Expressions For MSE Research Articles

An improved class of estimators for estimation of population distribution functions under stratified random sampling

The main objective of the current study is to suggest an enhanced family of log ratio-exponential type estimators for population distribution function (DF) using auxiliary information under stratified random sampling. Putting different choices in our suggested generalized class of estimators, we found some Specific estimators. The bias and MSE expressions of the estimators have been approximated up to the first order. By using the actual and simulated data sets, we measured the performance of estimators. Based on the results, the suggested estimators for DF show better performance as compared to the preliminary estimators considered here. The suggested estimators have a advanced efficiency than the other estimators examined with the estimators F‾ˆlogPR(st)2, and F‾ˆlogPR(st)4 for both the actual and simulated data sets. The magnitude of the improvement in efficiency is noteworthy, indicating the superiority of the proposed estimators in terms of MSE.

An Improved Estimation of Population Mean Using Auxiliary Attribute in Stratified Sampling

Stratification improves the efficiency when the variance between strata is much larger than the variances within strata. An exponential estimator for stratified random sampling was proposed. Data sets on the amount of Apple production taken at three different occasions in Turkey in 1999 were used to validate the proposed estimator. The data sets were stratified by regions of Turkey and from each region; we randomly selected the samples by using the Neyman allocation method. Expressions for MSE of the proposed estimators were derived, up to first order of approximation. The suggested estimator families' bias and MSE equations were computed, the MSE for the proposed estimators is unbiased and efficient. The results of the theoretical comparison and numerical analysis shows that the suggested estimator outperformed the existing estimator. Moreover, the derived exponential estimator performed better than existing estimators for estimating population mean in stratified random sampling with an auxiliary attribute.

Some Model Assisted Estimators Using Functional Form Calibration Approach

The Model assisted estimators are approximately design unbiased, consistent and provides robustness in the case of large sample sizes. The model assisted estimators result in reduction of the design variance if underlying model reasonably defines the regression relationship. If the model is misspecified, then model assisted estimators might result in an increase of the design variance but remain approximately design unbiased and show robustness against model-misspecification. The well-known model assisted estimators, generalized regression estimators are members of a larger class of calibration estimators. Calibration method generates calibration weights that meet the calibration constraints and have minimum distance from the sampling design weights. By using different distance measures, classical calibration approach generates different calibration estimators but with asymptotically identical properties. The constraint of distance minimization was reduced for studying the properties of calibration estimators by proposing a simple functional form approach. The approach generates calibration weights that prove helpful to control the changes in calibration weights by using different choices of auxiliary variable’s functions. This paper is an extended work on model assisted approach by using functional form of calibration weights. Some new model assisted estimators are considered to get efficient and stabilized regression weights by introducing a control matrix. The asymptotic un-biasedness of the proposed estimators is verified and the expressions for MSE are derived in three different cases. A simulation study is done to compare and evaluate the efficiency of the proposed estimators with some existing model assisted estimators.

On distribution function under two-phase stratified sampling using two auxiliary variables for mean estimation

We suggest a new class of difference-type estimators under two-phase stratified sampling when using information from two auxiliary variables including their distribution functions. The bias and MSE expressions are derived up to first degree of approximations. It is observed that the proposed class of difference-type estimators is more efficient as compared to the estimators: sample mean, ratio, exponential-ratio, traditional difference-types, Chand, Singh and Choudhury, and Anieting and Ezugwu regression type. The comparisons are made in Section 4 and results based on real data sets and simulation are given in Section 5. It is observed that the performances of the generalized class of estimators is remarkable.

