Asymptotic Joint Distribution Research Articles

Asymptotic confidence interval for R 2 in multiple linear regression

Following White's approach of robust multiple linear regression [White H. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 1980;48(4):817–838], we give asymptotic confidence intervals for the multiple correlation coefficient R 2 under minimal moment conditions. We also give the asymptotic joint distribution of the empirical estimators of the individual R 2 's. Through different sets of simulations, we show that the procedure is indeed robust (contrary to the procedure involving the near exact distribution of the empirical estimator of R 2 is the multivariate Gaussian case) and can be also applied to count linear regression. Several extensions are also discussed, as well as an application to robust screening.

Scalable Methods for Multiple Time Series Comparison in Second Order Dynamics

Statistical comparison of multiple time series in their underlying frequency patterns has many real applications. However, existing methods are only applicable to a small number of mutually independent time series, and empirical results for dependent time series are only limited to comparing two time series. We propose scalable methods based on a new algorithm that enables us to compare the spectral density of a large number of time series. The new algorithm helps us efficiently obtain all pairwise feature differences in frequency patterns between M time series, which plays an essential role in our methods. When all M time series are independent of each other, we derive the joint asymptotic distribution of their pairwise feature differences. The asymptotic dependence structure between the feature differences motivates our proposed test for multiple mutually independent time series. We then adapt this test to the case of multiple dependent time series by partially accounting for the underlying dependence structure. Additionally, we introduce a global test to further enhance the approach. To examine the finite sample performance of our proposed methods, we conduct simulation studies. The new approaches demonstrate the ability to compare a large number of time series, whether independent or dependent, while exhibiting competitive power. Finally, we apply our methods to compare multiple mechanical vibrational time series.

Simultaneous Inference of Multiple Binary Endpoints in Biomedical Research: Small Sample Properties of Multiple Marginal Models and a Resampling Approach.

In biomedical research, the simultaneous inference of multiple binary endpoints may be of interest. In such cases, an appropriate multiplicity adjustment is required that controls the family-wise error rate, which represents the probability of making incorrect test decisions. In this paper, we investigate two approaches that perform single-step -value adjustments that also take into account the possible correlation between endpoints. A rather novel and flexible approach known as multiple marginal models is considered, which is based on stacking of the parameter estimates of the marginal models and deriving their joint asymptotic distribution. We also investigate a nonparametric vector-based resampling approach, and we compare both approaches with the Bonferroni method by examining the family-wise error rate and power for different parameter settings, including low proportions and small sample sizes. The results show that the resampling-based approach consistently outperforms the other methods in terms of power, while still controlling the family-wise error rate. The multiple marginal models approach, on the other hand, shows a more conservative behavior. However, it offers more versatility in application, allowing for more complex models or straightforward computation of simultaneous confidence intervals. The practical application of the methods is demonstrated using a toxicological dataset from the National ToxicologyProgram.

Parameter Estimation and Testing for the Doubly Geometric Process with Lognormal Distribution: Application to Bladder Cancer Patients’ Data

The geometric process (GP) has been widely utilized as a stochastic monotone model in the fields of probability and statistics. However, its practical application is often limited by certain assumptions. To address this, [Wu (2018). Doubly geometric process and applications. Journal of the Operational Research Society, 69(1), 66–67] introduced the doubly geometric process (DGP) as an extension of the GP model, relaxing some of its assumptions. Due to its ability to overcome the limitations of the GP model, the DGP has gained significant popularity in recent times. This study focuses on the parameter estimation problem for the DGP when the distribution of the first interarrival time follows a lognormal distribution with parameters [Formula: see text] and [Formula: see text]. We employ the maximum likelihood method to obtain estimates for both the model parameters and the distribution parameters. Additionally, we investigate the asymptotic joint distribution and statistical properties such as asymptotic unbiasedness and consistency of the estimators. Furthermore, we propose a novel test procedure to distinguish between the GP and DGP models. To assess the performance of the estimators and the proposed test procedure, we conduct a simulation study involving various sample sizes and parameter values. Finally, we present an application of the developed methods in fitting data from bladder cancer patients.

