Expressions For Mean Square Error Research Articles

Improved estimation of population mean in simple random sampling using information on auxiliary attribute

This paper addresses the problem of estimating the population mean with known population proportion of an auxiliary variable. A class of estimators is defined which includes the estimators recently proposed by Shabbir and Gupta (2007) [10] and Abd-Elfattah et al. (2010) [1]. The usual unbiased estimator and Naik and Gupta (1996) [15] estimator are also the member of the proposed class of the estimators. The bias and mean square error (MSE) expressions of the proposed class are obtained up to first order of approximation. Asymptotically optimum estimator (AOE) in the class of estimators is identified alongwith its mean square error formula. The correct MSE and minimum MSE expressions of Shabbir and Gupta (2007) [10] estimator are also given. It has been shown that the proposed class of estimators is more efficient than the usual unbiased estimator, usual linear regression estimator and estimators/classes of estimators due to Naik and Gupta (1996) [15], Jhajj et al. (2008) [9], Shabbir and Gupta (2007) [10] estimator, Singh et al. (2008) [13] and Abd-Elfattah et al. (2010) [1]. The double sampling version of the proposed class of estimators is proposed alongwith its properties. Numerical illustrations are given in support of the present study.

A Generalized Difference-cum-Ratio Type Estimator for the Population Mean in Double Sampling

Using double sampling technique, a generalized difference-cum-ratio type estimator has been proposed for estimating the population mean. The expressions for bias and mean square error of proposed estimator have been obtained. The conditions under which proposed estimator is better than the regression estimator and mean per unit estimator have also been obtained. The gain in efficiency over the existing ones have been illustrated numerically as well as graphically. Simulation study has also been done to support the results by generating data.

Time Synchronization Errors in Loosely Coupled GPS-Aided Inertial Navigation Systems

The effects of data time synchronization errors in a loosely coupled Global-Positioning-System (GPS)-aided inertial navigation system (INS) are studied and quantified in terms of the increased mean square error (MSE) of the navigation solution. An expression for evaluating the MSE of the navigation solution, given the vehicle trajectory and the model of the INS error dynamics, is derived. Thereafter, a software-based time synchronization method, where the time synchronization error is included as a state to be estimated by the data integration filter, is proposed. A practical approach to the implementation of the proposed time synchronization method is also briefly described. Moreover, an expression for the MSE of the navigation solution in the system that employs the proposed synchronization method is derived. Finally, through simulations and tests with real-world data, the correctness of the derived MSE expressions is validated, and the application of the proposed synchronization method is shown. The test results show that, with the proposed synchronization approach, a data time synchronization, which is accurate to the order of a few milliseconds, can be achieved.

Distributed Estimation in a Wireless Sensor Network Using Hybrid MAC

In this paper, the distributed estimation of a scalar signal by a power-constrained wireless sensor network (WSN) with a fusion center is investigated over a hybrid multiple-access channel (MAC). The proposed hybrid MAC is a composite of conventional orthogonal and coherent MACs and is shown to offer more performance choices than both conventional channels in terms of mean square error (MSE) performance and system outage probability. Both random and deterministic scalar sources are considered for the hybrid MAC WSN in conjunction with linear minimum MSE and maximum-likelihood estimations, respectively. In particular, the optimal and suboptimal schemes are developed to allocate sensor power under the total power constraint, and these power allocation schemes are found to be similar in terms of algorithm and achievable estimation performance for both scalar sources. Moreover, closed-form expressions for sensor power are obtained for suboptimal power allocation, and two-step procedures are proposed to numerically solve the optimal power allocation problems. Furthermore, asymptotic expressions of MSE under various limiting parametric cases and system outage probability are analytically derived for suboptimal power allocation under a uniform sensor grouping strategy. It is analytically shown that, under a uniform sensor grouping strategy, the suboptimal power allocation scheme can asymptotically offer the same MSE performance as the optimal power allocation scheme when the number of sensors that transmit through the same coherent channel increases to infinity. Simulation results also show that the suboptimal power allocation scheme provides nearly optimal MSE performance for arbitrary sensor grouping strategies and that uniform sensor grouping is virtually the best grouping strategy. Moreover, for uniformly grouped hybrid MAC, the system outage probability for suboptimal power allocation is shown to provide a diversity order equal to the number of orthogonal channels when there are sufficiently many orthogonal channels.

