Weak Moment Conditions Research Articles

Convergence Rates of Regularized Huber Regression Under Weak Moment Conditions

Convergence Rates of Regularized Huber Regression Under Weak Moment Conditions

Least squares regression under weak moment conditions

In this paper we consider the robust regression problem when the output variable may be heavy-tailed. In such scenarios, the traditional least squares regression paradigm is usually thought to be not a good choice as it lacks robustness to outliers. By projecting the outputs onto an adaptive interval, we show the regularized least squares regression can still work well when the conditional distribution satisfies a weak moment condition. Fast convergence rates in various norm are derived by tuning the projection scale parameter and regularization parameter in according with the sample size and the moment condition.

Strong limit theorem for largest entry of large-dimensional random tensor

Suppose that [Formula: see text] are i.i.d. copies of random vector [Formula: see text]. Let [Formula: see text] then the random tensor product constructed by [Formula: see text] is defined by [Formula: see text] In this paper, we obtain the strong limit theorems of the largest entry of large-dimensional random tensor product [Formula: see text] under two high-dimensional settings the polynomial rate and the exponential rate. The conclusions are established under weaker moment condition than the exist papers and the relationship between [Formula: see text] and [Formula: see text] is more flexible.

An Excess Entropy Approach to Classify Long-Term and Short-Term Memory Stationary Time Series

Long-term memory behavior is one of the most important phenomena that has appeared in the time series analysis. Different from most definitions of second-order properties, an excess entropy approach is developed for stationary time series to classify long-term and short-term memory. A stationary sequence with finite block entropy is long-term memory if its excess entropy is infinite. The simulation results are graphically demonstrated after some theoretical results are simply presented by various stochastic sequences. Such an approach has advantages over the traditional ways that the excess entropy of stationary sequence with finite block entropy is invariant under instantaneous one-to-one transformation, and that it only requires very weak moment conditions rather than second-order moment conditions and thus can be applied to distinguish the LTM behavior of stationary sequences with unbounded second moment (e.g., heavy tail distribution). Finally, several applications on real data are exhibited.

The Marcinkiewicz–Zygmund law of large numbers for exchangeable arrays

We show a Marcinkiewicz–Zygmund law of large numbers for jointly, dissociated exchangeable arrays, in Lr (r∈(0,2)) and almost surely. Then, we obtain a law of iterated logarithm for such arrays under a weaker moment condition than the existing one.

Change point analysis of covariance functions: A weighted cumulative sum approach

We develop and study change point detection and estimation procedures for the covariance kernel of functional data based on the norms of a generally weighted process of partial sample estimates. It is shown under mild weak dependence and moment conditions on the data that in the absence of a change point a detector based on integrating such a process over the partial sample parameter is asymptotically distributed as the norm of a Gaussian process, which furnishes a consistent change point detection procedure. We further derive consistency and local asymptotic results for this detector in the presence of a change in the covariance function. The corresponding change point estimator based on such a process is also shown to be rate optimal for estimating an existing change point, and further is asymptotically distributed as the argument maximum of a Gaussian process under a local asymptotic framework. We study the detector and change point estimator in a small simulation study to detect changes in the covariance of functional autoregressive and generalized conditionally heteroscedastic processes, which demonstrate that the use of the weighted CUSUM statistics in this context generally improves performance over existing methods. These new statistics are demonstrated in an application to detecting changes in the volatility of high resolution intraday asset price curves derived from oil futures prices.

Learning theory of minimum error entropy under weak moment conditions

Minimum error entropy (MEE) is an information theoretic learning approach that minimizes the information contained in the prediction error, which is measured by entropy. It has been successfully used in various machine learning tasks for its robustness to heavy-tailed distributions and outliers. In this paper, we consider its use in nonparametric regression and analyze its generalization performance from a learning theory perspective by imposing a [Formula: see text]th order moment condition on the noise variable. To this end, we establish a comparison theorem to characterize the relation between the excess generalization error and the prediction error. A relaxed Bernstein condition and concentration inequalities are used to derive error bounds and learning rates. Note that the [Formula: see text]th moment condition is rather weak particularly when [Formula: see text] because the noise variable does not even admit a finite variance in this case. Therefore, our analysis explains the robustness of MEE in the presence of heavy-tailed distributions.

Strong convergence for weighted sums of END random variables under the sub-linear expectations

In this article, we study strong limit theorems for weighted sums of extended negatively dependent (END, for short) random variables under the sub-linear expectations. We establish some general complete convergence theorems for weighted sums of END random variables under the sub-linear expectations. Our results partly generalize and improve the corresponding ones previously obtained by Cai (Metrika, 68:323-331, 2008), Huang and Wang (J. Inequal. Appl. 233, doi:10.1186/1029-242X-2012-233, 2012) and Wang et al. (RACSAM, 106:235-245, 2012) in the classical probability space to the sub-linear expectation space under weaker moment conditions.

On distance covariance in metric and Hilbert spaces

Distance covariance is a measure of dependence between two random variables that take values in two, in general different, metric spaces, see Sz\'ekely, Rizzo and Bakirov (2007) and Lyons (2013). It is known that the distance covariance, and its generalization $\alpha$-distance covariance, can be defined in several different ways that are equivalent under some moment conditions. The present paper considers four such definitions and find minimal moment conditions for each of them, together with some partial results when these conditions are not satisfied. The paper also studies the special case when the variables are Hilbert space valued, and shows under weak moment conditions that two such variables are independent if and only if their ($\alpha$-)distance covariance is 0; this extends results by Lyons (2013) and Dehling et al. (2018+). The proof uses a new definition of distance covariance in the Hilbert space case, generalizing the definition for Euclidean spaces using characteristic functions by Sz\'ekely, Rizzo and Bakirov (2007).

