Asymptotic Normal Distribution Research Articles

An Adaptive Kernel-Based Structural Change Test for Copulas

This paper proposes a structural change test for copula models based on the kernel smoothing method. The proposed approach enables adaptable estimation of the dynamic marginal distributions, either parametrically or semi-parametrically. The test statistic is formulated via the weighted quadratic distance between the local smoothing copula and the empirical copula function, utilizing pseudo-observations of marginal distributions. The test statistic is pivotal with an asymptotic standard Normal distribution, irrespective of the marginal distributions, parameters, and estimations, and is consistent against a wide range of smoothly transitioning structural changes as well as abrupt structural breaks for copula models. Monte Carlo simulations show that the test performs well in finite samples and outperforms existing tests in the case of periodic changes.

Tail index estimation for tail adversarial stable time series with an application to high‐dimensional tail clustering

For stationary time series with regularly varying marginal distributions, an important problem is to estimate the associated tail index which characterizes the power‐law behavior of the tail distribution. For this, various results have been developed for independent data and certain types of dependent data. In this article, we consider the problem of tail index estimation under a recently proposed notion of serial tail dependence called the tail adversarial stability. Using the technique of adversarial innovation coupling and a martingale approximation scheme, we establish the consistency and central limit theorem of the tail index estimator for a general class of tail dependent time series. Based on the asymptotic normal distribution from the obtained central limit theorem, we further consider an application to cluster a large number of regularly varying time series based on their tail indices by using a robust mixture algorithm. The results are illustrated using numerical examples including Monte Carlo simulations and a real data analysis.

A New Mixture Model With Cure Rate Applied to Breast Cancer Data.

We introduce a new modelling for long-term survival models, assuming that the number of competing causes follows a mixture of Poisson and the Birnbaum-Saunders distribution. In this context, we present some statistical properties of our model and demonstrate that the promotion time model emerges as a limiting case. We delve into detailed discussions of specific models within this class. Notably, we examine the expected number of competing causes, which depends on covariates. This allows for direct modeling of the cure rate as a function of covariates. We present an Expectation-Maximization (EM) algorithm for parameter estimation, to discuss the estimation via maximum likelihood (ML) and provide insights into parameter inference for this model. Additionally, we outline sufficient conditions for ensuring the consistency and asymptotic normal distribution of ML estimators. To evaluate the performance of our estimation method, we conduct a Monte Carlo simulation to provide asymptotic properties and a power study of LR test by contrasting our methodology against the promotion time model. To demonstrate the practical applicability of our model, we apply it to a real medical dataset from a population-based study of incidence of breast cancer in São Paulo, Brazil. Our results illustrate that the proposed model can outperform traditional approaches in terms of model fitting, highlighting its potential utility in real-world scenarios.

Statistical Inference For Noisy Matrix Completion Incorporating Auxiliary Information

This article investigates statistical inference for noisy matrix completion in a semi-supervised model when auxiliary covariates are available. The model consists of two parts. One part is a low-rank matrix induced by unobserved latent factors; the other part models the effects of the observed covariates through a coefficient matrix which is composed of high-dimensional column vectors. We model the observational pattern of the responses through a logistic regression of the covariates, and allow its probability to go to zero as the sample size increases. We apply an iterative least squares (LS) estimation approach in our considered context. The iterative LS methods in general enjoy a low computational cost, but deriving the statistical properties of the resulting estimators is a challenging task. We show that our method only needs a few iterations, and the resulting entry-wise estimators of the low-rank matrix and the coefficient matrix are guaranteed to have asymptotic normal distributions. As a result, individual inference can be conducted for each entry of the unknown matrices. We also propose a simultaneous testing procedure with multiplier bootstrap for the high-dimensional coefficient matrix. This simultaneous inferential tool can help us further investigate the effects of covariates for the prediction of missing entries. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

The Reliability Inference for Multicomponent Stress–Strength Model under the Burr X Distribution

The reliability of the multicomponent stress–strength system was investigated under the two-parameter Burr X distribution model. Based on the structure of the system, the type II censored sample of strength and random sample of stress were obtained for the study. The maximum likelihood estimators were established by utilizing the type II censored Burr X distributed strength and complete random stress data sets collected from the multicomponent system. Two related approximate confidence intervals were achieved by utilizing the delta method under the asymptotic normal distribution theory and parametric bootstrap procedure. Meanwhile, point and confidence interval estimators based on alternative generalized pivotal quantities were derived. Furthermore, a likelihood ratio test to infer the equality of both scalar parameters is provided. Finally, a practical example is provided for illustration.

Depth-First Search Performance in a Random Digraph with Geometric Outdegree Distribution

We present an analysis of the depth-first search algorithm in a random digraph model with independent outdegrees having a geometric distribution. The results include asymptotic results for the depth profile of vertices, the height (maximum depth) and average depth, the number of trees in the forest, the size of the largest and second-largest trees, and the numbers of arcs of different types in the depth-first jungle. Most results are first order. For the height, we show an asymptotic normal distribution. This analysis, proposed by Donald Knuth in his next to appear volume of The Art of Computer Programming, gives interesting insight in one of the most elegant and efficient algorithm for graph analysis due to Tarjan.