Estimation of population mean using multi-auxiliary characters with subsampling the nonrespondents

The aim of this paper is to suggest a class of two phase sampling estimators for population mean using multi-auxiliary characters in presence of non-response on study character. The expressions for bias and mean square error are obtained. The condition for minimum mean square error of the proposed class of estimators has been given. The optimum values of the size of first phase sample, second phase sample and the sub sampling fraction of non-responding group have been determined for the fixed cost and for the specified precision. A comparative study of the proposed class of estimators has been carried out with an empirical study.

A cost effective stratified two-stage sampling design to estimate the forest land area of southern Chile

There is a growing demand for improving the measurement of forest resources, with more frequent updating and better information on environmental variables. We explore the cost efficiency of a stratified two-stage design using area sampling to estimate the forest plantation and native forest areas in southern Chile. Analytical expressions for the approximate mean square error of combined and separate ratio estimators are derived applying Taylor linearization. Under a unified framework, this procedure allows the evaluation of the precision of design and post-design estimators for unequal unit area sizes at both stages. Monte Carlo simulations were used to assess empirically the approximate analytical measures of the mean square error and the biases associated with the ratio estimators. Adopting proportional allocation among strata and clusters, the optimal allocation among the two stages is determined. A substantial improvement in sampling precision was achieved using the separate ratio estimator and the bias was found to be small. Post-stratification based on categorical information on growing zones also improved the precision of estimating the forest plantation area and a smaller extent the native forest area. The results of this paper support a wider adoption of sampling methods to estimate land use and land cover at regional or national levels.

A novel expression for the achievable MSE performance obtained by blind adaptive equalizers

In this paper, I propose for the noisy, real, and two independent quadrature carrier case, an approximated closed-form expression for the achievable minimum mean square error (MSE) performance obtained by blind equalizers where the error that is fed into the adaptive mechanism which updates the equalizer’s taps can be expressed as a polynomial function of the equalized output of order three like in Godard’s algorithm. The proposed closed-form expression for the achievable MSE is based on the step-size parameter, on the equalizer’s tap length, on the channel power, on the signal to noise ratio (SNR), on the nature of the chosen equalizer, and on the input signal statistics. Since the channel power is measurable or can be calculated if the channel coefficients are given, there is no need anymore to carry out any simulation with various step-size parameters, different values for the signal to noise ratio (SNR) and equalizer’s tap length for a given equalization method, and input signal statistics in order to find the MSE performance in the convergence state.

Joint Power Allocation and Interference Mitigation Techniques for Cooperative Spread Spectrum Systems with Multiple Relays

This paper presents joint power allocation and interference mitigation techniques for the downlink of spread spectrum systems which employ multiple relays and the amplify and forward cooperation strategy. We propose a joint constrained optimization framework that considers the allocation of power levels across the relays subject to an individual power constraint and the design of linear receivers for interference suppression. We derive constrained minimum mean-squared error (MMSE) expressions for the parameter vectors that determine the optimal power levels across the relays and the linear receivers. In order to solve the proposed optimization problem efficiently, we develop joint adaptive power allocation and interference suppression algorithms that can be implemented in a distributed fashion. The proposed stochastic gradient (SG) and recursive least squares (RLS) algorithms mitigate the interference by adjusting the power levels across the relays and estimating the parameters of the linear receiver. SG and RLS channel estimation algorithms are also derived to determine the coefficients of the channels across the base station, the relays and the destination terminal. The results of simulations show that the proposed techniques obtain significant gains in performance and capacity over non-cooperative systems and cooperative schemes with equal power allocation.

Convergence Performance of the Simplified Set-Membership Affine Projection Algorithm

Set-membership (SM) adaptive filtering is appealing in many practical situations, particularly those with inherent power and computational constraints. The main feature of the SM algorithms is their data-selective coefficient update leading to lower computational complexity and power consumption. The set-membership affine projection (SM-AP) algorithm does not trade convergence speed with misadjustment and computation complexity as many existing adaptive filtering algorithms. In this work analytical results related to the SM-AP algorithm are presented for the first time, providing tools to setup its parameters as well as some interpretation to its desirable features. The analysis results in expressions for the excess mean square error (MSE) in stationary environments and the transient behavior of the learning curves. Simulation results confirm the accuracy of the analysis and the good features of the SM-AP algorithms.

