Joint Asymptotic Normality Research Articles

Asymptotic normality of estimators for all parameters in the Vasicek model by discrete observations

We continue to study the joint estimation of all three parameters of the Vasicek model, based on equidistant observations of a trajectory. We consider strongly consistent estimators previously constructed in Prykhodko and Ralchenko (2022) and establish their joint asymptotic normality. The theoretical results are illustrated by a numerical study that confirms our theoretical findings.

Error Propagation in Asymptotic Analysis of the Data-Driven (s, S) Inventory Policy

Estimating Optimal Inventory Policies: Moving Beyond SAA and Empirical Process Theory In “Error Propagation in Asymptotic Analysis of the Data-Driven (s,S) Inventory Policy,” Zhang, Ye, and Haskell dive into the class multiperiod stochastic inventory control in a data-driven setting. They investigate the statistical properties of the data-driven $(s, S)$-policy obtained by recursively computing the empirical cost-to-go functions. In this setting, the empirical cost-to-go functions for the estimated parameters are not i.i.d. sums because of the error propagation. They establish a novel asymptotic representation for multi-sample $U$-processes in terms of i.i.d. sums. This representation enables them to apply empirical process theory to derive the influence functions of the estimated parameters and to establish joint asymptotic normality. Based on these results, they also propose an entirely data-driven estimator of the optimal expected cost and derive its asymptotic distribution. They demonstrate some useful applications of their asymptotic results, including sample size determination and interval estimation.

Joint Statistical Inference for the Area under the ROC Curve and Youden Index under a Density Ratio Model

The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve (AUC) and the Youden index (J) are common summary indices used to evaluate a biomarker’s diagnostic accuracy. Simultaneously examining AUC and J offers a more comprehensive understanding of the ROC curve’s characteristics. In this paper, we utilize a semiparametric density ratio model to link the distributions of a biomarker for healthy and diseased individuals. Under this model, we establish the joint asymptotic normality of the maximum empirical likelihood estimator of (AUC,J) and construct an asymptotically valid confidence region for (AUC,J). Furthermore, we propose a new test to determine whether a biomarker simultaneously exceeds prespecified target values of AUC0 and J0 with the null hypothesis H0:AUC≤AUC0 or J≤J0 against the alternative hypothesis Ha:AUC>AUC0 and J>J0. Simulation studies and a real data example on Duchenne Muscular Dystrophy are used to demonstrate the effectiveness of our proposed method and highlight its advantages over existing methods.

Modified least squares estimators for Ornstein–Uhlenbeck processes from low-frequency observations

We propose a modified least squares estimator for the drift parameter of the Ornstein–Uhlenbeck process when the observations are available at a discrete instant in a low-frequency level. Unlike in the past literature, this modified least squares estimator is asymptotically unbiased. This estimator is combined with the ergodic theorem to obtain joint estimators (θˆn,σˆn2) for both drift and diffusion parameters (θ,σ2). For any fixed observation time step h>0, the strong consistency and joint asymptotic normality of our estimators (θˆn,σˆn2) are obtained by using the linear model technology and the Delta method. Surprisingly, this linear model technology is not new but it is used for the first time to our model. It very significantly simplifies the arguments that researchers used in the literature. Numerical results illustrate the asymptotic behavior of the estimators.

Limit theorems for local polynomial estimation of regression for functional dependent data

<p>Local polynomial fitting exhibits numerous compelling statistical properties, particularly within the intricate realm of multivariate analysis. However, as functional data analysis gains prominence as a dynamic and pertinent field in data science, the exigency arises for the formulation of a specialized theory tailored to local polynomial fitting. We explored the intricate task of estimating the regression function operator and its partial derivatives for stationary mixing random processes, denoted as $ (Y_i, X_i) $, using local higher-order polynomial fitting. Our key contributions include establishing the joint asymptotic normality of the estimates for both the regression function and its partial derivatives, specifically in the context of strongly mixing processes. Additionally, we provide explicit expressions for the bias and the variance-covariance matrix of the asymptotic distribution. Demonstrating uniform strong consistency over compact subsets, along with delineating the rates of convergence, we substantiated these results for both the regression function and its partial derivatives. Importantly, these findings rooted in reasonably broad conditions that underpinned the underlying models. To demonstrate practical applicability, we leveraged our results to compute pointwise confidence regions. Finally, we extended our ideas to the nonparametric conditional distribution, and obtained its limiting distribution.</p>

Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations

The object of investigation is the mixed fractional Brownian motion of the form ${X_{t}}=\kappa {B_{t}^{{H_{1}}}}+\sigma {B_{t}^{{H_{2}}}}$, driven by two independent fractional Brownian motions ${B_{1}^{H}}$ and ${B_{2}^{H}}$ with Hurst parameters ${H_{1}}\lt {H_{2}}$. Strongly consistent estimators of unknown model parameters ${({H_{1}},{H_{2}},{\kappa ^{2}},{\sigma ^{2}})^{\top }}$ are constructed based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for $0\lt {H_{1}}\lt {H_{2}}\lt \frac{3}{4}$.