Testing high-dimensional multinomials with applications to text analysis.

Motivated by applications in text mining and discrete distribution inference, we test for equality of probability mass functions of K groups of high-dimensional multinomial distributions. Special cases of this problem include global testing for topic models, two-sample testing in authorship attribution, and closeness testing for discrete distributions. A test statistic, which is shown to have an asymptotic standard normal distribution under the null hypothesis, is proposed. This parameter-free limiting null distribution holds true without requiring identical multinomial parameters within each group or equal group sizes. The optimal detection boundary for this testing problem is established, and the proposed test is shown to achieve this optimal detection boundary across the entire parameter space of interest. The proposed method is demonstrated in simulation studies and applied to analyse two real-world datasets to examine, respectively, variation among customer reviews of Amazon movies and the diversity of statistical paper abstracts.

A nonparametric test for homogeneity of variances

. A nonparametric test is proposed for the two-sample variance equality problem based on the fact that the covariance of U = X + Y and W = X − Y is zero when X and Y have equal variances. It is shown that the test statistic has an asymptotic standard normal distribution, which is useful for obtaining critical values for moderate sample sizes. The results of a Monte Carlo simulation experiment conducted in order to study the power properties of the test are presented.

Functional-Coefficient Quantile Regression for Panel Data with Latent Group Structure

This article considers estimating functional-coefficient models in panel quantile regression with individual effects, allowing the cross-sectional and temporal dependence for large panel observations. A latent group structure is imposed on the heterogeneous quantile regression models so that the number of nonparametric functional coefficients to be estimated can be reduced considerably. With the preliminary local linear quantile estimates of the subject-specific functional coefficients, a classic agglomerative clustering algorithm is used to estimate the unknown group structure and an easy-to-implement ratio criterion is proposed to determine the group number. The estimated group number and structure are shown to be consistent. Furthermore, a post-grouping local linear smoothing method is introduced to estimate the group-specific functional coefficients, and the relevant asymptotic normal distribution theory is derived with a normalization rate comparable to that in the literature. The developed methodologies and theory are verified through a simulation study and showcased with an application to house price data from U.K. local authority districts, which reveals different homogeneity structures at different quantile levels.

Non‐parametric Estimator for Conditional Mode with Parametric Features*

AbstractWe in this paper propose a new approach for estimating conditional mode non‐parametrically to capture the ‘most likely’ effect built on local linear approximation, in which a parametric pilot modal regression is locally adjusted through a kernel smoothing fit to potentially reduce the bias asymptotically without affecting the variance of the estimator. Specifically, we first estimate a parametric modal regression utilizing prior information from initial studies or economic analysis, and then estimate the non‐parametric modal function based on the additive correction by eliminating the parametric feature. We derive the asymptotic normal distribution of the proposed modal estimator for both fixed and estimated parametric feature cases, and demonstrate that there is substantial room for bias reduction under certain regularity conditions. We numerically estimate the suggested modal regression model with the use of a modified modal‐expectation‐maximization (MEM) algorithm. Monte Carlo simulations and one empirical analysis are presented to illustrate the finite sample performance of the developed modal estimator. Several extensions, including multiplicative correction, generalized guidance, modal‐based robust regression and the incorporation of categorical covariates, are also discussed for the sake of completeness.

Determining confidence interval and asymptotic distribution for parameters of multiresponse semiparametric regression model using smoothing spline estimator

The multiresponse semiparametric regression (MSR) model is a regression model with more than two response variables that are mutually correlated, and its regression function is composed of parametric and nonparametric components. The study objectives are propose a new method for estimating the MSR model using smoothing spline. Also, find the confidence interval (CI) of parameters and the distribution asymptotically of the model parameters estimator. Methods used in this study are reproducing kernel Hilbert space (RKHS) method and a developed penalized weighted least squares (PWLS), and apply pivotal quantity, central limit theorem, and theorems of Cramer-Wold and Slutsky. The results are an 100(1–α)% CI estimate and an asymptotic normal distribution for the parameters of the MSR model. In conclusion, the estimated MSR model is a combined components estimate of parametric and nonparametric which is linear to observation, and CIs of parameters depend on t distribution and estimator of parameters is asymptotically normally distributed. Future time, this study results can be used as theoretical bases to design standard growth charts of the toddlers which can then be used to assess the nutritional status of the toddlers.

Nonparametric testing for the specification of spatial trend functions

Correct specification of a spatial trend constitutes an important topic in statistics because it facilitates spatial kriging and inference. This paper proposes a global integrated squared error (GISE) statistics between the nonparametric smoothing surface and the parametric hypothesized model to test for the goodness-of-fit of spatial trends. By virtue of the m-dependence approximation of a stationary random field, it is shown that under certain regularity conditions, the proposed GISE statistics has an asymptotic normal distribution. Further, a grid-based block bootstrap (GBBB) procedure is also proposed to deal with the complicated asymptotic variance involved in the limit distribution. Numerical studies are also presented to illustrate the performance of the proposed method.