Revisit of a Diaconis urn model

Let G be a finite Abelian group of order d. We consider an urn in which, initially, there are labeled balls that generate the group G. Choosing two balls from the urn with replacement, observe their labels, and perform a group multiplication on the respective group elements to obtain a group element. Then, we put a ball labeled with that resulting element into the urn. This model was formulated by P. Diaconis while studying a group theoretic algorithm called MeatAxe (Holt and Rees, 1994). Siegmund and Yakir (2005) partially investigated this model. In this paper, we further investigate and generalize this model. More specifically, we allow a random number of balls to be drawn from the urn at each stage in the Diaconis urn model. For such a case, we verify that the normalized urn composition converges almost surely to the uniform distribution on the group G. Moreover, we obtain the asymptotic joint distribution of the urn composition by using the martingale central limit theorem.

Inference for partial correlations of a multivariate Gaussian time series

Abstract We derive an asymptotic joint distribution and novel covariance estimator for the partial correlations of a multivariate Gaussian time series under mild regularity conditions. Using our derived asymptotic distribution, we develop a Wald confidence interval and testing procedure for inference of individual partial correlations for time series data. Through simulation we demonstrate that our proposed confidence interval attains higher coverage rates and our testing procedure achieves false positive rates closer to the nominal levels than approaches that assume independent observations when autocorrelation is present.

Fast and asymptotically-efficient estimation in an autoregressive process with fractional type noise

This paper considers the joint estimation of the parameters of a first-order fractional autoregressive model. A one-step procedure is considered in order to obtain an asymptotically-efficient estimator with an initial guess estimator with convergence speed lower than n and singular asymptotic joint distribution. This estimator is computed faster than the maximum likelihood estimator or the Whittle estimator and therefore allows for faster inference on large samples. The paper also illustrates the performance of this method on finite-size samples via Monte Carlo simulations.

Semiparametric estimation for the functional additive hazards model

AbstractWe propose a new functional additive hazards model to investigate the potential effects of functional and scalar predictors on mortality risks, and develop a penalized least squares estimation method for model parameters based on a pseudoscore estimating equation. A reproducing kernel Hilbert space approach is used to establish the consistency, convergence rate, and joint asymptotic distribution of the resulting estimators for finite‐dimensional and infinite‐dimensional parameters. Our simulation studies demonstrate that the proposed estimation procedure performs well. For illustration, we apply the proposed method to the Medical Information Mart for Intensive Care III dataset.

An adaptive test based on Kendall's tau for independence in high dimensions

We consider testing the mutual independency for high-dimensional data. It is known that L 2 -type statistics have lower power under sparse alternatives and L ∞ -type statistics have lower power under dense alternatives in high dimensions. In this paper, we develop an adaptive test based on Kendall's tau to compromise both situations of the alternative, which can automatically be adapted to the underlying data. An adaptive test is very useful in practice as the sparsity or density for a data set is usually unknown. In addition, we establish the asymptotic joint distribution of L 2 -type and L ∞ -type statistics based on Kendall's tau under mild assumptions and the asymptotic null distribution of the proposed statistic. Simulation studies show that our adaptive test performs well in either dense or sparse cases. To illustrate the usefulness and effectiveness of the proposed test, real data sets are also analysed.

Central limit theorems for local network statistics

Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model that generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.

Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes.

Regression models for continuous outcomes frequently require a transformation of the outcome, which is often specified a priori or estimated from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation by treating the continuous outcome as if it is ordered categorically. They thus represent a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where bounds, one lower and one upper, are chosen and the outcomes outside the bounds are set as two ordinal categories. We prove the uniform consistency of the estimated regression coefficients and of the estimated transformation function between the bounds. We also describe their joint asymptotic distribution, and show that the estimated regression coefficients attain the semiparametric efficiency bound. We show with simulations that results from this approach and those from using the CPM on the original data are very similar when a small fraction of the data are modified. We reanalyze a dataset of HIV-positive patients with CPMs to illustrate and compare the approaches.

