Finite Fourth Moment Research Articles

Estimation and tests for power-transformed and threshold GARCH models.

Consider a class of power-transformed and threshold GARCH (PTTGRACH) model, which is a natural generalization of power-transformed and threshold GARCH(1,1) model in Hwang and Basawa [2004. Stationarity and moment structure for Box–Cox transformed threshold GARCH(1,1) processes. Statistics & Probability Letters 68, 209–220.] and includes the standard GARCH model and many other models as special cases. We first establish the asymptotic normality for quasi-maximum likelihood estimators (QMLE) of the parameters under the condition that the error distribution has finite fourth moment. For the case of heavy-tailed errors, we propose a least absolute deviations estimation (LADE) for PTTGARCH model, and prove that the LADE is asymptotically normally distributed under very weak moment conditions. This paves the way for a statistical inference based on asymptotic normality for heavy-tailed PTTGARCH models. As a consequence, we can construct the Wald test for GARCH structure and discuss the order selection problem in heavy-tailed cases. Numerical results show that LADE is more accurate than QMLE for heavy-tailed errors. Furthermore, the theory is applied to the daily returns of the Hong Kong Hang Seng Index, which suggests that asymmetry and nonlinearity could be present in the financial time series and the PTTGARCH model is capable of capturing these characteristics. As for the probabilistic structure of PTTGARCH model, we give in the appendix a necessary and sufficient condition for the existence of a strictly stationary solution of the model, the existence of the moments and the tail behavior of the strictly stationary solution.

Model-free versus Model-based Volatility Prediction

The well-known ARCH/GARCH models for financial time series have been criticized of late for their poor performance in volatility prediction, that is, prediction of squared returns. Focusing on three representative data series, namely a foreign exchange series (Yen vs. Dollar), a stock index series (the S&P500 index), and a stock price series (IBM), the case is made that financial returns may not possess a finite fourth moment. Taking this into account, we show how and why ARCH/GARCH models when properly applied and evaluated actually do have nontrivial predictive validity for volatility. Furthermore, we show how a simple model-free variation on the ARCH theme can perform even better in that respect. The model-free approach is based on a novel normalizing and variance stabilizing transformation (NoVaS, for short) that can be seen as an alternative to parametric modeling. Properties of this transformation are discussed, and practical algorithms for optimizing it are given.

Parametric robust inference about regression parameters for the correlation coefficient

This article establishes a robust likelihood function about regression parameters for the correlation coefficients modeled in a generalized linear model fashion. The validity of the proposed likelihood requires no knowledge of the true underlying distributions, so long as they have finite fourth moments. The efficacy of the robust methodology is shown via simulations. The asymptotic variance of the maximum-likelihood estimate and the empirical error probabilities of the resultant robust likelihood ratio test are specifically exhibited.

Spectral norm of random large dimensional noncentral Toeplitz and Hankel matrices

Suppose $s_n$ is the spectral norm of either the Toeplitz or the Hankel matrix whose entries come from an i.i.d. sequence of random variables with positive mean $\mu$ and finite fourth moment. We show that $n^{-1/2}(s_n-n\mu)$ converges to the normal distribution in either case. This behaviour is in contrast to the known result for the Wigner matrices where $s_n-n\mu$ is itself asymptotically normal.

Parametric Robust Test for Several Variances with Unknown Underlying Distributions

Tsou (in comm Stat-Theor Math 32: 2013–2019, 2003) proposed a parametric robust procedure for testing the equality of two population variances. With large samples the proposed test remains valid under model misspecification. In this article the robust technique is further extended to the comparison of several population variances. More specifically the score test derived on the basis of normal models is adjusted to become robust. The adjusted robust test provides asymptotically valid inference so long as the true underlying distributions have finite fourth moments. Unlike most robust nonparametric approaches, this novel robust technique too provides legitimate variance estimates for estimators of the interested parameters.

