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Datasets that are big with regard to their volume, variety and velocity are becoming increasingly common. However, limitations in computer processing can often restrict analysis performed on them. Nonuniform subsampling methods are effective in reducing computational loads for massive data. However, the variance of the estimator of nonuniform subsampling methods becomes large when the subsampling probabilities are highly heterogenous. To this end, we develop two new algorithms to improve the estimation method for massive data logistic regression based on a chosen hard threshold value and combining subsamples, respectively. The basic idea of the hard threshold method is to carefully select a threshold value and then replace subsampling probabilities lower than the threshold value with the chosen value itself. The main idea behind the combining subsamples method is to better exploit information in the data without hitting the computation bottleneck by generating many subsamples and then combining estimates constructed from the subsamples. The combining subsamples method obtains the standard error of the parameter estimator without estimating the sandwich matrix, which provides convenience for statistical inference in massive data, and can significantly improve the estimation efficiency. Asymptotic properties of the resultant estimators are established. Simulations and analysis of real data are conducted to assess and showcase the practical performance of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract.The bilinear model has been frequently used in fields like control theory and economics to model seismic data. In this article, time-functional variance (TFV) noises are embedded into a specific bilinear model. We propose a generalized autoregressive conditional heteroskedasticity-type maximum likelihood estimator (GMLE) with a sieve method and then provide various inferential techniques based on this GMLE. It is shown that under the finite fourth moment of errors, the GMLE is consistent and asymptotically normally distributed. A simulation study and analysis of real data are additionally carried out to evaluate GMLE’s finite sample performance. [ABSTRACT FROM AUTHOR]

Summary: Data related to the counting of elements of variable character are frequently encountered in time series studies. This paper brings forward a new class of k$$ k $$th‐order dependence‐driven random coefficient mixed thinning integer‐valued autoregressive time series model (DDRCMTINAR(k$$ k $$)) to deal with such data. Stationarity and ergodicity properties of the proposed model are derived in detail. The unknown parameters are estimated by conditional least squares, and modified quasi‐likelihood and asymptotic normality of the obtained parameter estimators is established. The performances of the adopted estimate methods are checked via simulations, which present that modified quasi‐likelihood estimators perform better than the conditional least squares considering the proportion of within‐Ω$$ \Omega $$ estimates in certain regions of the parameter space. The validity and practical utility of the model are investigated by epileptic seizure data and COVID‐19 data of suspected cases in China. [ABSTRACT FROM AUTHOR]

Likelihood-based inference under nonconvex constraints on model parameters has become increasingly common in biomedical research. In this paper, we establish large-sample properties of the maximum likelihood estimator when the true parameter value lies at the boundary of a nonconvex parameter space. We further derive the asymptotic distribution of the likelihood ratio test statistic under nonconvex constraints on model parameters. A general Monte Carlo procedure for generating the limiting distribution is provided. The theoretical results are demonstrated by five examples in Anderson's stereotype logistic regression model, genetic association studies, gene-environment interaction tests, cost-constrained linear regression and fairness-constrained linear regression. [ABSTRACT FROM AUTHOR]

Existing methods for fitting spatial autoregressive models have various strengths and weaknesses. For example, the maximum likelihood estimation (MLE) approach yields efficient estimates but is computationally burdensome. Computationally efficient methods, such as generalized method of moments (GMMs) and spatial two‐stage least squares (2SLS), typically require exogenous covariates to be significant, a restrictive assumption that may fail in practice. We propose a new estimating equation approach, termed combined moment equation (COME), which combines the first moment with covariance conditions on the residual terms. The proposed estimator is less computationally demanding than MLE and does not need the restrictive exogenous conditions as required by GMM and 2SLS. We show that the proposed estimator is consistent and establish its asymptotic distribution. Extensive simulations demonstrate that the proposed method outperforms the competitors in terms of bias, efficiency, and computation. We apply the proposed method to analyze an air pollution study, and obtain some interesting results about the spatial distribution of PM2.5 concentrations in Beijing. [ABSTRACT FROM AUTHOR]

