Asymptotic Mean Square Error Research Articles

Joint 2-D DOA Estimation via Sparse L-shaped Array

In this paper, we address the problem of estimating the two-dimensional (2-D) directions of arrival (DOA) of multiple signals, by means of a sparse L-shaped array. The array consists of one uniform linear array (ULA) and one sparse linear array (SLA). The shift-invariance property of the ULA is used to estimate the elevation angles with low computational burden. The signal subspace is constructed by the cross-covariance matrix (CCM) of the received data without implementing eigendecomposition. The source waveforms are then obtained by the estimated elevation angles, which together with each sensor of the SLA, considered as a linear regression model, is used to estimate the azimuth angle by the modified total least squares (MTLS) technique. Our new algorithm yields correct parameter pairs without requiring the computationally expensive pairing operation, and therefore, has at least two advantages over the previous L-shaped array based algorithms: less computational load and better performance due to the use of SLA and CCM. Expressions for the asymptotic mean-squared error (MSE) of the 2-D DOA estimates are derived. Simulation results show that our method provides accurate and consistent 2-D DOA estimation results that could not be obtained by the existing methods with comparable computational complexity.

New recursive estimators of the time-average variance constant

Estimation of the time-average variance constant (TAVC) of a stationary process plays a fundamental role in statistical inference for the mean of a stochastic process. Wu (2009) proposed an efficient algorithm to recursively compute the TAVC with $$O(1)$$O(1) memory and computational complexity. In this paper, we propose two new recursive TAVC estimators that can compute TAVC estimate with $$O(1)$$O(1) computational complexity. One of them is uniformly better than Wu's estimator in terms of asymptotic mean squared error (MSE) at a cost of slightly higher memory complexity. The other preserves the $$O(1)$$O(1) memory complexity and is better then Wu's estimator in most situations. Moreover, the first estimator is nearly optimal in the sense that its asymptotic MSE is $$2^{10/3}3^{-2} \fallingdotseq 1.12$$210/33-2?1.12 times that of the optimal off-line TAVC estimator.

OPTIMAL BANDWIDTH SELECTION FOR ROBUST GENERALIZED METHOD OF MOMENTS ESTIMATION

A two-step generalized method of moments estimation procedure can be made robust to heteroskedasticity and autocorrelation in the data by using a nonparametric estimator of the optimal weighting matrix. This paper addresses the issue of choosing the corresponding smoothing parameter (or bandwidth) so that the resulting point estimate is optimal in a certain sense. We derive an asymptotically optimal bandwidth that minimizes a higher-order approximation to the asymptotic mean-squared error of the estimator of interest. We show that the optimal bandwidth is of the same order as the one minimizing the mean-squared error of the nonparametric plugin estimator, but the constants of proportionality are significantly different. Finally, we develop a data-driven bandwidth selection rule and show, in a simulation experiment, that it may substantially reduce the estimator’s mean-squared error relative to existing bandwidth choices, especially when the number of moment conditions is large.

Inverse probability weighted estimation of local average treatment effects: A higher order MSE expansion

We consider a new inverse probability weighted estimator for the local average treatment effect parameter where the instrument propensity score is estimated by local polynomial regression. We derive its asymptotics and provide a higher order expansion of its asymptotic MSE.

The Optimal Hard Threshold for Singular Values is <inline-formula> <tex-math notation="TeX">\(4/\sqrt {3}\) </tex-math></inline-formula>

We consider recovery of low-rank matrices from noisy data by hard thresholding of singular values, in which empirical singular values below a threshold λ are set to 0. We study the asymptotic mean squared error (AMSE) in a framework, where the matrix size is large compared with the rank of the matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. The AMSE-optimal choice of hard threshold, in the case of n-by-n matrix in white noise of level σ, is simply (4/√3)√nσ ≈ 2.309√nσ when σ is known, or simply 2.858 · y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">med</sub> when σ is unknown, where y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">med</sub> is the median empirical singular value. For nonsquare, m by n matrices with m ≠ n the thresholding coefficients 4/√3 and 2.858 are replaced with different provided constants that depend on m/n. Asymptotically, this thresholding rule adapts to unknown rank and unknown noise level in an optimal manner: it is always better than hard thresholding at any other value, and is always better than ideal truncated singular value decomposition (TSVD), which truncates at the true rank of the low-rank matrix we are trying to recover. Hard thresholding at the recommended value to recover an n-by-n matrix of rank r guarantees an AMSE at most 3 nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . In comparison, the guarantees provided by TSVD, optimally tuned singular value soft thresholding and the best guarantee achievable by any shrinkage of the data singular values are 5 nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , 6 nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , and 2 nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively. The recommended value for hard threshold also offers, among hard thresholds, the best possible AMSE guarantees for recovering matrices with bounded nuclear norm. Empirical evidence suggests that performance improvement over TSVD and other popular shrinkage rules can be substantial, for different noise distributions, even in relatively small n.

