Uniform Asymptotic Normality Research Articles

Local projection inference in high dimensions

Summary In this paper, we estimate impulse responses by local projections in high-dimensional settings. We use the desparsified (de-biased) lasso to estimate the high-dimensional local projections, while leaving the impulse response parameter of interest unpenalized. We establish the uniform asymptotic normality of the proposed estimator under general conditions. Finally, we demonstrate small sample performance through a simulation study and consider two canonical applications in macroeconomic research on monetary policy and government spending.

Locally optimal tests against periodic linear regression in short panels

This paper aims to detect the periodic coefficients of a Linear Regression model with Panel Data. Nonparametric locally and asymptotically optimal tests are proposed for testing the null hypothesis of a traditional linear regression model, against an alternative of a Periodic Coefficient Regression model. The Uniform Local Asymptotic Normality (ULAN) property is established for Periodic Linear Regression (PLR) model, to construct Locally asymptotically optimal tests. The rank-based version of the optimal parametric test is provided in particular cases. The performance of the proposed tests is valid with numerical simulations using the Monte Carlo study. Real data was studied for applying the periodic linear regression model which compared to both traditional regression and random coefficient regression models.

Tests in functional autoregressive processes via local asymptotic normality condition

Local asymptotic normality (LAN) and uniform local asymptotic normality (ULAN) conditions are obtained for a linear functional autoregressive process of order p, p≥1, using a martingale central limit theorem. As a consequence of the LAN property, asymptotically efficient tests for determining the order are constructed. An example is discussed as a special cases.

Complete subset averaging approach for high-dimensional generalized linear models

This study proposes a novel complete subset averaging (CSA) method for high-dimensional generalized linear models based on a penalized Kullback–Leibler (KL) loss. All models under consideration can be potentially misspecified, and the dimension of covariates is allowed to diverge to infinity. The uniform convergence rate and asymptotic normality of the proposed estimator are established. Moreover, it is asymptotically optimal in terms of achieving the lowest KL loss. To ease the computational burden, we randomly draw a fixed number of subsets from the complete subsets and show their asymptotic equivalence. The Monte Carlo simulation and empirical application demonstrate that the proposed CSA method outperforms popular model-averaging methods.

Minimax properties of Dirichlet kernel density estimators

This paper considers the asymptotic behavior in β-Hölder spaces, and under Lp losses, of a Dirichlet kernel density estimator proposed by Aitchison and Lauder (1985) for the analysis of compositional data. In recent work, Ouimet and Tolosana-Delgado (2022) established the uniform strong consistency and asymptotic normality of this estimator. As a complement, it is shown here that the Aitchison–Lauder estimator can achieve the minimax rate asymptotically for a suitable choice of bandwidth whenever (p,β)∈[1,3)×(0,2] or (p,β)∈Ad, where Ad is a specific subset of [3,4)×(0,2] that depends on the dimension d of the Dirichlet kernel. It is also shown that this estimator cannot be minimax when either p∈[4,∞) or β∈(2,∞). These results extend to the multivariate case, and also rectify in a minor way, earlier findings of Bertin and Klutchnikoff (2011) concerning the minimax properties of Beta kernel estimators.

Estimation in short-panel data models with bilinear errors

Many estimation methods have been proposed for the parameters of the regression models with serially correlated errors. In this work, we develop an asymptotic theory for estimation in the short panel data models with bilinear error. We propose a comparative study by simulation between several estimators (adaptive, ordinary and weighted least squares) for the coefficients of panel data models when the errors are bilinear serially correlated. As a consequence of the uniform local asymptotic normality property, we obtain adaptive estimates of the parameters. Finally, we illustrate the performance of the proposed estimators via Monte Carlo simulation study. We show that the adaptive estimates are more efficient than the weighted and ordinary least squares estimates.

Asymptotics in the Thurstone model with an increasing items

The Thurstone model is a common model for analyzing the paired comparison data. However, there is lack of asymptotic properties for the moment estimator of parameters in this model. Therefore, we prove uniform consistency and asymptotic normality of the moment estimator in the Thurstome model. Simulation studies and a data example illustrate the theoretical result.

Lasso inference for high-dimensional time series

In this paper we develop valid inference for high-dimensional time series. We extend the desparsified lasso to a time series setting under Near-Epoch Dependence (NED) assumptions allowing for non-Gaussian, serially correlated and heteroskedastic processes, where the number of regressors can possibly grow faster than the time dimension. We first derive an error bound under weak sparsity, which, coupled with the NED assumption, means this inequality can also be applied to the (inherently misspecified) nodewise regressions performed in the desparsified lasso. This allows us to establish the uniform asymptotic normality of the desparsified lasso under general conditions, including for inference on parameters of increasing dimensions. Additionally, we show consistency of a long-run variance estimator, thus providing a complete set of tools for performing inference in high-dimensional linear time series models. Finally, we perform a simulation exercise to demonstrate the small sample properties of the desparsified lasso in common time series settings.

Asymptotics in the Thurstone model with an increasing items

The Thurstone model is a common model for analyzing the paired comparison data. However, the asymptotic properties of the parameter moment estimator for this model are lacking. Therefore, we prove uniform consistency and asymptotic normality of the moment estimator in the Thurstome model. Simulation studies and a data example illustrate the theoretical result.

Cluster based inference for extremes of time series

We introduce a new type of estimator for the spectral tail process of a regularly varying time series. The approach is based on a characterizing invariance property of the spectral tail process, which is incorporated into the new estimator via a projection technique. We show uniform asymptotic normality of this estimator, both in the case of known and of unknown index of regular variation. In a simulation study the new procedure shows a more stable performance than previously proposed estimators.

