternative to those classic nonparametric tests for the two-sample problem.

Two-sample test for high-dimensional covariance matrices: A normal-reference approach

Testing the equality of the covariance matrices of two high-dimensional samples is a fundamental inference problem in statistics. Several tests have been proposed but they are either too liberal or too conservative when the required assumptions are not satisfied which attests that they are not always applicable in real data analysis. To overcome this difficulty, a normal-reference test is proposed and studied in this paper. It is shown that under some regularity conditions and the null hypothesis, the proposed test statistic and a chi-squared-type mixture have the same limiting distribution. It is then justified to approximate the null distribution of the proposed test statistic using that of the chi-squared-type mixture. The distribution of the chi-squared-type mixture can be well approximated using a three-cumulant matched chi-squared-approximation with its approximation parameters consistently estimated from the data. The asymptotic power of the proposed test under a local alternative is also established. Simulation studies and a real data example demonstrate that the proposed test works well in general scenarios and outperforms the existing competitors substantially in terms of size control.

Adjusted location‐invariant U‐tests for the covariance matrix with elliptically high‐dimensional data

AbstractThis paper analyzes several covariance matrix U‐tests, which are constructed by modifying the classical John‐Nagao and Ledoit‐Wolf tests, under the elliptically distributed data structure. We study the limiting distributions of these location‐invariant test statistics as the data dimension may go to infinity in an arbitrary way as the sample size does. We find that they tend to have unsatisfactory size performances for general elliptical population. This is mainly because such population often possesses high‐order correlations among their coordinates. Taking such kind of dependency into consideration, we propose necessary corrections for these tests to cope with elliptically high‐dimensional data. For computational efficiency, alternative forms of the new test statistics are also provided. We derive the universal asymptotic null distributions of the proposed test statistics under elliptical distributions and beyond. The powers of the proposed tests are further investigated. The accuracy of the tests is demonstrated by simulations and an empirical study.

Testing independence based on Spearman’s footrule in high dimensions

. Spearman’s footrule is a widely used tool to discover the relationship between two continuous random variables, mainly because of its robustness, natural interpretation, and simplicity of calculation. In this article, we propose a new test statistic based on the sum of squares of pairwise Spearman’s footrule, to test the mutual independence of a high dimensional random vector. The large sample properties of the proposed test method are established. Simulation results show that the new test has higher power than existing test methods under some circumstances. An example with real life data is discussed to illustrate the usefulness and effectiveness of the proposed test statistic.

Group inference of high-dimensional single-index models

For the supervised and semi-supervised settings, a group inference method is proposed for regression parameters in high-dimensional semi-parametric single-index models with an unknown random link function. The inference procedure is based on least squares, which can be extended to other general convex loss functions. The proposed test statistics are weighted quadratic forms of the regression parameter estimates, in which the weight could be a non-random matrix or the sample covariance matrix of the covariates. The proposed method could detect dense but weak signals and deal with high correlation of covariates inside the group. A ‘contaminated test statistic’ is established in the semi-supervised regime to decrease the variance. The asymptotic properties of the resulting estimators are established. The finite-sample behaviour of the proposed method is evaluated through extensive simulation studies. Applications to two genomic datasets are provided.

Specifications tests for count time series models with covariates

AbstractWe propose a goodness-of-fit test for a class of count time series models with covariates which includes the Poisson autoregressive model with covariates (PARX) as a special case. The test criteria are derived from a specific characterization for the conditional probability generating function, and the test statistic is formulated as a $$L_2$$ L 2 weighting norm of the corresponding sample counterpart. The asymptotic properties of the proposed test statistic are provided under the null hypothesis as well as under specific alternatives. A bootstrap version of the test is explored in a Monte–Carlo study and illustrated on a real data set on road safety.

Testing ARCH effect of high-dimensional time series data

Testing for the autoregressive conditional heteroscedasticity (ARCH) effect is an important problem in economic and financial time series. In this article, we consider the ARCH effect test for high-dimensional time series. Our test statistic is built based on the maximum element of Spearman’s rank correlation at different lags. The bootstrap methods are used to approximate the null distribution of the proposed test statistics. We also consider the goodness-of-fit test for ARCH models by applying the test to the residuals of fitted models. Simulation results show that the proposed test statistics enjoy good properties of size in finite samples. Compared to the existing methods, the new test possesses much better power. Moreover, the test is not sensitive to the underlying distribution of the time series. The proposed method is illustrated via two real data.

Inference for high-dimensional linear expectile regression with de-biasing method

The methodology for the inference problem in high-dimensional linear expectile regression is developed. By transforming the expectile loss into a weighted-least-squares form and applying a de-biasing strategy, Wald-type tests for multiple constraints within a regularized framework are established. An estimator for the pseudo-inverse of the generalized Hessian matrix in high dimension is constructed using general amenable regularizers, including Lasso and SCAD, with its consistency demonstrated through a novel proof technique. Simulation studies and real data applications demonstrate the efficacy of the proposed test statistic in both homoscedastic and heteroscedastic scenarios.

Fisher matrix based fault detection for PMUs data in power grids

In this paper, using the data collected from phasor measurement units (PMUs), two methods based on Fisher random matrix are proposed to detect faults in power grids. Firstly, the fault detection matrix is constructed and the event detection problem is reformatted as a testing problem for two-sample covariance matrices, which is related with the so-called Fisher matrix. To save computing resources, the screening step of fault interval based on the test statistic is designed to check the existence of faults. Then two point-by-point methods are proposed to determine the time of the fault in the selected interval. One method detects faults by the limiting spectral distribution of the standard Fisher matrix, which can detect the faults with higher accuracy. The other method tests the faults based on the proposed test statistic , which has a faster detection speed. Compared with existing works, the simulation results illustrate that two methods proposed in this paper cost less computational time and provide a higher degree of accuracy and sensitivity.

Testing distribution for multiplicative distortion measurement errors

. In this article, we study a goodness of fit test for a multiplicative distortion model under a uniformly distributed but unobserved random variable. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. The proposed k-th power test statistic is based on logarithmic transformed observations and a correlation coefficient-based estimator without distortion measurement errors. The proper choice of k is discussed through the empirical coverage probabilities. The asymptotic null distribution of the test statistics are obtained with known asymptotic variances. Next, we proposed the conditional mean calibrated test statistic when a variable is distorted in a multiplicative fashion. We conduct Monte Carlo simulation experiments to examine the performance of the proposed test statistics.