The Shanghai- Hong Kong Stock Connect: An Application of the Semi-CGARCH and Semi-EGARCH

This paper examines the impact on volatility on the Shanghai Stock Exchange and the Hong Kong Stock Exchange before and after the connection on November 17, 2014. We test whether this event led to a structural break. For this purpose, the volatility series are shown and analysed in more detail using two new models. We test both semiparametric GARCH extensions, the Semi-EGARCH based on the EGARCH model (Nelson, 1991) and the Semi-CGARCH model, based on the CGARCH (Engle & Lee, 1999). Both univariate models work with a constrained local linear estimator for the scale function and a fully data-driven algorithm developed under weak moment conditions. The proposed method is applied to the two major Asian financial indices and to stocks from the banking sector. Furthermore, our focus is to improve the quantitative risk management analysis. When a parametric GARCH model provides satisfactory results for the calculation of e.g the Value at Risk (VaR), semiparametric can also be applied to improve the quality of measurement. This article compares the results of various parametric and semiparametric approaches regarding the VaR to show how the extensions increase the performance.

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.

Short Communications Moderate Large Deviations Principles for Trajectories of Compound Renewal Processes

Moderate large deviations principles for trajectories of compound renewal processes are put forward assuming that either the Cramér condition is satisfied or weaker moment conditions are satisfied.MSC codescompound renewal processesmoderate large deviations principleCramér conditioncrude (logarithmic) invariance principle

Nonlinear Least Squares Estimation of Log-ACD Models

This paper studies a nonlinear least squares estimation method for the logarithmic autoregressive conditional duration (Log-ACD) model. We establish the strong consistency and asymptotic normality for our estimator under weak moment conditions suitable for applications involving heavy-tailed distributions. We also discuss inference for the Log-ACD model and Log-ACD models with exogenous variables. Our results can be easily translated to study Log-GARCH models. Both simulation study and real data analysis are conducted to show the usefulness of our results.

Asymptotics of the weighted least squares estimation for AR(1) processes with applications to confidence intervals

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.

Comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering.

Comparing large covariance matrices has important applications in modern genomics, where scientists are often interested in understanding whether relationships (e.g., dependencies or co-regulations) among a large number of genes vary between different biological states. We propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes. A distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices. Hence, the test is robust with respect to various complex dependence structures that frequently arise in genomics. We prove that the proposed procedure is asymptotically valid under weak moment conditions. As an interesting application, we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high-dimensional genomics data. Using an asthma gene expression dataset, we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets/pathways between the disease group and the control group, and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups. The proposed methods have been implemented in an R-package HDtest and are available on CRAN.

Local semicircle law under weak moment conditions

Symmetric random matrices are considered whose upper triangular entries are independent identically distributed random variables with zero mean, unit variance, and a finite moment of order 4 + δ, δ > 0. It is shown that the distances between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law are of order lnn/nv, where v is the distance to the real axis in the complex plane. Applications concerning the convergence rate in probability to the semicircle law, localization of eigenvalues, and delocalization of eigenvectors are discussed.

Exact convergence rates in central limit theorems for a branching random walk with a random environment in time

Chen (2001) derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process. We extend Chen’s results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen’s by relaxing the second moment condition used by Chen to a moment condition of the form EX(ln+X)1+λ<∞. In the rate functions that we find for a branching random walk, we figure out some new terms which did not appear in Chen’s work. The results are established in the more general framework, i.e. for a branching random walk with a random environment in time. The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment.

On Weak Invariance Principles for Partial Sums

Given a sequence of random functionals \(\bigl \{X_k(u)\bigr \}_{k \in \mathbb {Z}}\), \(u \in \mathbf{I}^d\), \(d \ge 1\), the normalized partial sums \(\check{S}_{nt}(u) = n^{-1/2}\bigl (X_1(u) + \cdots + X_{\lfloor n t \rfloor }(u)\bigr )\), \(t \in [0,1]\) and its polygonal version \({S}_{nt}(u)\) are considered under a weak dependence assumption and \(p > 2\) moments. Weak invariance principles in the space of continuous functions and cadlag functions are established. A particular emphasis is put on the process \(\check{S}_{nt}(\widehat{\theta })\), where \(\widehat{\theta } \xrightarrow {\mathbb {P}} \theta \), and weaker moment conditions (\(p = 2\) if \(d = 1\)) are assumed.

Two-dimensional renewal theorems with weak moment conditions and critical Bellman – Harris branching processes

AbstractWe consider critical Bellman – Harris processes with two types of particles. The tail of the lifetime distribution of the first type particles decreases as

A New Pearson-Type QMLE for Conditionally Heteroscedastic Models

This article proposes a novel Pearson-type quasi-maximum likelihood estimator (QMLE) of GARCH(p, q) models. Unlike the existing Gaussian QMLE, Laplacian QMLE, generalized non-Gaussian QMLE, or LAD estimator, our Pearsonian QMLE (PQMLE) captures not just the heavy-tailed but also the skewed innovations. Under strict stationarity and some weak moment conditions, the strong consistency and asymptotic normality of the PQMLE are obtained. With no further efforts, the PQMLE can be applied to other conditionally heteroscedastic models. A simulation study is carried out to assess the performance of the PQMLE. Two applications to four major stock indexes and two exchange rates further highlight the importance of our new method. Heavy-tailed and skewed innovations are often observed together in practice, and the PQMLE now gives us a systematic way to capture these two coexisting features.