Expressions For Mean Square Error Research Articles

Efficient estimation of population mean in the presence of non-response and measurement error

In real-world surveys, non-response and measurement errors are common, therefore studying them together seems rational. Some population mean estimators are modified and studied in the presence of non-response and measurement errors. Bias and mean squared error expressions are derived under different cases. For all estimators, a theoretical comparison is made with the sample mean per unit estimator. The Monte-Carlo simulation is used to present a detailed picture of all estimators’ performance.

An Efficient Class of Estimators in Stratified Random Sampling with an Application to Real Data

This research article addresses an efficient separate and combined class of estimators for the population mean estimation based on stratified random sampling (StRS). The first order approximated expressions of bias and mean square error of the proposed separate and combined class of estimators are obtained. A comparative study is conducted to determine the efficiency conditions in which the suggested class of estimators outperforms the contemporary estimators. These efficiency conditions are examined through an extensive simulation study by employing a hypothetically drawn symmetrical and asymmetrical populations. The simulation results have shown that the suggested class of estimators is more effective than the other available estimators. In addition, an application of the proposed methods is also presented by examining a real data set.

Analysis of the LMS and NLMS algorithms using the misalignment norm

This work describes the convergence of the misalignment square norm (MSN) of the NLMS and LMS algorithms. It is shown that the MSN decrease is almost proportional to the mean square error (MSE). This allows obtaining simple expressions for the steady-state MSE. Also, it allows limiting the amount of time that the MSE takes large values and a curve that limits the MSE of LMS at any given time, independent of the input and background noise signals’ properties. Finally, it is also shown that many complications in the analysis of the LMS and NLMS algorithms can come from variations in the input vector square norm. The proposed analysis becomes very simple for long filters or constant power signals.

Efficient class of ratio cum median estimators for estimating the population median.

In estimation theory, the use of auxiliary information significantly improves precision while estimating population parameters. In this paper, an efficient class of ratio cum median estimators of the population median is suggested using simple random sampling without replacement. The expressions for bias and mean square error of the proposed class are derived theoretically. The condition for the asymptotic optimum estimator is obtained with its bias and mean square error expressions. Under certain realistic conditions, the asymptotic optimum estimator is more proficient, based on analytical and numerical comparisons with some existing estimators that are members of the suggested class of estimators. The superiority of the proposed ratio cum median estimators is shown through real data applications. Such a new proposed estimator will be useful in the future for data analysis and making decisions.

Optimized inferences of finite population mean using robust parameters in systematic sampling.

In this article, we have proposed a generalized estimator for mean estimation by combining the ratio and regression methods of estimation in the presence of auxiliary information using systematic sampling. We incorporated some robust parameters of the auxiliary variable to obtain precise estimates of the proposed estimator. The mathematical expressions for bias and mean square error of proposed the estimator are derived under large sample approximation. Many other generalized ratio and product-type estimators are obtained from the proposed estimator using different choices of scalar constants. Some special cases are also discussed in which the proposed generalized estimator reduces to the usual mean, classical ratio, product, and regression type estimators. Mathematical conditions are obtained for which the proposed estimator will perform more precisely than the challenging estimators mentioned in this article. The efficiency of the proposed estimator is evaluated using four populations. Results showed that the proposed estimator is efficient and useful for survey sampling in comparison to the other existing estimators.

A Framework for Hardware Impairments-Aware Multi-Antenna Transceiver Design in IoT Systems via Majorization–Minimization

In view of the nonideality of communication links in the Internet of Things (IoT) originating from transceiver hardware impairments, in this article, we introduce a general framework for hardware impairments-aware multiantenna transceiver design, which considers different availabilities of CSI at the transmitter (CSIT) and the receiver (CSIR). The well-known Kronecker model is applied to characterize stochastic channel state information (CSI) errors. For each case, we aim to minimize the (average) total mean square error (MSE) of all data streams subject to the practical per-antenna power constraints. To address the nonconvexity of the formulated problem, we propose an efficient majorization–minimization (MM)-based iterative algorithm to transform the original problem into a series of convex subproblems with semiclosed-form optimal solutions. For low-complexity implementation, we also develop an alternative scheme for directly finding a high-quality suboptimal solution by considering both worst case hardware impairments and worst case CSI errors. In particular, since an explicit expression of the average total MSE for the perfect CSIR and imperfect CSIT case is hard to derive, we instead optimize its effective upper and lower bounds. The prospective applications of our work in the two currently popular multiple-input–multiple-output (MIMO) IoT scenarios are then discussed. Furthermore, we fundamentally reveal the MSE floor effect caused by both hardware distortion and CSI imperfection in the high-SNR regime. Numerical results illustrate the excellent average total MSE and average bit error rate (BER) performance of our proposed algorithms over the adopted benchmark schemes.

