L.C. Zhao

On detection of the number of signals in presence of white noise

In this paper, the authors propose procedures for detection of the number of signals in presence of Gaussian white noise under an additive model. This problem is related to the problem of finding the multiplicity of the smallest eigenvalue of the covariance matrix of the observation vector. The methods used in this paper fall within the framework of the model selection procedures using information theoretic criteria. The strong consistency of the estimates of the number of signals, under different situations, is established. Extensions of the results are also discussed when the noise is not necessarily Gaussian. Also, certain information-theoretic criteria are investigated for determination of the multiplicities of various eigenvalues.

On detection of the number of signals when the noise covariance matrix is arbitrary

In this paper, the authors proposed model selection methods for determination of the number of signals in presence of noise with arbitrary covariance matrix. This problem is related to finding the multiplicity of the smallest eigenvalue of [Sigma]2[Sigma]1-1, where [Sigma]2 = [Gamma] + [lambda][Sigma]1, [Sigma]1 and [Sigma]2 are covariance matrices, [lambda] is a scalar, and [Gamma] is non-negative definite matrix and is not of full rank. Also, the authors proposed methods for determination of the multiplicities of various eigenvalues of [Sigma]2[Sigma]1-1. The methods used in these procedures are based upon certain information theoretic criteria. The strong consistency of these criteria is established in this paper.

On rates of convergence of efficient detection criteria in signal processing with white noise

L.C. Zhao et al. (1986) proposed certain information-theoretic criteria for detection of the number of signals under an additive model with white noise when the noise variance is known or unknown. It was shown that these criteria are strongly consistent even when the underlying distribution is not necessarily Gaussian. Upper bounds on the probabilities of error detection are obtained here. >

Remarks on certain criteria for detection of number of signals

In this paper, we derive the asymptotic distribution of the logarithm of the likelihood ratio statistic for testing the hypothesis that the number of signals is equal to q against the alternative that it is equal to k (specified) for a special case. This distribution is not chi-square. The above statistic also rises in studying consistency property of the MDL criterion and AIC for detection of the number of signals.

Exponential bounds of mean error for the nearest neighbor estimates of regression functions

Abstract Let ( X , Y ), ( X 1 , Y 1 ),…, ( X n , Y n ) be i.i.d. ( R r × R )-valued random vectors with E | Y | m n ( x ) be a nearest neighbor estimate of the regression function m ( x ) = E ( YsβX = x ). We establish an exponential bound of the mean deviation between m n ( x ) and m ( x ) given the training sample Z n = ( X 1 , Y 1 ,…, X n , Y n ), under the conditions as weak as possible. This is a substantial improvement on Becks result.

Almost sure L 1 -norm convergence for data-based histogram density estimates

The main result of this paper is summarized in Theorem 1, which states that when certain conditions of a general nature are satisfied, the data-based histogram density estimator is strongly consistent in the sence that the mean absolute derivation of the estimator and the density function converges to zero almost surely for any density function, as the sample size increases to infinity.

Exponential bounds of mean error for the kernel estimates of regression functions

Abstract Let ( X , Y ), ( X 1 , Y 1 ), …, ( X n , Y n ) be i.d.d. R r × R -valued random vectors with E | Y | Q n ( x ) be a kernel estimate of the regression function Q ( x ) = E ( Y | X = x ). In this paper, we establish an exponential bound of the mean deviation between Q n ( x ) and Q ( x ) given the training sample Z n = ( X 1 , Y 1 , …, X n , Y n ), under conditions as weak as possible.

On estimation of the number of signals and frequencies of multiple sinusoids

In this paper, the authors review some recent developments on the estimation of the frequencies and the number of signals under a model involving multiple sinusoids. The main emphasis is on the recent results obtained by the authors.

On the determination of the order of an autoregressive model

To determine the order of an autoregressive model, a new method based on information theoretic criterion is proposed. This method is shown to be strongly consistent and the convergence rate of the probability of wrong determination is established.

Statistical analysis of dyadic stationary processes

In this paper we consider a multiple dyadic stationary process with the Walsh spectral density matrix fθ(λ), where θ is an unknown parameter vector. We define a quasi-maximum likelihood estimator % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiUdyaaja% aaaa!377D!\[{\rm{\hat \theta }}\] of θ, and give the asymptotic distribution of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiUdyaaja% aaaa!377D!\[{\rm{\hat \theta }}\] under appropriate conditions. Then we propose an information criterion which determines the order of the model, and show that this criterion gives a consistent order estimate. As for a finite order dyadic autoregressive model, we propose a simpler order determination criterion, and discuss its asymptotic properties in detail. This criterion gives a strong consistent order estimate. In Section 5 we discuss testing whether an unknown parameter θ satisfies a linear restriction. Then we give the asymptotic distribution of the likelihood ratio criterion under the null hypothesis.