We consider the most basic concept of bivariate symmetry, referred to as general symmetry or conditional symmetry and provide a necessary condition for it based on the joint probability density function and the associated cumulative distribution function. This condition is then used to provide a test of general bivariate symmetry by using the sample analog of the necessary condition as a test statistic. We derive the asymptotic distribution of the test statistic under the null hypothesis and exhibit the performance of the proposed test through simulations. The test is then applied to a couple of real datasets, one coming from an insurance company and the other from a medical trial of Mayo Clinic. [ABSTRACT FROM AUTHOR]

Inverse Weibull, lognormal, and inverse Gaussian are some commonly used statistical distributions for modeling positively skewed failure lifetime data. These distributions share some interesting properties among themselves like they all have uni‐modal hazard rates. In this paper, we address the problem of discriminating among these statistical distributions to consider a more appropriate lifetime model. We first consider the complete samples and make use of the maximized log‐likelihood approach for choosing the correct model. We also obtain the expressions for logarithmic of defined test statistics, and associated asymptotic distributions. We then extend our discussion based on the observed sample in the presence of some censoring. We perform a simulation study in both cases to compare the probabilities of correct selection. Furthermore, for a given probability of correct selection and user‐specified protection level, we present a discussion to determine the minimum sample size required to discriminate among the three lifetime models. Finally, two real data sets are analyzed to illustrate the proposed methodology. In our findings, we observed that model parameters and methods of estimating unknown parameters play very important roles in the discrimination process, and sample size and the proportion of censoring become key factors to ensure a high probability of selecting a correct model. [ABSTRACT FROM AUTHOR]

AbstractThis article studies the CUSUM estimators of the mean change-point based on the association errors and obtains some important consistency results. For example, the weak and strong convergence rates as well as the limit distribution are obtained for the estimators of the mean change-point and mean parameter. We extend the existing results of mean change points, and give some simulations with Normality and multivariate t distribution errors. As an application, two real data examples are provided to illustrate the analysis of the mean change-point. [ABSTRACT FROM AUTHOR]

The paper treats the modeling of stationary multivariate stochastic processes via frequency domain, and extends the notion of generalized autocovariance function, given by Proietti and Luati (2015) for univariate time series, to the multivariate setting. The generalized autocovariance matrices are defined for stationary multivariate stochastic processes as the Fourier transform of the power transformation of the spectral density matrix. Then we prove the consistency and derive the asymptotic distribution of frequency domain non-parametric estimators of the generalized autocovariance matrices, based on the power transformation of the periodogram matrix. Generalized autocovariance matrices are used to construct white noise hypothesis testing, to discriminate stochastic processes, and to introduce a generalized Yule–Walker estimator for the spectrum. A so-called λ–squared distance between two multivariate stochastic processes is also defined by using their generalized autocovariance matrices, and it serves for clustering time series and estimation by feature matching. Another use is in discriminant analysis. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Bayesian estimation of the parameters in the geometric process (GP) in which the first occurrence time of an event is assumed to have Weibull distribution. The Bayesian estimators are derived based on both symmetric (Squared Error) and asymmetric (General Entropy, LINEX) loss functions. Since the Bayesian estimators of unknown parameters cannot be obtained analytically, Lindley's approximation and the Markov Chain Monte Carlo (MCMC) methods are applied to compute the Bayesian estimates. Furthermore, by using the MCMC methods, credible intervals of the parameters are constructed. Maximum likelihood (ML) estimators are also derived for unknown parameters. The confidence intervals of the parameters are obtained based on an asymptotic distribution of ML estimators. Moreover, the performances of the proposed Bayesian estimators are compared with the corresponding ML, modified moment and modified maximum likelihood estimators through an extensive simulation study. Finally, analyses of two different real data sets are presented for illustrative purposes. [ABSTRACT FROM AUTHOR]

The parameters of generalized linear models (GLMs) are usually estimated by the maximum likelihood estimator (MLE) which is known to be asymptotically efficient. But the MLE is computed using a Newton-Raphson-type algorithm which is time-consuming for a large number of variables or modalities, or a large sample size. An alternative closed-form estimator is proposed in this paper in the case of categorical explanatory variables. Asymptotic properties of the alternative estimator is studied. The performances in terms of both computation time and asymptotic variance of the proposed estimator are compared with the MLE for a Gamma distributed GLM. [ABSTRACT FROM AUTHOR]

