Narrow Results

We provide the nonparametric estimators of the infinitesimal coefficients of the second-order continuous-time models with discontinuous sample paths of jump-diffusion models. Under the mild conditions, we obtain the weak consistency and the asymptotic normality of the estimators. A Monte Carlo experiment demonstrates the better small-sample performance of these estimators. In addition, the estimators are illustrated empirically through stock index of Shanghai Stock Exchange in high frequency data.

In this paper we study the complex interactions involved in the incoming stimulus, from a gamma (γ) and/or an alpha (α) motoneuron, and the outgoing response from the muscle spindle transmitted by the Ia sensory afferent neuron to the spinal cord. The most interesting case is the γ and α coactivation to the function of the muscle spindle, while the effect from a single (γ or α) motoneuron is also presented as a comparison. The mathematical background of this analysis is based on the theory of stationary point processes. A kernel type method of estimating second- and third-order conditional densities is used. Under certain conditions the asymptotic distributions of these estimates are Normal and the construction of 95% approximate confidence intervals is feasible. The application of these asymptotic results to the system of the muscle spindle enables us to detect and interpret its excitatory and/or inhibitory behavior.

We consider the solution to the stochastic heat equation driven by the time-space white noise and study the asymptotic behavior of its spatial quadratic variations with “moving time”, meaning that the time variable is not fixed and its values are allowed to be very big or very small. We investigate the limit distribution of these variations via Malliavin calculus.

A parameter estimation problem is considered for a diagonalizable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter H ≥ 1/2. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation.

We study parameter estimation problem for diagonalizable stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter H ∈ (0, 1). Two classes of estimators are investigated: traditional maximum likelihood type estimators, and a new class called closed-form exact estimators. Finally the general results are applied to stochastic heat equation driven by a fractional Brownian motion.

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators. It is also a powerful tool in understanding deeper properties of partial sums of independent random variables. In this paper we review some basic results of the theory and discuss new results in the central limit theory of trimmed sums. In particular, we show that for random variables in the domain of attraction of a stable law with parameter 0 < α < 2, the asymptotic behavior of modulus trimmed sums depends sensitively on the number of elements eliminated from the sample. We also show that under moderate trimming, the central limit theorem always holds if we allow random centering factors. Finally, we give an application to change point problems.

In this paper, we consider the generalized linear models (GLMs)

In this paper, we provide an asymptotic expression for mean integrated squared error (MISE) of nonlinear wavelet density estimator for a truncation model. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimator, the MISE expression of the nonlinear wavelet estimator is not affected by the presence of discontinuities in the curves. Also, we establish asymptotic normality of the nonlinear wavelet estimator.

This paper studies the parameter estimation of one dimensional linear errors-in-variables (EV) models in the case that replicated observations are available in some experimental points. Asymptotic normality is established under mild conditions, and the parameters entering the asymptotic variance are consistently estimated to render the result useable in construction of large-sample confidence regions.

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.

For the Generalized Linear Model (GLM), under some conditions including that the specification of the expectation is correct, it is shown that the Quasi Maximum Likelihood Estimate (QMLE) of the parameter-vector is asymptotic normal. It is also shown that the asymptotic covariance matrix of the QMLE reaches its minimum (in the positive-definte sense) in case that the specification of the covariance matrix is correct.

We report on some statistical regularity properties of greatest common divisors: for large random samples of integers, the number of coprime pairs and the average of the gcd's of those pairs are approximately normal, while the maximum of those gcd's (appropriately normalized) follows approximately a Fréchet distribution. We also consider r-tuples instead of pairs, and moments other than the average.

Change-point detection is an integral component of statistical modeling and estimation. For high-dimensional data, classical methods based on the Mahalanobis distance are typically inapplicable. We propose a novel testing statistic by combining a modified Euclidean distance and an extreme statistic, and its null distribution is asymptotically normal. The new method naturally strikes a balance between the detection abilities for both dense and sparse changes, which gives itself an edge to potentially outperform existing methods. Furthermore, the number of change-points is determined by a new Schwarz’s information criterion together with a pre-screening procedure, and the locations of the change-points can be estimated via the dynamic programming algorithm in conjunction with the intrinsic order structure of the objective function. Under some mild conditions, we show that the new method provides consistent estimation with an almost optimal rate. Simulation studies show that the proposed method has satisfactory performance of identifying multiple change-points in terms of power and estimation accuracy, and two real data examples are used for illustration.