Subspace-Based Estimation of Rapidly Varying Mobile Channels for OFDM Systems

It is well-known that time-varying channels can provide time diversity and improve error rate performance compared to time-invariant fading channels. However, exploiting time diversity requires very accurate channel estimates at the receiver. In order to reduce the number of unknown channel coefficients while estimating the time-varying channel, basis expansion models can be used along with long transmission frames that contain multiple orthogonal frequency division multiplexing (OFDM) symbols that experience the channel variation. The design of these OFDM frames need to judiciously incorporate training and data insertions in the transmitted signal while maintaining orthogonality. In this work, we propose an inter channel interference (ICI)-free training model depending on pilot symbols only and provide a corresponding time-varying channel estimation method. This scheme relies on an algorithm to determine the number of OFDM symbols per frame and the number of basis functions per path with minimal information about the Doppler bandwidth. As a performance benchmark, Bayesian Cramer Rao lower bound (CRLB) and the corresponding MSE bound are derived analytically for the proposed training model. Theoretical MSE expressions of the proposed estimation scheme are also derived as well as the MSE expressions in the presence of Doppler frequency mismatch. Simulations exhibit substantial MSE improvement and the corresponding Symbol Error Rate (SER) performances of the low complexity estimation scheme. They also corroborate that, unlike the common results in the literature, an OFDM system can perform better as the Doppler frequency increases with judicious design of training and channel estimation schemes.

An improved general class of estimators for finite population mean in simple random sampling

Recently Pal et al. introduced a mean estimator which generalized most of the known mean estimators. We introduce a class of estimators that performs better than Pal et al. estimator. Expressions for bias and MSE for the proposed estimator are derived up to first order of approximation. We use the six datasets for numerical comparison.

On improved class of difference type estimators for population median in survey sampling

We propose two new improved class of difference type estimators for population median under simple random sampling (SRS) and two-phase sampling (TPS). Expressions for bias and MSE are derived up to first order of approximation. We make a comparison of proposed estimators with all other commonly known estimators in literature. Numerical findings show that the proposed class of estimators perform better as compared to all other estimators considered here.

On estimating the population mean using auxiliary character in stratified random sampling

In this article, we propose an exponential estimator in the stratified random sampling taking an auxiliary attribute. Expressions for MSE of the proposed estimators are derived up to first order of approximation and comparison is made with the estimators mentioned herein. By these comparisons, we show that the suggested exponential estimators is found to perform better than the estimators in considered here. These theoretical findings are also supported with the aid of a numerical example.

A modified regression-cum-ratio estimator of population mean of a sensitive variable in the presence of non-response in simple random sampling

A generalized class of estimators has been proposed for finite population mean of a sensitive study variable in the presence of non-response under a Simple Random Sampling Without Replacement design (SRSWOR) using the Randomized Response Technique (RRT) in two phase sampling . Auxiliary information from a positively correlated non-sensitive variable is used. The expressions for MSE of the proposed estimator are derived. The performance of the proposed estimator is evaluated both theoretically and empirically in comparison to some of the existing estimators.

A new estimator for population mean using two auxiliary variables in stratified random sampling

In this paper, we suggest an estimator using two auxiliary variables in stratified random sampling. The propose estimator has an improvement over mean per unit estimator as well as some other considered estimators. Expressions for bias and MSE of the estimator are derived up to first degree of approximation. Moreover, these theoretical findings are supported by a numerical example with original data.

A modified ratio-product estimator of population mean using some known parameters of the auxiliary variable

The estimation of population mean is one of the challenging aspects in sampling theory and population study and much effort has been vigorously employed to improve the precision of estimates. In this research work, a modified ratio-product estimator of population mean Y of the study variable Y using median and coefficient of variation of the auxiliary variabl X , in simple random sampling scheme is proposed. The expression of bias and MSE of the proposed estimator have been obtained under large sample approximation, asymptotically optimum estimator (AOE) is identified with its approximate MSE formula. Estimator based on “estimated optimum values” was also investigated. Theoretical and empirical comparison of proposed estimator with some other ratio and product estimators justified the performance of the proposed estimators. A minimum of 20 percent reduction in the MSE were observed from each of the existing estimators considered. It is found that the proposed estimator were uniformly better than all other modified ratio and product estimators and thus most preferred over the existing estimators for the use in practical application. Keywords: Finite population mean, bias, mean square error, auxiliary variable, optimum estimator, study variable