Asymptotic values and expansions for the correlation between different measures of spread

For i.i.d. samples from a normal, uniform, or exponential distribution, we give exact formulas for the correlation between the sample variance and the sample range for all fixed n. These exact formulas are then used to obtain asymptotic expansions for the correlations. It is seen that the correlation converges to zero at the rate log n / n in the normal case, the 1 / n rate in the uniform case, and the ( log n ) 2 / n rate in the exponential case. In two of the three cases, we obtain higher-order expansions for the correlation. We then obtain the joint asymptotic distribution of the interquartile range and the standard deviation for any distribution with a finite fourth moment. This is used to obtain the nonzero limits of the correlation between them for some important distributions as well as some potentially useful practical diagnostics based on the interquartile range and the standard deviation. It is seen that the correlation is higher for thin tailed and smaller for thick tailed distributions. We also show graphics for the Cauchy distribution. The graphics exhibit interesting phenomena. Other numerics illustrate the theoretical results.

Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

We consider quadratic forms of the type $$ Q(F,{\bf A})=\sum_{\mathop{1\le j,k \le N}\limits_{j\ne k}}a_{jk} X_j X_k, $$ where X j are i.i.d. random variables with common distribution F and finite fourth moment, $${\bf A}=\{a_{jk}\}_{j,k=1}^N$$ denotes a symmetric matrix with eigenvalues λ1, ..., λ N ordered to be non-increasing in absolute value. We prove explicit bounds in terms of sums of 4th powers of entries of the matrix A and the size of the eigenvalue λ13 for the approximation of the distribution of Q(F,A) by the distribution of Q (φ, A) where φ is standard Gaussian distribution. In typical cases this error is of optimal order $${\cal {O}}(N^{-1})$$

LARGE SAMPLE PROPERTIES OF THE SIR IN CDMA WITH MATCHED FILTER RECEIVERS

The output signal-to-interference (SIR) of conventional matched filter receiver in random environment is considered. When the number of users and the spreading factors tend to infinity with their ratio fixed, the convergence of SIR is showed to be with probability one under finite fourth moment of the spreading sequences. The asymptotic distribution of the SIR is also obtained.

Parameter Estimation for Periodically Stationary Time Series

Abstract. The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper, we compute the asymptotic distribution for these estimates in the case, where the innovations have a finite fourth moment. These asymptotic results are useful to determine which model parameters are significant. In the process, we also develop asymptotics for the Yule–Walker estimates.

CLT for linear spectral statistics of large-dimensional sample covariance matrices

Let $B_n=(1/N)T_n^{1/2}X_nX_n^*T_n^{1/2}$ where $X_n=(X_{ij})$ is $n\times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. The limiting behavior, as $n\to\infty$ with $n/N$ approaching a positive constant, of functionals of the eigenvalues of $B_n$, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of $B_n$, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be $1/n$ by proving, after proper scaling, that they form a tight sequence. Moreover, if $\expp X^2_{11}=0$ and $\expp|X_{11}|^4=2$, or if $X_{11}$ and $T_n$ are real and $\expp X_{11}^4=3$, they are shown to have Gaussian limits.

Comparing Two Population Means and Variances: A Parametric Robust Way

This article introduces a parametric robust way of comparing two population means and two population variances. With large samples the comparison of two means, under model misspecification, is lesser a problem, for, the validity of inference is protected by the central limit theorem. However, the assumption of normality is generally required, so that the inference for the ratio of two variances can be carried out by the familiar F statistic. A parametric robust approach that is insensitive to the distributional assumption will be proposed here. More specifically, it will be demonstrated that the normal likelihood function can be adjusted for asymptotically valid inferences for all underlying distributions with finite fourth moments. The normal likelihood function, on the other hand, is itself robust for the comparison of two means so that no adjustment is needed.

Nonparametric covariance estimation in multivariate distributions

The estimation problem of the unknown covariance matrix of a multivariate distribution with the known mean is studied under a matrix-valued quadratic loss function. The conditions on the sample sizes for the best unbiased estimator to have a smaller risk than the sample covariance matrix is established. The former estimator is completely (without exceptional sets of Lebesgue measure zero) characterized by its expectation in the class of all multivariate distributions with zero mean and finite fourth moments.

