Proof Of Asymptotic Normality Research Articles

Intersections of Poisson [formula omitted]-flats in constant curvature spaces

Poisson processes in the space of k-dimensional totally geodesic subspaces (k-flats) in a d-dimensional standard space of constant curvature κ∈{−1,0,1} are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order m together with their (d−m(d−k))-dimensional Hausdorff measure within a geodesic ball of radius r. Asymptotic normality for fixed r is shown as the intensity of the underlying Poisson process tends to infinity for all m satisfying d−m(d−k)≥0. For κ∈{−1,0} the problem is also approached in the set-up where the intensity is fixed and r tends to infinity. Again, if 2k≤d+1 a central limit theorem is shown for all possible values of m. However, while for κ=0 asymptotic normality still holds if 2k>d+1, we prove for κ=−1 convergence to a non-Gaussian infinitely divisible limit distribution in the special case m=1. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin–Stein method. We also show for general κ∈{−1,0,1} that, roughly speaking, the variances within a general observation window W are maximal if and only if W is a geodesic ball having the same volume as W. Along the way we derive a new integral-geometric formula of Blaschke–Petkantschin type in a standard space of constant curvature.

T分布的渐进正态性证明及其特点分析

t分布的渐进正态性证明及其特点分析

Capacity Analysis of Distributed Antenna Systems in a Composite Fading Channel

In this paper, we investigate a behavior of the cell average ergodic capacity for distributed antenna systems (DAS) in a composite fading channel model which contains small-scale and large-scale fadings. For small-scale Rayleigh fadings, based on the proof of asymptotic normality, the mean and the variance of the instantaneous capacity were recently presented as a closed form solution. However, this solution is too complicated to be applied directly for obtaining the average ergodic capacity over a cell. In this work, we derive a simple and accurate expression for the ergodic capacity by utilizing the high signal to noise ratio (SNR) analysis. Our simple solution provides meaningful insight on how the ergodic capacity is affected as SNR, pathloss and antenna configurations change. Also it is useful for capturing quantitative performance measures such as the multiplexing gain and the power offset. In addition, we analyze the cell average ergodic capacity by taking into account the shadowing effect and the pathloss on the basis of our results on the small-scale fadings. These expressions lead to some insights on the performance of DAS under practical environments. Finally, numerical results confirm the validity of our analytical results.

Asymptotic expansion of the minimum covariance determinant estimators

In Cator and Lopuhaä ( arXiv:math.ST/0907.0079) [3], an asymptotic expansion for the minimum covariance determinant (MCD) estimators is established in a very general framework. This expansion requires the existence and non-singularity of the derivative in a first-order Taylor expansion. In this paper, we prove the existence of this derivative for general multivariate distributions that have a density and provide an explicit expression, which can be used in practice to estimate limiting variances. Moreover, under suitable symmetry conditions on the density, we show that this derivative is non-singular. These symmetry conditions include the elliptically contoured multivariate location-scatter model, in which case we show that the MCD estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent identically distributed vector and matrix valued random elements, respectively. This provides a proof of asymptotic normality and a precise description of the limiting covariance structure for the MCD estimators.

The proportional hazards model with covariate measurement error

The proportional hazards regression model is commonly used to evaluate the relationship between survival and covariates. Covariates are frequently measured with error. Substituting mismeasured values for the true covariates leads to biased estimation. Hu et al. (Biometrics 88 (1998) 447) have proposed to base estimation in the proportional hazards model with covariate measurement error on a joint likelihood for survival and the covariate variable. Nonparametric maximum likelihood estimation (NPMLE) was used and simulations were conducted to assess the asymptotic validity of this approach. In this paper, we derive a rigorous proof of asymptotic normality of the NPML estimators.

A proof of asymptotic normality for some VARX models

Here we present a proof of the asymptotic normality of least squares estimates for stable multivariate autoregressive models excited by a deterministic second order input signal.

An Asymptotic Theory for Sliced Inverse Regression

Sliced inverse regression [Li (1989), (1991) and Duan and Li (1991)] is a nonparametric method for achieving dimension reduction in regression problems. It is widely applicable, extremely easy to implement on a computer and requires no nonparametric smoothing devices such as kernel regression. If $Y$ is the response and $X \in \mathbf{R}^p$ is the predictor, in order to implement sliced inverse regression, one requires an estimate of $\Lambda = E\{\operatorname{cov}(X\mid Y)\} = \operatorname{cov}(X) - \operatorname{cov}\{E(X\mid Y)\}$. The inverse regression of $X$ on $Y$ is clearly seen in $\Lambda$. One such estimate is Li's (1991) two-slice estimate, defined as follows: The data are sorted on $Y$, then grouped into sets of size 2, the covariance of $X$ is estimated within each group and these estimates are averaged. In this paper, we consider the asymptotic properties of the two-slice method, obtaining simple conditions for $n^{1/2}$-convergence and asymptotic normality. A key step in the proof of asymptotic normality is a central limit theorem for sums of conditionally independent random variables. We also study the asymptotic distribution of Greenwood's statistics in nonuniform cases.

Testing whether new better than used in expectation for random censorship for random censorship

Kumazawa (1986) introduced a class of test statistics for testing whether a life distribution is new better than used in expectation under a random censorship model. The purpose of this paper is to give a proof of asymptotic normality of test statistics under contiguous sequence of alternatives by using a counting-process approach. For uncensored case the optimal test statistic in this class is seen to lead a locally most powerful tests for contiguous sequence of particular parametric alternatives.

Linear Functions of Order Statistics

Linear Functions of Order Statistics