The monitoring of high-dimensional data streams has become increasingly important for real-time detection of abnormal activities in many statistical process control (SPC) applications. Although the multivariate SPC has been extensively studied in the literature, the challenges associated with designing a practical monitoring scheme for high-dimensional processes when between-streams correlation exists are yet to be addressed well. Classical $T^2$-test-based schemes do not work well because the contamination bias in estimating the covariance matrix grows rapidly with the increase of dimension. We propose a test statistic which is based on the “divide-and-conquer” strategy, and integrate this statistic into the multivariate exponentially weighted moving average charting scheme for Phase II process monitoring. The key idea is to calculate the $T^2$ statistics on low-dimensional sub-vectors and to combine them together. The proposed procedure is essentially distribution-free and computation efficient. The control limit is obtained through the asymptotic distribution of the test statistic under some mild conditions on the dependence structure of stream observations. Our asymptotic results also shed light on quantifying the size of a reference sample required. Both theoretical analysis and numerical results show that the proposed method is able to control the false alarm rate and deliver robust change detection.

We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It provides a decomposition of the largest $K$ singular values into the deterministic rate of growth, random centered fluctuations given as explicit linear combinations of the entries of the matrix, and a term negligible in probability. We use this representation to establish asymptotic normality of the largest singular values for random matrices with means that have block structure. We also deduce asymptotic normality for the largest eigenvalues of a random matrix arising in a model of population genetics.

In this paper, for the generalized estimating equation (GEE) with diverging number of covariates, the asymptotic properties of GEE estimator are considered. Under the weaker assumption on the minimum eigenvalue of Fisher information matrix and some other regular conditions, we prove the asymptotic existence, consistency and asymptotic normality of the GEE estimator and the asymptotic distribution of the test statistics of linear combination of the unknown parameters. The results are illustrated by Monte-Carlo simulations.

When the conditional normality assumption on the regression disturbance does not hold, the OLS estimator no longer has the finite sample normal distribution, and the t-test and F-test statistics no longer follow the Student’s t-distribution and F-distribution in finite samples respectively. In this chapter, we show that under the assumption of IID observations with conditional homoskedasticity, the classical t-test and F-test are approximately applicable in large samples. However, under conditional heteroskedasticity, the t-test and F-test statistics are not applicable even when the sample size goes to infinity. Instead, White’s (1980) heteroskedasticity-consistent variance-covariance matrix estimator should be used, which yields asymptotically valid confidence interval estimation and hypothesis test procedures. A direct test for conditional heteroskedasticity due to White (1980) is presented. To facilitate asymptotic analysis in this and subsequent chapters, we introduce some basic tools for asymptotic analysis in this chapter.

Varying-coefficient partially linear (VCPL) models are very useful tools. This chapter focuses on inferences for the VCPL model when the errors are serially correlated and modeled as an AR process. A penalized spline least squares (PSLS) estimation is proposed based on the penalized spline technique. This approach is then improved by a weighted PSLS estimation. We investigate the asymptotic theory under the assumption that the number of knots is fixed, though potentially large. The weighted PSLS estimators of all parameters are shown to be -consistent, asymptotically normal and asymptotically more efficient than the un-weighted ones. The proposed method can be used to make simultaneous inference for the parametric and nonparametric components by virtue of the sandwich formula for the joint covariance matrix. Simulations are conducted to demonstrate the finite sample performance of the proposed estimators. A real data analysis is used to illustrate the application of the proposed method.

We consider L₁-norm kernel estimator of the conditional median θ(x), for a wide class of stationary processes. Asymptotic normality of the resulting estimator θ_n(x) is established under different regularity conditions on bandwidths. Applications of our main results are also given.

After a quick introduction to some basic properties of U statistics with examples, we discuss M_m estimators and their asymptotic properties under easily verifiable conditions. In particular, these estimators are approximately U statistics and as a consequence, a huge collection of commonly used estimators are consistent and asymptotically normal. We also establish some higher order asymptotic properties of these estimates. The material is more or less self contained.