Maximizing Spectral Efficiency for High Mobility Systems with Imperfect Channel State Information

This paper studies the optimum system design that can maximize the spectral efficiency of high mobility wireless communication systems with imperfect channel state information (CSI). The fast time-varying fading in high mobility systems can be tracked with pilot-assisted channel estimation. The percentage of pilot symbols in the transmitted symbols plays a critical role on the system performance: a higher pilot percentage yields a more accurate channel estimation, but also more overhead. The effects of pilot percentage are quantified through the derivation of the channel estimation mean squared error (MSE), which is expressed as a closed-form expression of various system parameters through asymptotic analysis. It is discovered that, if the pilots sample the channel above its Nyquist rate, then the estimation of the channel coefficients of data symbols through temporal interpolation yields the same asymptotic MSE as the direct estimation of the channel coefficients of the pilot symbols. Based on the statistical properties of the channel estimation error, we quantify the impacts of the imperfect CSI on the system performance by developing the analytical symbol error rate (SER) and a spectral efficiency lower bound of the communication system. The optimum pilot percentage that can maximize the spectral efficiency lower bound is identified through both analytical and simulation results.

Performance bounds of direction finding and its applications for multiple‐input multiple‐output radar

In this study, the authors derive an explicit expression for Cramer–Rao lower bound (CRLB) on direction parameters with the antenna locations of a single point target in the far-field scenario for multiple-input multiple-output (MIMO) radar, firstly. Since there are two unknown nuisance parameters (target radar cross section and interference-plus-noise strength), the authors use the concentrated CRLB to simplify the analytical computation. Then, the authors derive two sets of necessary and sufficient geometrical constraints for direction estimation with both bistatic and monostatic MIMO radars. Then, the authors consider the bounds on the asymptotic normalised mean-square angular error (ANMSAE), which are useful measures in three-dimensional bearing estimation problems. The authors derive the expressions for ANMSAE in both bistatic and monostatic MIMO radars, and discuss them in terms of the antenna locations. The authors then show that the uncoupled geometrical conditions are also sufficient to ensure that ANMSAE is independent of azimuth. Finally, the authors extend those conditions to obtain two sets of geometry constraints for bistatic and monostatic MIMO radars, respectively, such as some kinds of symmetry on the geometry, that ensure the optimal performance is isotropic, that is, the ANMSAE bounds are independent of all direction parameters. In numerical examples section, the authors give several representative antenna geometries to illustrate the derived antenna geometry strategy.

Third‐order complex amplitudes tracking loop for slow flat fading channel online estimation

This study deals with channel estimation in tracking mode over a flat Rayleigh fading channel with Jakes’ Doppler spectrum. Many estimation algorithms exploit the time-domain correlation of the channel by employing a Kalman filter based on a first-order (or sometimes second-order) approximation model of the time-varying channel. However, the nature of the approximation model itself degrades the estimation performance for slow to moderate varying channel scenarios. Furthermore, the Kalman-based algorithms exhibit a certain complexity. Hence, a different model and approach has been investigated in this study to tackle all of these issues. A novel phase-locked-loop-structured third-order tracking loop estimator with a low complexity is proposed. The connection between a steady-state Kalman filter based on a random walk approximation model and the proposed estimator is first established. Then, a suboptimal mean-squared-error (MSE) is given in a closed-form expression as a function of the tracking loop parameters. The parameters that minimise this suboptimal MSE are also given in a closed-form expression. The asymptotic MSE and bit-error-ratio simulation results demonstrate that the proposed estimator outperforms the first- and second-order Kalman-based filters reported in literature. The robustness of the proposed estimator is also verified by a mismatch simulation.

Histogram Model with Plausible Parametric Detection Function for Line Transect Data

This article develops a new model that combines between the histogram and plausible parametric detection function to estimate the population density (abundance) by using line transects technique. A parametric detection function is introduced to improve the properties of the classical histogram estimator. Asymptotic properties of the resulting estimator are derived and an expression for the asymptotic mean square error (AMSE) is given. A general formula for the optimal choice of the histogram bin width based on AMSE is derived. Moreover, other possible alternative procedures to select the bin width are suggested and studied via simulation technique. The results show the superiority of the proposed estimators over both the classical histogram and the usual kernel estimators in most reasonable cases. In addition, the simulation results indicate that the choice of a plausible detection function is less sensitive than the choice of a bin width on the performance of the proposed estimator.