Noncentral Wishart matrices, asymptotic normality of vec and smooth statistics

Wishart matrices play an important role in normal multivariate statistical analysis. In this work, we present an approach that has been already used for normal vectors and is now applied to noncentral Wishart matrices. We show that, under general conditions, the vec of the Wishart matrix and a large class of its statistics have asymptotic normal distributions when the norm of the noncentrality parameter diverges ∞. These statistics are called smooth and are given by functions whose component functions have continuous second-order partial derivatives in a neighbourhood of a ‘pivot’ point. Moreover, we derive the application domain of the asymptotic normal distributions for the vec of the Wishart matrix and its smooth statistics. Thus we have an attraction to the normal model, for the increasing predominance of noncentrality and not for increasing sample sizes. A simulation study shows that the threshold for the use of asymptotic normal distributions is quite acceptable.

Response adaptive randomization design for a two-stage study with binary response

ABSTRACT Response adaptive randomization has the potential to treat more participants in better treatments in a trial to benefit participants. We propose optimal response adaptive randomization designs for a two-stage study with binary response, having the smallest expected sample size or the fewest expected number of failures. Equal randomization is used in the first stage, and data from the first stage is used to determine the adaptive sample size ratio in the second stage. In the proposed optimal designs, the type I error rate and the statistical power are calculated from the asymptotic normal distributions. The new designs that minimize the expected number of failures have the advantage over the existing optimal randomized designs to substantially reduce the number of failures.

Asymptotic distribution of one‐component partial least squares regression estimators in high dimensions

AbstractIn 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.

Asymptotic and finite sample properties of Hill-type estimators in the presence of errors in observations

We establish asymptotic and finite sample properties of the Hill and Harmonic Moment estimators applied to heavy-tailed data contaminated by errors. We formulate conditions on the errors and the number of upper order statistics under which these estimators continue to be asymptotically normal. We specify analogous conditions which must hold in finite samples for the confidence intervals derived from the asymptotic normal distribution to be reliable. In the large sample analysis, we specify conditions related to second-order regular variation and divergence rates for the number of upper order statistics, k, used to compute the estimators. In the finite sample analysis, we examine several data-driven methods of selecting k, and determine which of them are most suitable for confidence interval inference. The results of these investigations are applied to interarrival times of internet traffic anomalies, which are available only with a round-off error.

Kernel density estimation for undirected dyadic data

We study nonparametric estimation of density functions for undirected dyadic random variables (i.e., random variables defined for all n≡defN2 unordered pairs of agents/nodes in a weighted network of order N). These random variables satisfy a local dependence property: any random variables in the network that share one or two indices may be dependent, while those sharing no indices in common are independent. In this setting, we show that density functions may be estimated by an application of the kernel estimation method of Rosenblatt (1956) and Parzen (1962). We suggest an estimate of their asymptotic variances inspired by a combination of (i) Newey’s (1994) method of variance estimation for kernel estimators in the “monadic” setting and (ii) a variance estimator for the (estimated) density of a simple network first suggested by Holland and Leinhardt (1976). More unusual are the rates of convergence and asymptotic (normal) distributions of our dyadic density estimates. Specifically, we show that they converge at the same rate as the (unconditional) dyadic sample mean: the square root of the number, N, of nodes. This differs from the results for nonparametric estimation of densities and regression functions for monadic data, which generally have a slower rate of convergence than their corresponding sample mean.

Specification tests for time-varying coefficient models

Time-varying coefficient models have been widely used to characterize changing relationships among economic and financial variables. The existing literature usually specifies the time-varying coefficient vector as a stationary stochastic process, a deterministic function of time, and a unit root process, respectively. In this paper, we propose two tests to distinguish these three specifications. Both test statistics follow asymptotic normal distributions under the respective null hypotheses and diverge to infinity in probability under the corresponding alternatives. To improve the finite sample performance of the tests, we propose a dependent wild bootstrap to obtain the bootstrap critical value (or P-value) and establish its asymptotic validity. Simulation studies show that our bootstrap-based tests perform reasonably well in finite samples. We apply the proposed tests to the time-varying specifications of the equity return’s predictive model, the U.S. Taylor rule, and inflation persistence, respectively. The results suggest that a unit root process is favored for the first application, whereas a deterministic function of time should be adopted for the latter two applications.

Bias-Correction and Test for Mark-Point Dependence with Replicated Marked Point Processes

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. Supplementary materials for this article are available online.

Inferences of the Multicomponent Stress–Strength Reliability for Burr XII Distributions

Multicomponent stress–strength reliability (MSR) is explored for the system with Burr XII distributed components under Type-II censoring. When the distributions of strength and stress variables have Burr XII distributions with common or unequal inner shape parameters, the existence and uniqueness of the maximum likelihood estimators are investigated and established. The associated approximate confidence intervals are obtained by using the asymptotic normal distribution theory along with the delta method and parametric bootstrap procedure, respectively. Moreover, alternative generalized pivotal quantities-based point and confidence interval estimators are developed. Additionally, a likelihood ratio test is presented to diagnose the equivalence of both inner shape parameters or not. Conclusively, Monte Carlo simulations and real data analysis are conducted for illustration.