Smoothness: Bias and Efficiency of Nonparametric Kernel Estimators

Semivarying coefficient models are frequently used in statistical models. In this paper, under the condition that the coefficient functions possess different degrees of smoothness, a two-step method is proposed. In the case, one-step method for the smoother coefficient functions cannot be optimal. This drawback can be repaired by using the two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies are conducted to evaluate the proposed estimation methods.

The asymptotic error rate performance of orthogonal frequency division multiplexing (OFDM) signals in frequency-selective Nakagami-m fading channels is considered. A novel approach is provided to study the asymptotic error rate performance of a large class of modulations in general fading channels. This approach is applied to an OFDM signal on two-tap multipath Nakagami-m channels. Our analysis and numerical results show that the asymptotic error performance of an OFDM signal is similar to that of a single carrier signal transmitted over a Rayleigh fading channel. Our analysis further shows that a channel with two constant taps has similar asymptotic error rate performance to that of a one-sided Gaussian fading channel.

Varying-coefficient models with longitudinal observations are very useful in epidemiology and some other practical fields. In this paper, a reducing component procedure is proposed for estimating the unknown functions and their derivatives in very general models, in which the unknown coefficient functions admit different or the same degrees of smoothness and the covariates can be time-dependent. The asymptotic properties of the estimators, such as consistency, rate of convergence and asymptotic distribution, are derived. The asymptotic results show that the asymptotic variance of the reducing component estimators is smaller than that of the existing estimators when the coefficient functions admit different degrees of smoothness. Finite sample properties of our procedures are studied through Monte Carlo simulations.

The mean squared error (MSE) performance prediction of maximum-likelihood (ML) direction-of-arrival (DOA) angle estimation has been studied extensively. Previous analyses consider Cramer-Rao Bounds, sensitivity/asymptotic [in signal-to-colored noise ratio (SNR)] local error performance prediction that includes the impact of finite samples effects and additive signal modeling errors (mismatch), and prediction of the low SNR threshold region performance of ML DOA (without mismatch). Analysis of the adaptive array ML DOA (without mismatch) scenario has also been considered. The goals of this present analysis include the following: (i) to extend prediction of the asymptotic and threshold region MSE performance of ML to include a general form of deterministic signal model mismatch, and (ii) to begin looking at the threshold region performance of ML DOA estimation from an information-theoretic perspective, (iii) to determine if the classic work of Huber on model misspecification, although primarily asymptotic in nature, provide new insights into this finite sample problem. This initial work will focus on the DOA estimation of a single deterministic planewave signal in known colored noise and brief consideration will be given to the more complex scenario of an adaptive array in which the colored noise covariance must be estimated

The finite sample properties of simultaneous equations' estimates and estimators Bayesian and non-Bayesian approaches

A family of kernel-based estimators parametrised by a bandwidth is used to estimate the location of the discontinuity in discretely observed data consisting of a discontinuous regression function with added noise. Asymptotic bias and variance calculations for the estimated location are given as a function of the bandwidth and the sample size. For functions Which are Smooth apart from a single discontinuity, bias is of order h 2, and is not in general negligible compared with variance, which is of order h/n. The asymptotic mean square error rate MSE = O(n −4/3), obtained by balancing squared bias and variance, can usually be improved to O(n −2+δ ) with δ > 0 arbitrarily small by choosing a smaller bandwidth h = O(n −1+δ ). Special cases are treated and bias and variance are calculated explicitly. Only bias depends on the type of discontinuous function considered: For a step function with added noise and for any function which is antisymmetric about the discontinuity point, squared bias becomes negligible compared to variance, and MSE can therefore converge at the faster rate O(n −2+δ )1 For a smooth function with one added step and added white noise, bias is of order h 3 showing that bias decreases more rapidly as the function becomes simpler.

Some tests of the consumption-based asset pricing model

formula omitted]-deficiency of the sample quantile estimator with respect to a kernel quantile estimator

Based on an Edgeworth expansion for two kernel quantile estimators, the performance of the kernel quantile estimators is examined versus sample quantile estimator under the criterion of equal covering probability. The asymptotically optimal bandwidths is obtained corresponding to the maximum rate of convergence of the deficiency of the sample quantile estimator with respect to the kernel quantile estimators. A data-based procedure for optimal bandwidth selection is also investigated.