Third-Order Kalman Filter: Tuning and Steady-State Performance

This letter deals with the Kalman filter (KF) based on a third-order integrated random walk model (RW3). The resulting filter, noted as RW3-KF, is well suited to track slow time-varying parameters with strong trend behaviour. We first prove that the RW3-KF in steady-state admits an equivalent structure to the third-order digital phase-locked loops (DPLL). The approximate asymptotic mean-squared-error (MSE) is obtained by solving the Riccati equations, which is given in a closed-form expression as a function of the RW3 model parameter: the state noise variance. Then, the closed-form expression of the optimum state noise variance is derived to minimize the asymptotic MSE. Simulation results are given for the particular case where the parameter to be estimated is a Rayleigh channel coefficient with Jakes' Doppler spectrum.

Design of Asymptotically Optimal Unrestricted Polar Quantizer for Gaussian Source

In this letter, in order to outperform the existing method for unrestricted polar quantizer (UPQ) design in terms of signal to quantization noise ratio, the asymptotic approximations of Rayleigh distributed function are applied to all magnitude regions of the UPQ, except to the last one. Given the constraint, the UPQ is designed to provide the minimum of the asymptotic mean-squared error distortion for the Gaussian source of unit variance. The effects of this constraint are studied for different bit rates R. The accuracy of the derived formulas is assessed and the reasonable accuracy is observed for R ≥ 2.5 bit/sample.

On an improvement of Hill and some other estimators

In the paper, we propose a new idea in the tail-index estimation. This idea allows us to improve the asymptotic performance of the classical Hill estimator and other most popular estimators over the range of the parameters present in the second-order regular-variation condition. We prove the asymptotic normality of the introduced estimators and provide a comparison (using the asymptotic mean-squared error) with other estimators of the tail index.

Joint Probability Mass Function Estimation From Asynchronous Samples

A common approach to study the relationship between different signals is to model them as random processes and estimate their joint probability distribution from the observed data. When synchronous samples of the random processes are available, then the empirical distribution gives a reliable estimate. However, in several situations, such as sensors spread over a vast region or a software probing smart phone sensors for readings, synchronous samples are either not available, or are difficult/costly to obtain. In such cases, we have to depend on non-periodic, asynchronous samples to obtain good estimates of the joint distribution. In this paper, we consider independent Poisson sampling of the individual random processes and we propose a kernel based estimate of the joint probability mass function. We prove that our estimate is consistent (in the mean-square sense) for strong mixing processes, which is a wide class of random processes including Markov processes. We also provide expressions for the asymptotic mean-square error (MSE), study the bias-variance tradeoff, and discuss the choice of the kernel bandwidth. By appropriately choosing the kernel, we show that we can obtain an asymptotic rate of T-4/5 for the MSE, where T is the interval of observation. We also present several numerical results to discuss the accuracy of our asymptotic approximations for finite T.

UWB Sparse/Diffuse Channels, Part II: Estimator Analysis and Practical Channels

In this two-part paper, the problem of channel estimation in Ultra Wide-Band (UWB) systems is investigated. In Part I, a novel Hybrid Sparse/Diffuse (HSD) model is proposed for the UWB channel, and new channel estimation strategies are designed for this model. In this paper (Part II), a Mean-Squared Error (MSE) analysis of the Generalized MMSE and Generalized Thresholding Estimators developed in Part I is performed, for the asymptotic regimes of low and high SNR. The analysis quantifies the achievable MSE performance of these schemes over unstructured estimators. Specifically, we prove that it is beneficial to be conservative in the estimation of the sparse component, i.e., to assume that the sparse component is sparser than it actually is. Moreover, we analyze the scenario with a non-orthogonal pilot sequence, and establish a connection between the Generalized Thresholding estimator and conventional sparse approximation algorithms proposed in the literature. In addition to the theoretical analysis, these channel estimation schemes are evaluated in a more realistic geometry-based channel emulator, for which the HSD model developed in Part I is an approximation. The numerical results are shown to match the expected asymptotic MSE behavior. Moreover, the proposed estimation techniques are shown to outperform conventional unstructured and purely sparse estimators, from both an MSE and a bit error rate perspectives, even for the realistic geometry-based channel model.

Cramer–Rao bound based mean‐squared error and throughput analysis of superimposed pilots for semi‐blind multiple‐input multiple‐output wireless channel estimation

SUMMARYThis work presents a study of the mean‐squared error (MSE) and throughput performance of superimposed pilots (SP) for the estimation of a multiple‐input multiple‐output (MIMO) wireless channel. The Cramer–Rao bound (CRB) is derived for SP based estimation of the MIMO channel matrix. Employing the CRB analysis, it is proved that the asymptotic MSE bound is potentially 3 dB lower than the MSE performance of the existing SP mean based estimation (SPME) schemes. Motivated by this observation, a novel SP semi‐blind scheme is presented for MIMO channel estimation. This scheme asymptotically achieves the CRB and hence has a lower MSE of estimation when compared with SPME schemes. We also derive closed form expressions for the optimal source‐pilot power allocation in SP by maximizing the post‐processing signal‐to‐noise power ratio at the receiver. In the final part, a new result is presented for the worst‐case capacity of a communication channel with correlated information symbols and noise. This framework is employed to quantify the throughput performance of SP and also to demonstrate the bandwidth efficiency of SP compared with that of a conventional pilot based system. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Dual-Diagonal LMMSE Channel Estimation for OFDM Systems