In this article, we consider the asymptotic distribution of residual distribution estimator in the first order autoregressive models with positively associated or negatively associated random errors. Under mild regularity assumptions, some asymptotic normality results for residual distribution estimator are obtained when the autoregressive model is a stationary process or an explosive process. In addition, some simulations of estimated curves and mean integrated square errors are illustrated, which agree with our theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we consider the k-Nearest Neighbors (k-NN) method in a single index regression model when the explanatory variable is valued in functional space in the setting of the quasi-association dependence condition. The main result of this work is the establishment of the asymptotic distribution for the k-NN kernel single index estimator. These results are established under fairly general conditions on the underlying models. As an application, the asymptotic confidence bands for the regression model based on the single-index model are presented. Some simulation studies are carried out to show the finite sample performances of the k-NN estimator. [ABSTRACT FROM AUTHOR]

In this paper, the empirical likelihood and estimations problems in varying coefficient models with right censored data are investigated by using a bias correction method. Naive and residual-adjusted empirical likelihood ratios and estimators of the coefficient functions are constructed, their asymptotic distributions are obtained, and a consistent estimator of the asymptotic variance is given. The obtained results can be directly used to construct the confidence regions of the coefficient functions. Furthermore, a new method for constructing pointwise confidence intervals and simultaneous confidence bands for each coefficient function is also proposed. The proposed method is to directly calibrate the empirical log-likelihood ratio so that the obtained ratio is asymptotically chi-squared. Undersmoothing of each coefficient function is avoided, and existing data-driven methods can also effectively select the optimal bandwidth. Simulation study is performed to compare the empirical likelihood with the normal approximation-based method. An example of real data is used to illustrate our approach. [ABSTRACT FROM AUTHOR]

When large amounts of data continuously arrive in streams, online updating is an effective way to reduce storage and computational burden. The key idea of online updating is that the previous estimators are sequentially updated only using the current data and summary statistics of historical data. In this article, we develop a renewable learning method for the multiplicative regression model with streaming data, where the parameter estimator based on a least product relative error (LPRE) criterion is renewed without revisiting historical data. Under some regularity conditions, we establish the consistency and asymptotic normality of the renewable estimator. Moreover, our proposed renewable estimator has an identical asymptotic distribution with that of the full data LPRE estimator. Numerical studies and two real-world datasets are provided to evaluate the performance of our proposed method. [ABSTRACT FROM AUTHOR]

The log-normal and the log-logistic distributions are two of the most commonly used distributions for studying positively skewed lifetime data. Both the distributions share number of interesting properties, and for a certain range of parameters their cumulative and hazard functions can also be similar in nature. However, selecting a more appropriate distribution and discriminating among them for a given data to best fit is an important issue. Further, when the data are observed in the presence of some censoring scheme the problem becomes more challenging. In this paper, we address the problem of selecting a more appropriate distribution by discriminating based on the random samples drawn in the presence of type-II censoring. We consider the difference of the maximized log-likelihood functions, and compute the asymptotic distribution of the discrimination statistic. We further propose a modified discriminating approach, and compute the probabilities of correct selection to check the performance of the discrimination procedure. Finally, simulation study is conducted, and two real data sets are analysed for the illustration purpose. [ABSTRACT FROM AUTHOR]

This paper extends the threshold regression to threshold effect in varying coefficient model. We allow for either cross-section or time series observations. Estimation of the regression parameters is considered. An asymptotic distribution theory for the regression estimates (the threshold and the regression slopes) is developed. The distribution of threshold estimates is found to be non-standard. Under some sufficient conditions, we show that the proposed estimator for regression slopes is root-n consistent and asymptotically normally distributed, and that the proposed estimator for the varying coefficient is consistent and also asymptotically normal distributed but at a rate slower than root-n. Consistent estimators for the asymptotic variances of the proposed estimators are provided. Monte Carlo simulations are presented to assess the performance of the asymptotic approximations. The empirical relevance of the theory is illustrated through an application to the relationship between environmental regulation and regional technological innovation study. [ABSTRACT FROM AUTHOR]

In this article, we develop additive autoregressive models (Add‐ARM) for the time series data with matrix valued predictors. The proposed models assume separable row, column and lag effects of the matrix variables, attaining stronger interpretability when compared with existing bilinear matrix autoregressive models. We utilize the Gershgorin's circle theorem to impose some certain conditions on the parameter matrices, which make the underlying process strictly stationary. We also introduce the alternating least squares estimation method to solve the involved equality constrained optimization problems. Asymptotic distributions of the parameter estimators are derived. In addition, we employ hypothesis tests to run diagnostics on the parameter matrices. The performance of the proposed models and methods is further demonstrated through simulations and real data analysis. [ABSTRACT FROM AUTHOR]