Fluctuations, bias, variance and ensemble of learners: exact asymptotics for convex losses in high-dimension *

From the sampling of data to the initialisation of parameters, randomness is ubiquitous in modern Machine Learning practice. Understanding the statistical fluctuations engendered by the different sources of randomness in prediction is therefore key to understanding robust generalisation. In this manuscript we develop a quantitative and rigorous theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features in high-dimensions. In particular, we provide a complete description of the asymptotic joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit. Our result encompasses a rich set of classification and regression tasks, such as the lazy regime of overparametrised neural networks, or equivalently the random features approximation of kernels. While allowing to study directly the mitigating effect of ensembling (or bagging) on the bias-variance decomposition of the test error, our analysis also helps disentangle the contribution of statistical fluctuations, and the singular role played by the interpolation threshold that are at the roots of the ‘double-descent’ phenomenon.

The Limit Properties of Maxima of Stationary Gaussian Sequences Subject to Random Replacing

In applications, missing data may occur randomly and some relevant datum are often used to replace the missing ones. This article mainly explores the influence of the degree of dependence of stationary Gaussian sequences on the joint asymptotic distribution of the maximum of the Gaussian sequence and its maximum when the sequence is subject to random replacing.

Tensor Stein-rules in a generalized tensor regression model

In this paper, we consider an estimation problem in a generalized tensor regression model with multi-mode covariates. We generalize the main results in recent literature in four ways. First, we weaken assumptions underlying the main results of the previous works. In particular, the dependence structure of the error and covariates are as weak as an L2− mixingale array, and the error term does not need to be uncorrelated with regressors. Second, we consider a more general constraint than the one considered in previous works. Third, we establish the asymptotic properties of the tensor estimators. Specifically, we derive the joint asymptotic distribution of the unrestricted tensor estimator (UE) and restricted tensor estimator (RE). Fourth, we propose a class of shrinkage-type estimators in the context of tensor regression, and under a general loss function, we derive sufficient conditions for which the shrinkage estimators dominate the UE. In addition to these interesting contributions, we derive a kind of functional central limit theorem for mixingale vector-valued and we establish some identities which are useful in studying the risk dominance of shrinkage-type tensor estimators. Finally, to illustrate the application of the proposed methods, we corroborate the results by some simulation studies of binary, Normal and Poisson data and we analyze a multi-relational network and neuro-imaging datasets.

Power enhancement for testing multi-factor asset pricing models via Fisher’s method

Testing multi-factor asset pricing models is instrumental for asset pricing theory and practice. However, due to the accumulation of errors in estimating high-dimensional parameters, traditional quadratic-form tests such as the Wald test perform poorly against the sparse alternative hypothesis, i.e., a few mispriced assets. Fan et al. (2015b) introduced a powerful testing procedure by adding a power enhancement component to the Wald test statistic and proved power enhancement properties. To provide an alternative to their methodology, we first instantiate the power enhancement component by introducing a new maximum-form test statistic and then study the asymptotic joint distribution of the Wald test statistic and the maximum test statistic. We prove that these two test statistics are asymptotically independent. Given their asymptotic independence, we propose a new power-enhanced testing procedure to combine their respective power based on Fisher’s method (Fisher, 1925). Theoretically, we prove that the new power-enhanced test retains the desired nominal significance level and achieves asymptotically consistent power against more general alternatives. Furthermore, we demonstrate the finite-sample performance of our proposed power-enhanced test in both simulation studies and an empirical study of testing market efficiency using asset returns of the Russel-2000 portfolio.

Asymptotic Properties of Random Restricted Partitions

We study two types of probability measures on the set of integer partitions of n with at most m parts. The first one chooses the partition with a chance related to its largest part only. We obtain the limiting distributions of all of the parts together and that of the largest part as n tending to infinity for m fixed or tending to infinity with m=o(n1/3). In particular, if m goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general and includes the Dirichlet and uniform distributions as special cases. The joint asymptotic distributions of the parts are derived by taking limits of n and m in the same manner as that in the first probability measure.