This paper analyzes the performance of ATR algorithms in clutter. The variability of target type and pose is accommodated by introducing a deformable template for every target type, with low-dimensional groups of geometric transformations representing position and pose. Signature variation of targets is taken into account by expanding deformable templates into robust deformable templates generated from the template and a linear combination of PCA elements, spanning signature intensities. Detection and classification performance is characterized using ROC analysis. Asymptotic expressions for probabilities of recognition errors are derived, yielding asymptotic error rates. The results indicate that the asymptotic error probabilities depend upon a parameter, which characterizes the separation between the true target and the most similar but incorrect one. It is shown that the asymptotic expressions derived almost accurately predict performance of detection and identification of targets occluded by natural clutter.

In this paper, we construct the wavelet eigenvalue regression methodology (Abry and Didier in J Multivar Anal 168:75–104, 2018a; in Bernoulli 24(2):895–928, 2018b) in high dimensions. We assume that possibly non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate ( $$r \ll p$$ ) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. Building upon the asymptotic and large scale properties of wavelet random matrices in high dimensions, the wavelet eigenvalue regression is shown to be consistent and, under additional assumptions, asymptotically Gaussian in the estimation of the fractal structure of the system. We further construct a consistent estimator of the effective dimension r of the system that significantly increases the robustness of the methodology. The estimation performance over finite samples is studied by means of simulations.

Linear Least Squares is a very well known technique for parameter estimation, which is used even when sub-optimal, because of its very low computational requirements and the fact that exact knowledge of the noise statistics is not required. Surprisingly, bounding the probability of large errors with finitely many samples has been left open, especially when dealing with correlated noise with unknown covariance. In this paper we analyze the finite sample performance of the linear least squares estimator under sub-Gaussian martingale difference noise. In order to analyze this important question we used concentration of measure bounds. When applying these bounds we obtained tight bounds on the tail of the estimator's distribution. We show the fast exponential convergence of the number of samples required to ensure a given accuracy with high probability. We provide probability tail bounds on the estimation error's norm. Our analysis method is simple and uses simple $L_{\infty}$ type bounds on the estimation error. The tightness of the bounds is tested through simulation. The proposed bounds make it possible to predict the number of samples required for least squares estimation even when least squares is sub-optimal and used for computational simplicity. The finite sample analysis of least squares models with this general noise model is novel.

We propose a criterion for the comparison of different sampling strategies (synchronous, asynchronous and random) and filtering algorithms used in digital instruments which provide the estimate of the time average of a signal processed with a nonlinear conversion of multiple inputs (e.g. wattmeters, RMS voltmeters,…). This criterion uses the Bayesian approach to incorporate, for every sampling strategy, any prior information on the influences of each incidental quantity which can vary the output of the instrument, transforming this output into a statistic. The asymptotic mean-squared error of the measurements has been assumed as an estimator of the error and its general expression, valid for the most common sampling strategies used in practice, has been deduced. This asymptotic error is a function of the frequency response of the digital filter used and, eventually, of the characteristic function of the probability distribution selected for the random variables generating the sampling instants. Finally, the particular formulae for different sampling strategies and filtering algorithms are discussed and compared.

Exact error rate expressions for equal-gain combining over arbitrarily correlated Rician channels involve intractable integrals. Thus, asymptotic error rate expression was derived to provide accurate approximation in large signal-to-noise ratio region. However, the existing asymptotic analysis cannot evaluate the accuracy of the asymptotic error rate expression for equal-gain combining. In this work, we derive closed-form and asymptotically tight upper and lower bounds for the error rate of equal-gain combining over Rician fading channels with arbitrary correlation. These bounds can be used to evaluate the accuracy of the asymptotic error rate expression for equal-gain combining at high signal-to-noise ratio without using the time-consuming Monte Carlo simulation.

Exact error rate expressions for equal-gain combining over arbitrarily correlated Rician channels involve intractable integrals. Thus, asymptotic error rate expression was derived to provide accurate approximation in large signal-to-noise ratio region. However, the existing asymptotic analysis cannot evaluate the accuracy of the asymptotic error rate expression for equal-gain combining. In this work, we derive closed-form and asymptotically tight upper and lower bounds for the error rate of equal-gain combining over Rician fading channels with arbitrary correlation. These bounds can be used to evaluate the accuracy of the asymptotic error rate expression for equal-gain combining at high signal-to-noise ratio without using the time-consuming Monte Carlo simulation.