Winding number and density of states are two fundamental physical quantities for non-self-adjoint quasi-periodic Schrödinger operators, which reflect the asymptotic distribution of zeros of the characteristic determinants of the truncated operators under Dirichlet boundary condition, with respect to complexified phase and the energy, respectively. We will prove that the winding number is in fact Avila's acceleration, it is also closely related to the density of states by a generalized Thouless formula for non-self-adjoint Schrödinger operators and Avila's global theory. Communicating Editor: Prof Svetlana Jitomirskaya [ABSTRACT FROM AUTHOR]

Missing data in covariates can result in biased estimates and loss of power to detect associations. We consider Cox regression in which some covariates are subject to missing. The inverse probability weighted approach is often applied to regression analysis with missing covariates. Inverse probability weighted estimators typically are less efficient than likelihood‐based estimators, but in general are more robust against model misspecification. In this article, we propose a robust best linear weighted estimator for Cox regression with missing covariates. Our proposed estimator is the projection of the simple inverse probability weighted estimator onto the orthogonal complement of the score space based on a working regression model of the observed data. The efficiency gain is from the use of the association between the survival outcome variable and the available covariates, which is the working regression model. The asymptotic distribution is derived, and the finite sample performance of the proposed estimator is examined via extensive simulation studies. The methods are applied to a colorectal cancer study to assess the association of the microsatellite instability status with colorectal cancer‐specific mortality. [ABSTRACT FROM AUTHOR]

We consider the balanced two-way mixed effects design with some empty cells. A test procedure for the hypothesis of no main fixed effects is developed under violation of the assumption of variance homogeneity and symmetry. The asymptotic null distribution of the test statistics is studied under the condition that the number of levels of the random effects tends to infinity as both the number of complete and incomplete observations tend to infinity. An illustrative example is given. [ABSTRACT FROM AUTHOR]

This study examines interpoint distance-based tests for random elements from probability distributions in infinite-dimensional space. Some asymptotic properties such as the limiting distributions and the asymptotic power of the Biswas–Ghosh type test (Biswas and Ghosh in J Multivar Anal 123: 160–171, 2014) are presented using the theory of U-statistic. In addition, the p-value approximation based on the jackknife variance estimators and the Welch–Satterthwaite equation is proposed. Simulation studies are conducted to evaluate the performance of the proposed test statistics in functional data analysis. The proposed tests are shown to have better power than the existing method for functional data in some situations. [ABSTRACT FROM AUTHOR]

AbstractMotivated by the pressing needs for dissecting heterogeneous relationships in gene expression data, here we generalize the squared Pearson correlation to capture a mixture of linear dependences between two real-valued variables, with or without an index variable that specifies the line memberships. We construct the generalized Pearson correlation squares by focusing on three aspects: variable exchangeability, no parametric model assumptions, and inference of population-level parameters. To compute the generalized Pearson correlation square from a sample without a line-membership specification, we develop a K-lines clustering algorithm to find K clusters that exhibit distinct linear dependences, where K can be chosen in a data-adaptive way. To infer the population-level generalized Pearson correlation squares, we derive the asymptotic distributions of the sample-level statistics to enable efficient statistical inference. Simulation studies verify the theoretical results and show the power advantage of the generalized Pearson correlation squares in capturing mixtures of linear dependences. Gene expression data analyses demonstrate the effectiveness of the generalized Pearson correlation squares and the K-lines clustering algorithm in dissecting complex but interpretable relationships. The estimation and inference procedures are implemented in the R package gR2 (https://github.com/lijy03/gR2). Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