Fisher’s Combined Probability Test for High-Dimensional Covariance Matrices

Testing large covariance matrices is of fundamental importance in statistical analysis with high-dimensional data. In the past decade, three types of test statistics have been studied in the literature: quadratic form statistics, maximum form statistics, and their weighted combination. It is known that quadratic form statistics would suffer from low power against sparse alternatives and maximum form statistics would suffer from low power against dense alternatives. The weighted combination methods were introduced to enhance the power of quadratic form statistics or maximum form statistics when the weights are appropriately chosen. In this article, we provide a new perspective to exploit the full potential of quadratic form statistics and maximum form statistics for testing high-dimensional covariance matrices. We propose a scale-invariant power-enhanced test based on Fisher’s method to combine the p-values of quadratic form statistics and maximum form statistics. After carefully studying the asymptotic joint distribution of quadratic form statistics and maximum form statistics, we first prove that the proposed combination method retains the correct asymptotic size under the Gaussian assumption, and we also derive a new Lyapunov-type bound for the joint distribution and prove the correct asymptotic size of the proposed method without requiring the Gaussian assumption. Moreover, we show that the proposed method boosts the asymptotic power against more general alternatives. Finally, we demonstrate the finite-sample performance in simulation studies and a real application. Supplementary materials for this article are available online.

Infinite-Server Resource Queueing Systems with Different Types of Markov-Modulated Poisson Process and Renewal Arrivals

In this paper, we propose models that significantly expand the scope of practical applications, namely, queueing systems with various nodes for processing heterogeneous data that require arbitrary resource capacities for their service. When a customer arrives in the system, the customer typeis randomly selected according to a set of probabilities. Then the customer goes to the server of the corresponding device type, where its service is performed during a random time period with a distribution function depending on the type of customer. Moreover, each customer requires a random amount of resources, of which the distribution function also depends on the customer type, but is independent of its service time. The aim of this research was to develop a heterogeneous queueing resource system with an unlimited number of servers and an arrival process in the form of a Markov-modulated Poisson process or stationary renewal process, and with requests for a random number of heterogeneous resources. We have performed analysis under conditions of growing intensity of the arrival process. Here we formulate the theorems and prove that under high-load conditions, the joint asymptotic probability distribution of the n-dimensional process of the total amounts of the occupied resources in the system is a multidimensional Gaussian distribution with parameters that are dependent on the type of arrival process. As a result of numerical and simulation experiments, conclusions are drawn on the limits of the applicability of the obtained asymptotic results. The dependence of the convergence of experimental results on the type of distribution of the system parameters (including the distributions of the service time and of the customer capacity) are also studied. The results of the approximations may be applied to estimating the optimal total number of resources for a system with a limited amount of resources.

Estimation of the parameters of the gamma geometric process

There is no doubt that finding the estimators of model parameters accurately and efficiently is very important in many fields. In this study, we obtain the explicit estimators of the unknown model parameters in the gamma geometric process (GP) via the modified maximum likelihood (MML) methodology. These estimators are as efficient as maximum likelihood (ML) estimators. The marginal and joint asymptotic distributions of the MML estimators are also derived and efficiency comparisons between ML and MML estimators are made through an extensive Monte Carlo simulations. Moreover, a real data example is considered to illustrate the performances of the MML estimators together with their ML counterparts. According to simulation results, the performances of MML and ML estimators are close to each other even for small sample sizes.

The Joint Asymptotic Distribution of Probability Density Function in a Finite Number of Points Under φ -mixing Samples

The joint asymptotic distribution of kernel estimator of probability density function in a finite number of points is studied and proved under φ-mixing samples. By applying the blockwise technique and moment inequalities, it is shown that the joint asymptotic distribution is asymptotically multivariate normal distributed. As its application, the asymptotic distribution of the difference of density function between any two points is given.