Generalized Family of Exponential type Estimators for the Estimation of Population Coefficient of Variation

In this article, we proposed an improved family of estimators for population coefficient of variation (CV) under simple random sampling, which needed two helping variables. The expression for bias and mean square error (MSE) are derived up to first order of approximation, and investigated the performance of the proposed family with existing estimators in both actual and simulated conditions, found that new estimators showing lower mean square errors as compare to the existing once, it is concluded that the suggested family of estimators achieved better results.

An efficient class of estimators for finite population mean in the presence of non-response under ranked set sampling (RSS).

In this study, we address the problem of estimating the finite population mean when the non-response occurs on the characteristics under study. We propose a class of Rao-regression type estimators when ranked set sampling (RSS) procedure is used to collect the data from non-response group only and from both, the response and non-response groups. The information provided on the auxiliary variable is used at both stages i.e., at designing stage and the estimation stage. Expressions for bias and mean square error of the estimators are obtained up to first order of approximation. A comprehensive simulation study is carried out to observe the performances of the estimators under non-response.

Modified Regression Estimators for Improving Mean Estimation -Poisson Regression Approach

In this article, a class of Poisson-regression based estimators has been proposed for estimating the finite population mean in simple random sampling without replacement (SRSWOR). The Poisson-regression model is the most common method used to model count responses in many studies. The expression for bias and mean square error (MSE) of proposed class of estimators are obtained up to first order of approximation. The proposed estimators have been compared theoretically with the existing estimators, and the condition under which the proposed class of estimators perform better than existing estimators have been obtained. Two real data sets are considered to assess the performance of the proposed estimators. Numerical findings confirms that the proposed estimators dominate over the existing estimators such as Koc (2021) and Usman et al. (2021) in terms of mean squared error.

An Approximate MSE Expression for Maximum Likelihood and Other Implicitly Defined Estimators of Non-Random Parameters

An approximate mean square error (MSE) expression for the performance analysis of implicitly defined estimators of non-random parameters is proposed. An implicitly defined estimator (IDE) declares the minimizer/maximizer of a selected cost/reward function as the parameter estimate. The maximum likelihood (ML) and the least squares estimators are among the well known examples of this class. In this paper, an exact MSE expression for implicitly defined estimators with a symmetric and unimodal objective function is given. It is shown that the expression reduces to the Cramer-Rao lower bound (CRLB) and misspecified CRLB in the large sample size regime for ML and misspecified ML estimation, respectively. The expression is shown to yield the Ziv-Zakai bound (without the valley filling function) for the maximum a posteriori (MAP) estimator when it is used in a Bayesian setting, that is, when an a-priori distribution is assigned to the unknown parameter. In addition, extension of the suggested expression to the case of nuisance parameters is studied and some approximations are given to ease the computations for this case. Numerical results indicate that the suggested MSE expression not only predicts the estimator performance in the asymptotic region; but it is also applicable for the threshold region analysis, even for IDEs whose objective functions do not satisfy the symmetry and unimodality assumptions. Advantages of the suggested MSE expression are its conceptual simplicity and its relatively straightforward numerical calculation due to the reduction of the estimation problem to a binary hypothesis testing problem, similar to the usage of Ziv-Zakai bounds in random parameter estimation problems.

Variance estimation under an efficient class of estimators in simple random sampling

This study acquaints an efficient class of estimators for variance estimation in simple random sampling. Some existing prominent estimators are established to be the particular cases of the suggested estimators. The mathematical expressions of bias and mean square error of the suggested estimators are determined to the approximation of order one. The efficacious execution of the suggested estimators is examined against all distinguished estimators available till date. Further, numerical illustrations and Monte Carlo simulations are accomplished to extend the findings of the study.