Contingency tables, data represented as counts matrices, are ubiquitous across quantitative research and data-science applications. Existing statistical tests are insufficient however, as none are simultaneously computationally efficient and statistically valid for a finite number of observations. In this work, motivated by a recent application in reference-free genomic inference [K. Chaung et al., Cell 186, 5440-5456 (2023)], we develop Optimized Adaptive Statistic for Inferring Structure (OASIS), a family of statistical tests for contingency tables. OASIS constructs a test statistic which is linear in the normalized data matrix, providing closed-form P-value bounds through classical concentration inequalities. In the process, OASIS provides a decomposition of the table, lending interpretability to its rejection of the null. We derive the asymptotic distribution of the OASIS test statistic, showing that these finite-sample bounds correctly characterize the test statistic's P-value up to a variance term. Experiments on genomic sequencing data highlight the power and interpretability of OASIS. Using OASIS, we develop a method that can detect SARS-CoV-2 and Mycobacterium tuberculosis strains de novo, which existing approaches cannot achieve. We demonstrate in simulations that OASIS is robust to overdispersion, a common feature in genomic data like single-cell RNA sequencing, where under accepted noise models OASIS provides good control of the false discovery rate, while Pearson's X² consistently rejects the null. Additionally, we show in simulations that OASIS is more powerful than Pearson's X² in certain regimes, including for some important two group alternatives, which we corroborate with approximate power calculations. [ABSTRACT FROM AUTHOR]

The purpose of this work is to perform the generalized secrecy capacity (i.e., secrecy rate and secrecy outage capacity) analysis of the scheduled transmit antenna selection scheme over non-identical Nakagami fading channels. Especially, this work considers maximal ratio combining (MRC) and selection combining (SC) at the authorized receiver and eavesdropper. To derive the secrecy capacity, this work derives accurate and asymptotic distribution functions (i.e., probability density function and cumulative distribution function) of the received signal to noise ratio (SNR) of the proposed scheme using MRC and SC receivers over non-identical Nakagami fading channels. Using the accurate distribution functions, this work derives the accurate average secrecy rate and tight average secrecy outage capacity of the proposed scheme for various combining scenarios over non-identical Nakagami fading channels. Using the asymptotic distribution functions, this work derives the asymptotic average secrecy capacity and quantifies the high SNR slope and the high SNR power offset for the average secrecy capacity over non-identical Nakagami fading channels. From numerical results, this work shows that the high SNR slope for the average secrecy capacity is one regardless of combining type. It means that neither the authorized parameters nor the eavesdropper parameters affect the high SNR slope over non-identical Nakagami fading channels. On the contrary, the high SNR power offset improves proportionally to the authorized parameters and inversely proportional to the eavesdropper parameters over non-identical Nakagami fading channels. [ABSTRACT FROM AUTHOR]

The representability of the Burr distribution as a mixture of Weibull distribution is studied in order to justify its utility for modelling the statistical regularities in extreme values registered in non-stationary flows of informative events. A result of [24] is improved by extending the domain of admissible values of the parameters which provide the representability of the (generalized) Burr distribution as a scale mixture of the Weibull distribution. This result gives an argument in favour of application of the Burr distribution as a model of statistical regularities of extreme values registered within moderate regular time intervals, say, daily (short-term) extremes. In turn, if we are interested in the statistical regularities of the behaviour of the absolute extreme observation over a long period, say, a decade (the long-term extreme), then it can be noted that the daily extreme values form a sample of the Burr-distributed random variables. As is known, the Burr distribution belongs to the domain of max-attraction of the Fréchet distribution. The problem of improving the accuracy of the approximation of the distribution of the absolute extreme by the Fréchet distribution by the construction of an asymptotic expansion for the distribution of the extreme order statistics in the sample of independent identically Burr-distributed random variables is also considered. These results are illustrated by an example of fitting the Burr distribution to the data representing the extreme values of characteristics of the interplanetary magnetic field. [ABSTRACT FROM AUTHOR]

Many real-world networks exhibit the phenomenon of edge clustering,which is typically measured by the average clustering coefficient. Recently,an alternative measure, the average closure coefficient, is proposed to quantify local clustering. It is shown that the average closure coefficient possesses a number of useful properties and can capture complementary information missed by the classical average clustering coefficient. In this paper, we study the asymptotic distribution of the average closure coefficient of a heterogeneous Erdős–Rényi random graph. We prove that the standardized average closure coefficient converges in distribution to the standard normal distribution. In the Erdős–Rényi random graph,the variance of the average closure coefficient exhibits the same phase transition phenomenon as the average clustering coefficient. [ABSTRACT FROM AUTHOR]