Analysis of Online Secondary-Path Modeling With Auxiliary Noise Scaled by Residual Noise Signal

Online secondary-path modeling of active noise control (ANC) systems may be effectively implemented by injecting an auxiliary white noise whose magnitude is scaled by a function of residual noise signal. In this paper, a filtered-X LMS (FXLMS) based narrowband ANC system is analyzed, whose online secondary-path modeling is based on the use of an auxiliary white noise scaled by one-sample-delayed residual noise signal. Difference equations governing the dynamics of the entire system and closed-form expressions for steady-state mean-square errors (MSE) as well as the residual noise power are derived and discussed in detail. Extensive simulations are conducted to confirm the validity of the analytical findings.

Joint Time Synchronization and Localization of an Unknown Node in Wireless Sensor Networks

Time synchronization and localization are two important issues in wireless sensor networks. Although these two problems share many aspects in common, they are traditionally treated separately. In this paper, we present a unified framework to jointly solve time synchronization and localization problems at the same time. Furthermore, since the accuracy of synchronization and localization is very sensitive to the accuracy of anchor timings and locations, the joint time synchronization and localization problem with inaccurate anchors is also considered in this paper. For the case with accurate anchors, the joint maximum likelihood estimator and a more computationally efficient least squares (LS) estimator are proposed. When the anchor timings and locations are inaccurate, a generalized total least squares (GTLS) scheme is proposed. Crame¿r-Rao lower bounds (CRLBs) and the analytical mean square error (MSE) expressions of the LS based estimators are derived for both accurate and inaccurate anchor cases. Results show that the proposed joint estimators exhibit performances close to their respective CRLBs and outperform the separate time synchronization and localization approach. Furthermore, the derived analytical MSE expressions predict the performances of the proposed joint estimators very well.

A Correction Note on Improvement in Variance Estimation Using Auxiliary Information

Shabbir and Gupta (2007) introduced an exponential ratio-type estimator for the population variance and compare its mean square error with mean square error of some of the existing estimators. But unfortunately, the expression of mean square error of their proposed estimator is incorrect and hence the comparison based on incorrect mean square error, carried by them, is wrong. In the present article, the author provides the correct expression of the mean square error, up to first order of approximation, of the same estimator. A comparison has been made by taking the corrected expression of mean square error. The corresponding exponential ratio-type estimator for the population variance, under double sampling technique, is also proposed which followed by a comparison.

The Performance Analysis of Error Saturation Nonlinearity LMS in Impulsive Noise based on Weighted-Energy Conservation

This paper introduces a new approach for the performance analysis of adaptive filter with error saturation nonlinearity in the presence of impulsive noise. The performance analysis of adaptive filters includes both transient analysis which shows that how fast a filter learns and the steady-state analysis gives how well a filter learns. The recursive expressions for mean-square deviation(MSD) and excess mean-square error(EMSE) are derived based on weighted energy conservation arguments which provide the transient behavior of the adaptive algorithm. The steady-state analysis for co-related input regressor data is analyzed, so this approach leads to a new performance results without restricting the input regression data to be white. Keywords—Error saturation nonlinearity, transient analysis, impulsive noise.

Adaptive Channel Estimation for Multiple Antenna OFDM Systems

We propose a simple adaptive algorithm for channel estimation of multiple antenna OFDM systems to provide competitive performance with low computational complexity and tracking ability. The derivation of the mean-squared error (MSE) for the proposed scheme is presented and the optimum selection of the parameter in the scheme is analyzed by utilizing the derived MSE expression.

An Addendum to the Estimators of Murthy–Sarma and Anis–Pandey of the Mean of the Normal Distribution

Anis and Pandey [Economic Quality Control 18: 83–90, 2003] have pointed out that the bias and mean squared error expressions of the estimators of the mean μ of a normally distributed random variable envisaged by Murthy and Sarma [Assam Statistical Review 12: 1–5, 1998] for small samples were found to be erroneous. Keeping this in view, Anis and Pandey [Economic Quality Control 18: 83–90, 2003] have derived an alternative expression for the mean squared error (MSE) of one of the estimators proposed in Murthy and Sarma [Assam Statistical Review 12: 1–5, 1998] claiming that the MSE expression obtained by them is correct. Unfortunately the MSE expression of the estimator given by both Murthy and Sarma [Assam Statistical Review 12: 1–5, 1998] and Anis and Pandey [Economic Quality Control 18: 83–90, 2003] are found incorrect for small samples. In this paper we have derived the correct expressions for biases and MSEs of the estimators suggested in Murthy and Sarma [Assam Statistical Review 12: 1–5, 1998] for small samples of a normally distributed random variable. We also propose two new estimators of the mean μ when the coefficient of variation is known and their properties are studied for small samples. Numerical illustration is given in support of the present study. It is shown both theoretically and empirically that the proposed optimum estimators are more efficient than those considered by Murthy and Sarma [Assam Statistical Review 12: 1–5, 1998].