General Parameters Estimation in the Presence of Nonresponse Using Difference-cum Exponential Estimators

Non response is a common problem in a survey process. Therefore, it is necessary to find a way out of handling non response whenever arises. The current study proposed difference-cum exponential ratio-type estimator for estimation of general parameters using auxiliary information which is defined in two situations of non response. A conventional estimator t*(a,b) is used to define population constants including population mean, standard deviation, coefficient of variation and mean square. The expression of bias and mean square error of the proposed estimators are obtained up to the first order of approximation for situation I and situation II. To compare the efficiency of the proposed estimators over the existing ones, an empirical study is carried out using real and simulated data sets. Both the theoretical and empirical study shows that the proposed class of estimators outperformed other existing estimators.

Performance analysis of the augmented complex-valued least mean kurtosis algorithm

The augmented complex-valued least mean kurtosis (ACLMK) algorithm was proposed to enhance the performance of adaptive filters by utilizing the negated kurtosis of the error signal. This paper provides a comprehensive physical insight into its stochastic behaviors and a more thorough performance analysis. By using Isserlis’ theorem, the evolutions of both the covariance and complementary covariance matrices of the weight error vector are established, based on which the expressions for transient mean-square deviation (MSD) and excess mean-square error (EMSE) are derived. Moreover, the higher-order moments of the measurement noise are also taken into account. Different from the existing framework which treats the ACLMK as a variable step-size augmented complex-valued least mean square (ACLMS) algorithm, the steady-state mean-square performance of the ACLMK is analyzed through the approximate uncorrelating transform (AUT) to yield the theoretical steady-state MSD and EMSE. The proposed framework of performance analysis is shown to provide more accurate models for the theoretical MSD and MSE. Simulation results are provided to validate the theoretical findings for second-order noncircular input signals in second-order circular and noncircular noisy environments.

Computing the effect of measurement errors on ranked set sampling estimators of the population mean

SummaryThe effect of measurement errors shows the seriousness of the estimators or the estimation procedures and cannot be avoided in sensitive cases because “a miss is as good as a mile.” In this article, we have proposed the very first ranked set estimators like unbiased, ratio, product, difference, ratio exponential, and product exponential estimators under both correlated and uncorrelated measurement errors. The expressions for bias and mean square error of the proposed estimators are obtained up to the first order of approximation. The proposed ranked set estimators have been compared through relative efficiencies over unbiased estimators and the effect of measurement errors has been given through percentage computation of measurement errors. The theoretical conditions are obtained under which the proposed estimator has performed better. The performance of the proposed ranked set estimators under measurement errors are shown by efficiency comparisons, Monte‐Carlo simulation study, and real data study.

Some Improved Classes of Estimators in Stratified Sampling Using Bivariate Auxiliary Information

This manuscript considers some improved combined and separate classes of estimators of population mean using bivariate auxiliary information under stratified simple random sampling. The expressions of bias and mean square error of the proposed classes of estimators are determined to the first order of approximation. It is exhibited that under some particular conditions, the proposed classes of estimators dominate the existing prominent estimators. The theoretical findings are supported by a simulation study performed over a hypothetically generated population.

An efficient application of scrambled response approach to estimate the population mean of the sensitive variables

In the presence of one auxiliary variable and two auxiliary variables, we analyze various exponential estimators. The ranks of the auxiliary variables are also connected with the study variables, and there is a linkage between the study variables and the auxiliary variables. These ranks can be used to improve an estimator's accuracy. The Optional Randomized Response Technique (ORRT) and the Quantitative Randomized Response Technique are two techniques we utilize to estimate the sensitive variables from the population mean (QRRT). We used the scrambled response technique and checked the proposed estimators up to the first-order of approximation. The mean square error (MSE) equations are obtained for all the proposed ratio exponential estimators and show that our proposed exponential type estimator is more efficient than ratio estimators. The expression of mean square error is obtained up to the first degree of approximation. The empirical and theoretical comparison of the proposed estimators with existing estimators is also be carried out. We have shown that the proposed optional randomized response technique and quantitative randomized response model are always better than existing estimators. The simulation study is also carried out to determine the performance of the estimators. Few real-life data sets are also be applied in support of proposed estimators. It is observed that our suggested estimator is more efficient as compared to an existing estimator.