Innovations algorithm for periodically stationary time series

Periodic ARMA, or PARMA, time series are used to model periodically stationary time series. In this paper we develop the innovations algorithm for periodically stationary processes. We then show how the algorithm can be used to obtain parameter estimates for the PARMA model. These estimates are proven to be weakly consistent for PARMA processes whose underlying noise sequence has either finite or infinite fourth moment. Since many time series from the fields of economics and hydrology exhibit heavy tails, the results regarding the infinite fourth moment case are of particular interest.

Exact Separation of Eigenvalues of Large Dimensional Sample Covariance Matrices

Let $B _n = (1/N) T_n^{1/2} X _n X _n^*T_n^{1/2}$ where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. It was shown in an earlier paper by the authors that, under certain conditions on the eigenvalues of $T_n$, with probability 1 no eigenvalues lie in any interval which is outside the support of the limiting empirical distribution (known to exist) for all large $n$. For these $n$ the interval corresponds to one that separates the eigenvalues of $T_n$. The aim of the present paper is to prove exact separation of eigenvalues; that is, with probability 1, the number of eigenvalues of $B_n$ and $T_n$ lying on one side of their respective intervals are identical for all large $n$.

Remarks on the Convergence Rate of the Spectral Distributions of Wigner Matrices

In this remark, we show that under the assumption of the finite fourth moment elements in the empirical spectral distribution of a large Wigner matrix converges to the Wigner semicircular law in probability of the order O(n−1/3).

The homogenization problem for the Vicsek set

We discuss the homogenization problem for a particular class of nested fractals, called “tree-like” Vicsek sets, which are not covered by the class considered in Kumagai and Kusuoka (1996). Random irreducible conductivities are assigned to each cell in the infinite fractal lattice and then we consider the rescaled effective resistance. We only assume that the effective resistance across the unit cell has finite first moment and prove that the rescaled effective resistance converges. We also consider the limiting behaviour of the corresponding Markov chains under fractal “Brownian motion” scaling. By assuming finite first moment for the conductivity and finite fourth moments of the effective resistance across the unit cell, we show that for almost every environment, under the measure on the conductivities, the Markov chains converge weakly to a diffusion on the fractal. The limit process does not, in general, coincide with the “Brownian motion” on the original Vicsek set.

No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices

Let $B_n = (1/N)T_n^{1/2}X_n X_n^* T_n^{1/2}$, where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. It is known that, as $n \to \infty$, if $n/N$ converges to a positive number and the empirical distribution of the eigenvalues of $T_n$ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of $T_n$, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all $n$ sufficiently large.

Uniform convergence of the empirical spectral distribution function

Let X be a linear process having a finite fourth moment. Assume F is a class of square-integrable functions. We consider the empirical spectral distribution function J n, X based on X and indexed by F . If F is totally bounded then J n, X satisfies a uniform strong law of large numbers. If, in addition, a metric entropy condition holds, then J n, X obeys the uniform central limit theorem.

On Estimating the Dimensionality in Canonical Correlation Analysis

In canonical correlation analysis the number of nonzero population correlation coefficients is called the dimensionality. Asymptotic distributions of the dimensionalities estimated by Mallows's criterion and Akaike's criterion are given for nonnormal multivariate populations with finite fourth moments. These distributions have a simple form in the case of elliptical populations, and modified criteria are proposed which adjust for nonzero kurtosis. An estimation method based on a marginal likelihood function for the dimensionality is introduced and the asymptotic distribution of the corresponding estimator is derived for multivariate normal populations. It is shown that this estimator is not consistent, but that a simple modification yields consistency. An overall comparison of the various estimation methods is conducted through simulation studies.

The stability of LMS

New and weak conditions are given under which the LMS algorithm is exponentially convergent with probability one in a stochastic setting. These results show that LMS works under very broad conditions: a small gain, input signal with finite fourth moments; time varying persistence of excitation; and weak assumptions on the correlation structure of the input signal. Previous results at this level of generality linked convergence to peak signal amplitude rather than average amplitude. Under the same conditions stochastic boundedness of a forced LMS system is also established.