The random coefficient autoregressive (RCAR) processes are very useful to model time series in applications. It is commonly observed that the random autoregressive coefficient is assumed to be an independent identically distributed (i.i.d.) random variable sequence. To make the RCAR model more practical, this paper considers a new RCAR(1) model driven by explanatory variable and observations. We use the conditional least squares, the quantile regression and the conditional maximum likelihood methods to estimate the model parameters. The consistency and asymptotic normality of the proposed estimates are established. Simulation studies are conducted for the evaluation of the developed approaches and two applications to real-data examples are provided. The results show that the proposed procedures perform well for the simulations and application. [ABSTRACT FROM AUTHOR]

In this paper we extend the family of test proposed by Majumder and Mitra [Detecting trend change in failure functions-an L-statistic approach. Stat Pap. 2019;62:31–52. doi: ] for testing exponentiality against bathtub (BT) failure rate (upside down bathtub (UBT)) to randomly right censored case. We derive the asymptotic distribution of the test statistic and using Monte Carlo simulation we obtain the empirical powers for certain alternative distributions. As a special case the asymptotic distribution of the test statistic proposed by Park [Testing whether failure rate changes its trend. IEEE Trans Reliab. 1988;37(4):375–378. doi: ] is obtained. Finally some examples of real life data are given for illustrative purpose. [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 object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of such models related to connectivity structures have been studied extensively. In this article, we study the random graph from the perspective of extreme value theory and focus on the occurrence of single long edges. The model we investigate has non-periodic boundary and is parameterized by a positive constant α, which is the power for the polynomial decay of the probabilities determining the presence of an edge. As a main result, we provide a precise description of the magnitude of the longest edge in terms of asymptotic behavior in distribution. Thereby we illustrate a crucial dependence on the power α and we recover a phase transition which coincides with exactly the same phases in Benjamini and Berger. [ABSTRACT FROM AUTHOR]

Let { A n } n be a sequence of square matrices such that size (A n) = d n → ∞ as n → ∞. We say that { A n } n has an asymptotic spectral distribution described by a Lebesgue measurable function f : D ⊂ R k → C if, for every continuous function F : C → C with bounded support, lim n → ∞ ⁡ 1 d n ∑ i = 1 d n F (λ i (A n)) = 1 μ k (D) ∫ D F (f (x)) d x , where μ k is the Lebesgue measure in R k and λ 1 (A n) , ... , λ d n (A n) are the eigenvalues of A n. In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices A n arising from the discretization of differential equations, whose size d n diverges to ∞ along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as "Which is the largest set of test functions F for which the above limit relation is satisfied?" is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called "inner outliers" of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures. [ABSTRACT FROM AUTHOR]

In this article, we introduce a novel discrepancy called the maximum variance discrepancy for the purpose of measuring the difference between two distributions in Hilbert spaces that cannot be found via the maximum mean discrepancy. We also propose a two-sample goodness of fit test based on this discrepancy. We obtain the asymptotic null distribution of this two-sample test, which provides an efficient approximation method for the null distribution of the test. [ABSTRACT FROM AUTHOR]

This article deals with the asymptotic distribution for the extremely heavy-tailed distributions with infinite mean or variance by using a truncated sample mean. We obtain three necessary and sufficient conditions under which the asymptotic distribution of the truncated test statistics converges to normal, neither normal nor stable or converges to − ∞ or the combination of stable distributions, respectively. The numerical simulation illustrates an application of the theoretical results above in the hypothesis testing. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the asymptotic form of the joint distribution of maxima and minima of independent observations, when the sample size is a random variable. We also discuss the asymptotic distribution of the range. [ABSTRACT FROM AUTHOR]

Many financial time series not only have varying structures at different quantile levels and exhibit the phenomenon of conditional heteroscedasticity at the same time but also arrive in the stream. Quantile double‐autoregression is very useful for time series analysis but faces challenges with model fitting of streaming data sets when estimating other quantiles in subsequent batches. This article proposes a renewable estimation method for quantile double‐autoregression analysis of streaming time series data due to its ability to break with storage barrier and computational barrier. Moreover, the proposed flexible parametric structure of the quantile function enables us to predict any interested quantile value without quantile curve crossing problem or keeping the desirable monotone property of the conditional quantile function. The proposed methods are illustrated using current data and the summary statistics of historical data. Theoretically, the proposed statistic is shown to have the same asymptotic distribution as the standard version computed on an entire data stream with the data batches pooled into one data set, without additional condition. Simulation studies and an empirical example are presented to illustrate the finite sample performance of the proposed methods. [ABSTRACT FROM AUTHOR]