A Framework for Predictor Antennas in Practice

Channel predictions are important to achieve high spectral efficiency for high-mobility vehicles. Channel extrapolation, used by many prediction methods, suffers from a limited prediction horizon in difficult radio environments. The predictor antenna (PA) concept provides the prediction horizons required for efficient transmission to fast-moving vehicles by measuring the channel ahead of time with an extra antenna placed on the vehicle. This paper presents a general framework that addresses the practical signal processing challenges of the PA concept. It is adaptable to a vast variety of vehicular deployment, mobility, and communication scenarios. A new theoretical prediction normalized mean-squared error (NMSE) expression is derived based on the presented framework. The framework is demonstrated by applying it to extensive channel measurements and comparing the PA predictions to Kalman-based channel predictions and outdated channel estimates. By studying the impact of vehicular velocity and radio environment on the prediction performance, it is shown that PA prediction is weaker at low velocities, where Kalman prediction methods are sufficient, but is uncontested at high velocities in environments without a dominating path. At high velocities in dominating path environments, the Kalman predictor provides usable predictions, but it is still outperformed by the PA predictions.

Diffusion augmented complex-valued LMS algorithm with shared measurements and its performance analysis

The diffusion augmented complex-valued LMS (DACLMS) algorithm was proposed for distributed estimation over networks to deal with noncircular signals. This paper develops an improved DACLMS by sharing measurements between nodes within the same neighborhood. To predict its stochastic behavior, including the transient mean weight error, standard mean-square error (MSE), and complementary MSE (CMSE), is analyzed under some frequently-used statistical assumptions. A joint consideration of the MSE and CMSE performance is also analyzed to provide extra insight to the stochastic behavior of the proposed algorithm. Furthermore, the closed-form expressions for the steady-state MSE and CMSE are subsequently derived based on the transient analysis results. Since the proposed steady-state performance analysis does not assume that the network has the same data statistics at all nodes, it is applicable to the DACLMS without shared measurements and more general than the existing steady-state analysis. Simulation results are finally provided to verify the accuracy of the theoretical expressions and the effectiveness of the proposed algorithm.

Statistical analysis of multichannel FxLMS algorithm for narrowband active noise control

This paper analyzes the statistical performance of a multichannel narrowband active noise control (NANC) system based on the filtered-x least mean square (FxLMS) algorithm. The difference equations for the mean and mean square transient convergence behavior of the multichannel NANC system are derived. Steady-state expressions for weight-errors and mean square errors are then developed in closed forms. Stability boundaries for the multichannel FxLMS algorithm in the mean and mean square sense are also derived. When considering the mean square convergence, the coupling between the cross secondary paths and different frequencies is considered skillfully by matrixing and vectorization operations. Extensive simulations verify the accuracy of the theoretical analysis. Our findings deepen the understanding of the transient and steady-state behaviors of the multichannel NANC system and provide guidance for algorithm design, facilitate its improvement, and assist in parameter selection.

Kernel Recursive Maximum Versoria Criterion Based Post-Distorter for VLC Using Kernel-Width Sampling

Visible light communication (VLC) has emerged as a potential candidate for next generation wireless communication systems. However, nonlinear characteristics of light emitting diode (LED), user-mobility, and DC-bias fluctuations are the major factors that limit the throughput of a VLC link, and makes the overall additive distortion as non-Gaussian distributed. To mitigate this non-Gaussian noise processes encountered in VLC systems due to LED nonlinearity, and user-mobility, recently a random Fourier features (RFF) based kernel recursive maximum Versoria criterion (KRMVC) based post-distortion algorithm is proposed, which delivers better performance as compared to the classical polynomial series, and kernel recursive least squares (KRLS) algorithms due to the incorporation of higher order statistics of error. However, the performance of RFF-KRMVC algorithm is sensitive to the choice of kernel-width, and results in approximation errors due to imperfect choice of kernel-width. This paper proposes a novel RFF-KRMVC algorithm using a kernel-width sampling (KWS) technique called as RFF-KWS-KRMVC, which implements the post-distortion under a hyperparameter-free finite memory budget. Furthermore, analytical expressions for mean square error, and error rate are quantified for the proposed RFF-KWS-KRMVC post-distorter, and corroborated by Monte-Carlo simulations performed over standard VLC channel models.