On the performance of data-dependent superimposed training without Cyclic Prefix for SISO/MIMO systems

Compared with channel estimation method based on explicit training sequences, bandwidth is saved for those methods using superimposed training sequences, while it is wasted when Cyclic Prefix (CP) is added. In previous work of McLernon, the Mean Square Error (MSE) performance of Data-Dependent Superimposed Training (DDST) without CP for Single-Input Single-Output (SISO) system was analyzed under the assumption that the data-dependent sequence matrix was a circulant matrix and not interfered by others. In fact, for the system without CP, the data-dependent sequence matrix is not circulant any more and will be interfered. This paper derives the exact expression of MSE for the system without CP and also gives its extension to Multiple-Input Multiple-Output (MIMO) system without CP.

A new method of imputation in survey sampling

In this paper, an alternative estimator of population mean in the presence of non-response has been suggested which comes in the form of Walsh's estimator. The estimator of mean obtained from the proposed technique remains better than the estimators obtained from ratio or mean methods of imputation. The mean-squared error (MSE) of the resultant estimator is less than that of the estimator obtained on the basis of ratio method of imputation for the optimum choice of parameters. An estimator for estimating a parameter involved in the process of new method of imputation has been discussed. A suggestion to form ‘warm deck’ method of imputation has been suggested. The MSE expressions for the proposed estimators have been derived analytically and compared empirically. The work has been extended to the case of multi-auxiliary information to be used for imputation. Numerical illustrations are also provided.

Properties of FXLMS-Based Narrowband Active Noise Control With Online Secondary-Path Modeling

Rotating machines such as diesel engines, cutting machines, fans, motors, etc., generate sinusoidal noise signals that may be effectively reduced by narrowband active noise control (ANC) systems. In this paper, a typical filtered-X LMS (FXLMS) based narrowband ANC system equipped with an online secondary-path modeling subsystem is analyzed in detail. First, difference equations governing the dynamics of the FXLMS algorithm for secondary source synthesis and the LMS algorithm for secondary-path estimation are derived in terms of convergence in both mean and mean square. Steady-state expressions for mean-square error (MSE) as well as the residual noise power are then developed in closed form. Extensive simulations are performed to demonstrate the validity of the analytical results.

Convergence Behavior of NLMS Algorithm for Gaussian Inputs: Solutions Using Generalized Abelian Integral Functions and Step Size Selection

This paper studies the mean and mean square convergence behaviors of the normalized least mean square (NLMS) algorithm with Gaussian inputs and additive white Gaussian noise. Using the Price's theorem and the framework proposed by Bershad in IEEE Transactions on Acoustics, Speech, and Signal Processing (1986, 1987), new expressions for the excess mean square error, stability bound and decoupled difference equations describing the mean and mean square convergence behaviors of the NLMS algorithm using the generalized Abelian integral functions are derived. These new expressions which closely resemble those of the LMS algorithm allow us to interpret the convergence performance of the NLMS algorithm in Gaussian environment. The theoretical analysis is in good agreement with the computer simulation results and it also gives new insight into step size selection.

Kernel Method Starting with Half-Normal Detection Function for Line Transect Density Estimation

In this article, we introduce the nonparametric kernel method starting with half-normal detection function using line transect sampling. The new method improves bias from O(h 2), as the smoothing parameter h → 0, to O(h 3) and in some cases to O(h 4). Properties of the proposed estimator are derived and an expression for the asymptotic mean square error (AMSE) of the estimator is given. Minimization of the AMSE leads to an explicit formula for an optimal choice of the smoothing parameter. Small-sample properties of the estimator are investigated and compared with the traditional kernel estimator by using simulation technique. A numerical results show that improvements over the traditional kernel estimator often can be realized even when the true detection function is far from the half-normal detection function.