Asymptotic Normality of Parameter Estimators for~Mixed Fractional Brownian Motion with Trend

We investigate the mixed fractional Brownian motion of the form Xt = θt+σWt +κBtH , driven by a standard Brownian motion W and a fractional Brownian motion B H with Hurst parameter H. We consider strongly consistent estimators of unknown model parameters (θ, H, σ, κ) based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for H ∈ (0, 12 ).

Asymptotically normal estimation of parameters of mixed fractional Brownian motion

We investigate the mixed fractional Brownian motion of the form Xt = σ Wt + κ BtH, driven by a standard Brownian motion W and a fractional Brownian motion BH with Hurst parameter H. We consider strongly consistent estimators of unknown model parameters (H, κ, σ) based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for H ∈ (0, 1/2) ∪ (1/2, 3/4).

Refined moderation analysis with categorical outcomes inprecision medicine.

Moderation analysis is an integral part of precision medicine research. Concerning moderation analysis with categorical outcomes, we start with an interesting observation, which shows that heterogeneous treatment effects could be equivalently estimated via a role exchange between the outcome and the treatment variable in logistic regression models. Hence two estimators of moderating effects can be obtained. We then established the joint asymptotic normality for the two estimators, on which basis refined inference can be made for moderation analysis. The improved precision is helpful in addressing the lack-of-power problem that is common in search of moderators. The above-mentioned results hold for both experimental and observational data. We investigate the proposed method by simulation and provide an illustration with data from a randomized trial on wart treatment.

High Dimensional Normality of Noisy Eigenvectors

We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to a multidimensional Gaussian distribution. The proof involves analyzing the stochastic eigenstate equation (SEE) (Bourgade and Yau in Comm Math Phys, 2013) which describes the Lie group valued flow of eigenvectors induced by matrix valued Brownian motion. We consider the associated colored eigenvector moment flow defining an SDE on a particle configuration space. This flow extends the eigenvector moment flow first introduced in Bourgade and Yau (Comm Math Phys, 2013) to the multicolor setting. However, it is no longer driven by an underlying Markov process on configuration space due to the lack of positivity in the semigroup kernel. Nevertheless, we prove the dynamics admit sufficient averaged decay and contractive properties. This allows us to establish optimal time of relaxation to equilibrium for the colored eigenvector moment flow and prove joint asymptotic normality for eigenvectors. Applications in random matrix theory include the explicit computations of joint eigenvector distributions for general Wigner type matrices and sparse graph models when corresponding eigenvalues lie in the bulk of the spectrum, as well as joint eigenvector distributions for Lévy matrices when the eigenvectors correspond to small energy levels.

Nonparametric regression for locally stationary random fields under stochastic sampling design

In this study, we develop an asymptotic theory of nonparametric regression for locally stationary random fields (LSRFs) {Xs,An:s∈Rn} in Rp observed at irregularly spaced locations in Rn=[0,An]d⊂Rd. We first derive the uniform convergence rate of general kernel estimators, followed by the asymptotic normality of an estimator for the mean function of the model. Moreover, we consider additive models to avoid the curse of dimensionality arising from the dependence of the convergence rate of estimators on the number of covariates. Subsequently, we derive the uniform convergence rate and joint asymptotic normality of the estimators for additive functions. We also introduce approximately mn-dependent RFs to provide examples of LSRFs. We find that these RFs include a wide class of Lévy-driven moving average RFs.

Robust Two-Step Wavelet-Based Inference for Time Series Models

Latent time series models such as (the independent sum of) ARMA(p, q) models with additional stochastic processes are increasingly used for data analysis in biology, ecology, engineering, and economics. Inference on and/or prediction from these models can be highly challenging: (i) the data may contain outliers that can adversely affect the estimation procedure; (ii) the computational complexity can become prohibitive when the time series are extremely large; (iii) model selection adds another layer of (computational) complexity; and (iv) solutions that address (i), (ii), and (iii) simultaneously do not exist in practice. This paper aims at jointly addressing these challenges by proposing a general framework for robust two-step estimation based on a bounded influence M-estimator of the wavelet variance. We first develop the conditions for the joint asymptotic normality of the latter estimator thereby providing the necessary tools to perform (direct) inference for scale-based analysis of signals. Taking advantage of the model-independent weights of this first-step estimator, we then develop the asymptotic properties of two-step robust estimators using the framework of the generalized method of wavelet moments (GMWM). Simulation studies illustrate the good finite sample performance of the robust GMWM estimator and applied examples highlight the practical relevance of the proposed approach.

Stochastic approximation algorithms for superquantiles estimation

This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile, also known as conditional value-at-risk, estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.