AbstractThis work investigates the problem of testing whether d functional random variables are jointly independent, using a modified estimator of the d-variable Hilbert Schmidt Indepedence Criterion (dHSIC). We then get the asymptotic normality of this estimator both under the joint independence hypothesis and under the alternative hypothesis. A simulation study shows the good performance of the proposed test on a finite sample. [ABSTRACT FROM AUTHOR]

In this paper, we proposed a parametric hypothesis test of the multiplicative distortion model under the exponentially distributed but unobserved random variable. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. Firstly, some new test statistics are proposed to checking the exponential distribution assumption without distortion effects. Next, we proposed several test statistics when the variable is distorted in the multiplicative fashion. For the latter, the proposed test statistics automatically eliminate the distortion effects involved in the unobserved variable. The proposed test statistics with or without distortions are all asymptotical free, and the asymptotic null distribution of the test statistics are obtained with known asymptotic variances. We conduct Monte Carlo simulation experiments to examine the performance of the proposed test statistics. These methods are applied to analyze four real datasets for illustrations. [ABSTRACT FROM AUTHOR]

Parameter estimation of chirp signal model with time-varying amplitude and additive noise is a critical problem in statistical signal processing. A statistically efficient algorithm is proposed to estimate the nonlinear parameters of frequency rate (FR) and initial frequency (IF). Initial estimates are first deduced from the ambiguity function (AF) and then refined through a statistics-based iterative procedure that improves their convergence rates from O p (N − 1) and O p (N − 2) to O p (N − 3 / 2) and O p (N − 5 / 2) , respectively. It is worth noting that the convergence rates of FR and IF double while being controlled by each other at each iteration, causing the convergence rates of FR and IF to jump up and down before the algorithm converges. Four iterative processes are considered involving crossovers between two types of initial estimators and two types of iterative coefficients. The asymptotic distribution of the refined estimators is obtained and demonstrated to be bivariate normal, and the Cramér–Rao lower bound (CRLB) is derived for each. Monte Carlo simulations were performed to verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the three-dimensional standard flat torus, in particular the asymptotic distribution of the nodal intersection length against a fixed regular reference surface. Expectation and variance have been addressed by Maffucci [Ann. Henri Poincaré 20(11), 3651–3691 (2019)] who found that the expected length is proportional to the square root of the eigenvalue times the area of the surface, while the asymptotic variance only depends on the geometry of the surface, the projected lattice points being equidistributed on the two-dimensional unit sphere in the high-energy limit. He also noticed that there are "special" surfaces, so-called static, for which the variance is of smaller order; however he did not prescribe the precise asymptotic law in this case. In this paper, we study second order fluctuations of the nodal intersection length. Our first main result is a Central Limit Theorem for "generic" surfaces, while for static ones, a sphere or a hemisphere e.g., our main results are a non-Central Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of so-called well-separated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota [Trans. Am. Math. Soc. 372(5), 3539–3564 (2019)] is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the Kac-Rice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest. [ABSTRACT FROM AUTHOR]

For the direct problem, we give the asymptotic distribution of the resonances in the complex plane. For the inverse problem, we prove several uniqueness theorems of recovering the potential on the whole interval from partial resonances and the information of the potential on a subinterval. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the tail behavior of randomly weighted sums and their maxima of dependent subexponential random variables, in which the primary random variables X 1 , ... , X n are real-valued and dependent following two general dependence structures, respectively, and the random weights θ 1 , ... , θ n are another n positive and arbitrarily dependent random variables, but independent of X 1 , ... , X n.. Under some technical conditions, we derive some asymptotic formulas for the tail probability of the randomly weighted sums and their maxima, which coincide with some existing ones in the literature. The merit of our results is that unbounded supports for the random weights are allowed and the distributions of primary random variables can be different. [ABSTRACT FROM AUTHOR]