We propose a low-complexity dual-diagonal (DD-) linear-minimum-mean-square-error (LMMSE) channel estimation algorithm for orthogonal frequency-division multiplexing (OFDM) systems involving iterative channel estimation and signal detection. Computational complexity and mean-square-error (MSE) analysis are presented to evaluate the efficiency of the proposed algorithm. A closed-form expression is derived for the asymptotic MSE of the DD-LMMSE channel estimator. Both analysis and numerical results show that DD-LMMSE performs close to the well-known LMMSE estimator with much lower complexity.

A Plug-In Averaging Estimator for Regressions with Heteroskedastic Errors

This paper proposes a new model averaging estimator for the linear regression model with heteroskedastic errors. We address the issues of how to optimally assign the weights for candidate models and how to make inference based on the averaging estimator. We derive the asymptotic mean squared error (AMSE) of the averaging estimator in a local asymptotic framework, and then choose the optimal weights by minimizing the AMSE. We propose a plug-in estimator of the optimal weights and use these estimated weights to construct a plug-in averaging estimator of the parameter of interest. We derive the asymptotic distribution of the plug-in averaging estimator and suggest a plug-in method to construct confidence intervals. Monte Carlo simulations show that the plug-in averaging estimator has much smaller expected squared error, maximum risk, and maximum regret than other existing model selection and model averaging methods. As an empirical illustration, the proposed methodology is applied to cross-country growth regressions.

A two-parameter estimator in the negative binomial regression model

In this article, a two-parameter estimator is proposed to combat multicollinearity in the negative binomial regression model. The proposed two-parameter estimator is a general estimator which includes the maximum likelihood (ML) estimator, the ridge estimator (RE) and the Liu estimator as special cases. Some properties on the asymptotic mean-squared error (MSE) are derived and necessary and sufficient conditions for the superiority of the two-parameter estimator over the ML estimator and sufficient conditions for the superiority of the two-parameter estimator over the RE and the Liu estimator in the asymptotic mean-squared error (MSE) matrix sense are obtained. Furthermore, several methods and three rules for choosing appropriate shrinkage parameters are proposed. Finally, a Monte Carlo simulation study is given to illustrate some of the theoretical results.

Twicing local linear kernel regression smoothers

It is known that the local cubic smoother (LC) has a faster consistency rate than the popular local linear smoother (LL). However, LC often has a bigger mean squared error (MSE) than LL numerically for samples of finite size. By extending the idea of Stuetzle and Mittal [1979, ‘Some Comments on the Asymptotic Behavior of Robust Smoothers’, in Smoothing Techniques for Curve Estimation: Proceedings (chap. 11), eds. T. Gasser and M. Rosenbalatt, Berlin: Springer, pp. 191–195], we propose a new version of LC by ‘twicing’ the local linear smoother (TLL). Both asymptotic theory and finite sample simulations suggest that TLL has better efficiency than LL. Compared with LC, TLL has about the same asymptotic MSE (AMSE) as LC at the interior points and has a much smaller AMSE than LC at the boundary points. The TLL is also more stable than LC and has better performance than LC numerically. The application of TLL to estimate the first-order derivative of the regression function and other extensions are considered.

Asymptotic MSE Distortion of Mismatched Uniform Scalar Quantization

Asymptotic formulas are derived for the mean-squared error (MSE) distortion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -level uniform scalar quantizers designed to be MSE optimal for one density function, but applied to another, as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> → ∞. These formulas, which are based on the Euler-Maclaurin formula, are then applied with generalized gamma, Bucklew-Gallagher, and Hui-Neuhoff density functions as the designed-for and applied-to densities. It is found that the mismatch between the designed-for and applied-to densities can disturb the delicate balance between granular and overload distortions in optimal quantization, with the result that, generally speaking, the granular or overload distortion dominates, respectively, depending on whether the applied-to density function has a lighter or heavier tail than the designed-for density. Specifically, in the case of generalized gamma densities, a variance mismatch makes overload distortion dominate for an applied-to source with a slightly larger variance, whereas a shape mismatch can tolerate a wider variance difference while retaining the dominance of the granular distortion. In addition, for the studied density functions, the Euler-Maclaurin approach is used to derive asymptotic formulas for the optimal quantizer step size in a simpler, more direct, way than previous approaches.