Rank dynamics for functional data

The study of the dynamic behavior of cross-sectional ranks over time for functional data and the ranks of the observed curves at each time point and their temporal evolution can yield valuable insights into the time dynamics of functional data. This approach is of interest in various application areas. For the analysis of the dynamics of ranks, estimation of the cross-sectional ranks of functional data is a first step. Several statistics of interest for ranked functional data are proposed. To quantify the evolution of ranks over time, a model for rank derivatives is introduced, where rank dynamics are decomposed into two components. One component corresponds to population changes and the other to individual changes that both affect the rank trajectories of individuals. The joint asymptotic normality for suitable estimates of these two components is established. The proposed approaches are illustrated with simulations and three longitudinal datasets: Growth curves obtained from the Zürich Longitudinal Growth Study, monthly house price data in the US from 1996 to 2015 and Major League Baseball offensive data for the 2017 season.

Generalised regression estimation via the bootstrap

SummaryA generalised regression estimation procedure is proposed that can lead to much improved estimation of population characteristics, such as quantiles, variances and coefficients of variation. The method involves conditioning on the discrepancy between an estimate of an auxiliary parameter and its known population value. The key distributional assumption is joint asymptotic normality of the estimates of the target and auxiliary parameters. This assumption implies that the relationship between the estimated target and the estimated auxiliary parameters is approximately linear with coefficients determined by their asymptotic covariance matrix. The main contribution of this paper is the use of the bootstrap to estimate these coefficients, which avoids the need for parametric distributional assumptions. First‐order correct conditional confidence intervals based on asymptotic normality can be improved upon using quantiles of a conditional double bootstrap approximation to the distribution of the studentised target parameter estimate.

Risk forecasting in the context of time series*

We propose an approach for forecasting risk contained in future observations in a time series. We take into account both the shape parameter and the extremal index of the data. This significantly improves the quality of risk forecasting over methods that are designed for i.i.d. observations and over the return level approach. We prove functional joint asymptotic normality of the common estimators of the shape parameter and and extremal index estimators, based on which statistical properties of the proposed forecasting procedure can be analyzed.

Empirical Characteristic Functions‐Based Estimation and Distance Correlation for Locally Stationary Processes

In this article, we propose a kernel‐type estimator for the local characteristic function of locally stationary processes. Under weak moment conditions, we prove joint asymptotic normality for local empirical characteristic functions. For time‐varying linear processes, we establish a central limit theorem under the assumption of finite absolute first moments of the process. Additionally, we prove weak convergence of the local empirical characteristic process. We apply our asymptotic results to parameter estimation. Furthermore, by extending the notion of distance correlation to locally stationary processes, we are able to provide asymptotic theory for local empirical distance correlations. Finally, we provide a simulation study on minimum distance estimation for α‐stable distributions and illustrate the pairwise dependence structure over time of log returns of German stock prices via local empirical distance correlations.

Inference in a multivariate generalized mean-reverting process with a change-point

In this paper, we study inference problem about the drift parameter matrix in multivariate generalized Ornstein–Uhlenbeck processes with an unknown change-point. In particular, we consider the case where the parameter matrix may satisfy some restrictions. Thus, we generalize in five ways some recent findings about univariate generalized Ornstein–Uhlenbeck processes. First, the target parameter is a matrix and we derive a sufficient condition for the existence of the unrestricted estimator (UE) and the restricted estimator (RE). Second, we establish the joint asymptotic normality of the UE and the RE under a collection of local alternatives. Third, we construct a test for testing the uncertain restriction. The proposed test is also useful for testing the absence of the change-point. Fourth, we derive the asymptotic power of the proposed test and we prove that it is consistent. Fifth, we propose the shrinkage estimators (SEs) and we prove that SEs dominate the UE. Finally, we conduct some simulation studies which corroborate our theoretical findings.

Joint asymptotic normality of granulometric moments under multiple structuring elements

For a binary image composed of randomly sized disjoint grains, morphological granulometries produce a pattern spectrum containing textural information that can be used for image classification, image segmentation and model estimation. It has been shown that as the number of grains per image increases, any finite collection of granulometric moments based on the same structuring element are jointly asymptotically normal with a known analytic form for the moments. However, many applications require the use of multiple structuring elements, for instance horizontal and vertical linear generators, to draw out important textural characteristics. In this work, we prove the joint asymptotic normality of granulometric moments from multiple structuring elements. We also derive analytic expressions for the asymptotic mean vector and covariance matrix of the granulometric moments, including the previously unknown off-diagonal elements of the asymptotic covariance matrix corresponding to granulometric moments from distinct structuring elements.

Asymptotic normality of the local linear estimation of the conditional density for functional time-series data

ABSTRACTThis article focuses on the conditional density of a scalar response variable given a random variable taking values in a semimetric space. The local linear estimators of the conditional density and its derivative are considered. It is assumed that the observations form a stationary α-mixing sequence. Under some regularity conditions, the joint asymptotic normality of the estimators of the conditional density and its derivative is established. The result confirms the prospect in Rachdi et al. (2014) and can be applied in time-series analysis to make predictions and build confidence intervals. The finite-sample behavior of the estimator is investigated by simulations as well.