When dealing with very high-dimensional and functional data, rank deficiency of sample covariance matrix often complicates the tests for population mean. To alleviate this rank deficiency problem, Munk et al. (J Multivar Anal 99:815–833, 2008) proposed neighborhood hypothesis testing procedure that tests whether the population mean is within a small, pre-specified neighborhood of a known quantity, M. How could we objectively specify a reasonable neighborhood, particularly when the sample space is unbounded? What should be the size of the neighborhood? In this article, we develop the modified neighborhood hypothesis testing framework to answer these two questions. We define the neighborhood as a proportion of the total amount of variation present in the population of functions under study and proceed to derive the asymptotic null distribution of the appropriate test statistic. Power analyses suggest that our approach is appropriate when sample space is unbounded and is robust against error structures with nonzero mean. We then apply this framework to assess whether the near-default sigmoidal specification of dose-response curves is adequate for widely used CCLE database. Results suggest that our methodology could be used as a pre-processing step before using conventional efficacy metrics, obtained from sigmoid models (for example: IC 50 or AUC), as downstream predictive targets. [ABSTRACT FROM AUTHOR]

Mark-point dependence plays a critical role in research problems that can be fitted into the general framework of marked point processes. In this work, we focus on adjusting for mark-point dependence when estimating the mean and covariance functions of the mark process, given independent replicates of the marked point process. We assume that the mark process is a Gaussian process and the point process is a log-Gaussian Cox process, where the mark-point dependence is generated through the dependence between two latent Gaussian processes. Under this framework, naive local linear estimators ignoring the mark-point dependence can be severely biased. We show that this bias can be corrected using a local linear estimator of the cross-covariance function and establish uniform convergence rates of the bias-corrected estimators. Furthermore, we propose a test statistic based on local linear estimators for mark-point independence, which is shown to converge to an asymptotic normal distribution in a parametric n -convergence rate. Model diagnostics tools are developed for key model assumptions and a robust functional permutation test is proposed for a more general class of mark-point processes. The effectiveness of the proposed methods is demonstrated using extensive simulations and applications to two real data examples. for this article are available online. [ABSTRACT FROM AUTHOR]

This paper introduces a Nearly Unstable INteger‐valued AutoRegressive Conditional Heteroscedastic (NU‐INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU‐INARCH process weakly converges to a Cox–Ingersoll–Ross diffusion in the Skorohod topology. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to nonstationarity. A unit root test is proposed and its Type‐I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID‐19 in the United Kingdom. [ABSTRACT FROM AUTHOR]

We develop a group of robust, nonparametric hypothesis tests that detect differences between the covariance operators of several populations of functional data. These tests, called functional Kruskal–Wallis tests for covariance, or FKWC tests, are based on functional data depth ranks. FKWC tests work well even when the data are heavy‐tailed, which is shown both in simulation and theory. FKWC tests offer several other benefits: they have a simple asymptotic distribution under the null hypothesis, they are computationally cheap, and they possess transformation‐invariance properties. We show that under general alternative hypotheses, these tests are consistent under mild, nonparametric assumptions. As a result, we introduce a new functional depth function called L2‐root depth that works well for the purposes of detecting differences in magnitude between covariance kernels. We present an analysis of the FKWC test based on L2‐root depth under local alternatives. Through simulations, when the true covariance kernels have an infinite number of positive eigenvalues, we show that these tests have higher power than their competitors while maintaining their nominal size. We also provide a method for computing sample size and performing multiple comparisons. [ABSTRACT FROM AUTHOR]

Focusing on polygenic signal detection in high‐dimensional genetic association studies of complex traits, we develop a stable and adaptive test for generalized linear models to accommodate different alternatives. To facilitate valid post‐selection inference for high‐dimensional data, our study here adheres to the original sample‐splitting principle but does so repeatedly to increase stability of the inference. We show the asymptotic null distribution of the proposed test for both fixed and diverging numbers of variants. We also show the asymptotic properties of the proposed test under local alternatives, providing insights on why power gain attributed to variable selection and weighting can compensate for efficiency loss due to sample splitting. We support our analytical findings through extensive simulation studies and two applications. The proposed procedure is computationally efficient and has been implemented as the R package DoubleCauchy. [ABSTRACT FROM AUTHOR]

In a one‐component partial least squares fit of a linear regression model, we find the asymptotic normal distribution, as the sample size and number of predictors approach infinity, of a user‐selected univariate linear combination of the coefficient estimator and give corresponding asymptotic confidence and prediction intervals. Simulation studies and an analysis of a dopamine dataset are used to support our theoretical asymptotic results and their practical application. [ABSTRACT FROM AUTHOR]