Optimal tests for random effects in linear mixed models

In the past decade, mixed-effects modeling has received a great deal of attention in the applied and theoretical statistical literature. They are very flexible tools in analyzing repeated measures, panel data, cross-sectional data, and hierarchical data. However, the complex nature of these models has motivated researchers to study different aspects of this problem. One of which is to test the significance of random effects used to model unobserved heterogeneity in the population. The method of likelihood ratio test based on the normality assumption of the error term and random effects has been proposed. However, this assumption does not necessarily hold in practice. In this paper, we propose an optimal test based on the so-called uniform local asymptotic normality to detect the possible presence of random effects in linear mixed models. We show that the proposed test statistic is, consistent, locally asymptotically optimal even for a model that does not require the traditional assumption of normality and is comparable to the classical L.ratio-test when the standard assumptions are met. Finally, simulation studies and real data analysis are also conducted to empirically examine the performance of this procedure.

Estimation of nonstationary nonparametric regression model with multiplicative structure

Summary This paper presents a multiplicative nonstationary nonparametric regression model which allows for a broad class of nonstationary processes. We propose a three-step estimation procedure to uncover the conditional mean function and establish uniform convergence rates and asymptotic normality of our estimators. The new model can also be seen as a dimension-reduction technique for a general two-dimensional time-varying nonparametric regression model, which is especially useful in small samples and for estimating explicitly multiplicative structural models. We consider two applications: estimating a pricing equation for the US aggregate economy to model consumption growth and estimating the shape of the monthly risk premium for S&P 500 Index data.

The uniform asymptotic normality of a matrix-T distribution

Using the Kullback-Leibler distance between two density functions about a matrix T distribution and a matrix normal distribution, we obtain a Berry-Esseen boundary for the T distribution. Further, we give the condition under which a matrix T is uniformly asymptotically matrix normal distribution, and point out the convergence rate.

Efficient Estimation for Varying-Coefficient Mixed Effects Models with Functional Response Data

In this article, we focus on the estimation of varying-coefficient mixed effects models for longitudinal and sparse functional response data, by using the generalized least squares method coupling a modified local kernel smoothing technique. This approach provides a useful framework that simultaneously takes into account the within-subject covariance and all observation information in the estimation to improve efficiency. We establish both uniform consistency and pointwise asymptotic normality for the proposed estimators of varying-coefficient functions. Numerical studies are carried out to illustrate the finite sample performance of the proposed procedure. An application to the white matter tract dataset obtained from Alzheimer’s Disease Neuroimaging Initiative study is also provided.

Uniform asymptotic normality of weighted sums of short-memory linear processes

AbstractLet $X_1, X_2,\dots$ be a short-memory linear process of random variables. For $1\leq q<2$ , let ${\mathcal{F}}$ be a bounded set of real-valued functions on [0, 1] with finite q-variation. It is proved that $\{n^{-1/2}\sum_{i=1}^nX_i\,f(i/n)\colon f\in{\mathcal{F}}\}$ converges in outer distribution in the Banach space of bounded functions on ${\mathcal{F}}$ as $n\to\infty$ . Several applications to a regression model and a multiple change point model are given.

Empirical tail copulas for functional data

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula, uniformly over all finite subsets of at most D points (D fixed). An application for testing tail copula stationarity is presented. The main tool for deriving the result is the uniform asymptotic normality of all the D-variate tail empirical processes. The proof of the main result is nonstandard.

Empirical Tail Copulas for Functional Data

For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula,uniformly over all finite subsets of at most D points (D fixed). As a special case we obtain the uniform asymptotic normality of all estimated uppertail dependence coeffcients. The main tool for deriving the result is the uniform asymptotic normality of all the D-variate tail empirical processes. The proof of the main result is non-standard.

A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence

By using some inequalities for linearly negative quadrant dependent random variables, Berry-Ess$\acute{e}$en bound of wavelet estimation for a nonparametric regression model is investigated under linear process errors based on linearly negative quadrant dependent sequence. The rate of uniform asymptotic normality is presented and the rate of convergence is near $O(n^{-\frac{1}{6}})$ under mild conditions, which generalizes or extends the corresponding results of Li et al.(2008) under associated random samples in some sense.

Uniform asymptotic normality of self-normalized weighted sums of random variables*

Let X, X1, X2, . . . be a sequence of nondegenerate i.i.d. random variables, let μ = {μni : n ∈ ℕ+, i = 1, …, n} be a triangular array of possibly random probabilities on the interval [0, 1], and let $$ \mathcal{F} $$ be a class of functions with bounded q-variation on [0, 1] for some q ∈ [1, 2). We prove the asymptotic normality uniformly over $$ \mathcal{F} $$ of self-normalized weighted sums $$ {\sum}_{i=1}^n{X}_i{\mu}_{ni} $$ when μ is the array of point measures, uniform probabilities, and their random versions. Also, we prove a weak invariance principle in the Banach space of functions of bounded p-variation with p > 2 for partial-sum processes, polygonal processes, and their adaptive versions.

Improving precipitation forecasts using extreme quantile regression

Aiming to estimate extreme precipitation forecast quantiles, we propose a nonparametric regression model that features a constant extreme value index. Using local linear quantile regression and an extrapolation technique from extreme value theory, we develop an estimator for conditional quantiles corresponding to extreme high probability levels. We establish uniform consistency and asymptotic normality of the estimators. In a simulation study, we examine the performance of our estimator on finite samples in comparison with a method assuming linear quantiles. On a precipitation data set in the Netherlands, these estimators have greater predictive skill compared to the upper member of ensemble forecasts provided by a numerical weather prediction model.