Modified ratio-product estimator of population mean in the presence of median and coefficient of variation of the auxiliary variable in stratified random sampling

For the past decades, the estimation of population mean is one of the challenging aspects in sampling survey techniques and much effort has been employed to improve the precision of estimates. In this research work, we proposed a modified ratio-product estimator of population mean of the variable of interest using median and coefficient of variation of the auxiliary variable in stratified random sampling scheme. The expression of bias and MSE of the proposed estimator have been obtained under large sample approximation, asymptotically optimum estimator (AOE) is identified with its approximate MSE formula. Estimator based on “estimated optimum values” was also investigated. Theoretical and empirical comparison of proposed estimator with some other ratio and product estimator justified the performance of the proposed estimator. There is a minimum of 15 percent reduction in the MSE from each of the existing ratio and product estimators considered. Thus most preferred over the existing estimators for the use in practical application.Keywords: bias, mean square error, auxiliary variable, optimum estimator, stratified random sampling, study variable

Generalized P-phased Regression Estimators with Single and Two Auxiliary Variables

Multiphase sampling has been the concept not being utilized is estimation of ratio and regression estimator widely. In the recent study we have proposed new dimension of sampling survey of estimations by proposing two generalized p-phase regression estimators with single and two auxiliary variables for estimating population mean. The proposed estimators are the generalized p-phase cases of Hanif et al (2015) and Hanif (2007) respectively. Both the estimators from which we took motivation are now special cases of our proposed estimators. We have derived unbiasedness, expression of Mean Square Errors along with family of estimators based upon p-phased generalization. We have derived expression of MSE in such a way that these expression can be used to obtained results for every phase we desire. By conducting empirical study on proposed estimators we have shown many situation in which MSE can be reduced by increasing number of phases. Hence, our study will open new horizon in the field of multiphase sampling where a lot of challenges are waiting to be resolved by proposing new estimators for phases above 2 nd phase.

Generalized Class of Estimators for Population Variance Using Information on Two Auxiliary Variables

Yasmeen et al. (Int J Appl Comput Math 1(2), 1–10, 2015) introduced an exponential ratio cum dual to ratio type of estimator for the population mean and compared the performance of the several estimators using the derived expression of MSE. But unfortunately, the expressions of bias and MSE of their proposed estimator \(\hbox {t}_{\mathrm{n}/2\mathrm{G}}\) is found to be incorrect and hence the comparison based on incorrect MSE, carried by them, is wrong. In the present article, the author provides the correct expressions of the bias and MSE of estimator proposed by Yasmeen et al. (Int J Appl Comput Math 1(2), 1–10, 2015), up to the first order of approximation and an attempt is made to provide an estimate for the population variance \(\hbox {S}_{\mathrm{y}}^2\) of the study variable y under SRWOR using information on two auxiliary variables x and z. Bias and MSE expressions have been derived up to the first order of approximation. An empirical study is carried out in the support of theoretical results.

Modified chain regression type estimator for population mean in the presence of non- response

A modified chain regression type estimator for population mean in the presence of non-response have been proposed replacing Hansen & Hurwitz (1946) estimator for population mean by Searls (1964) type improved estimator and using Hansen & Hurwitz (1946) estimator for based on available information comparing to the study character in the second phase sample. The expressions for MSE for fixed sample size  and also fixed cost  have been obtained. The empirical studies show that the proposed estimator is more efficient than the relevant estimators in the case of fixed sample size as well as for fixed cost.

A Note on Generalized Exponential Type Estimator for Population Variance in Survey Sampling

<p>Recently a new generalized estimator for population variance using information on the auxiliary variable has been introduced by Asghar, Sanaullah & Hanif (2014). In that paper there was some inaccuracy in the bias and MSE expressions. In this paper, we provide the correct expressions for bias and MSE of the Asghar et al. (2014) estimator, up to the first order of approximation. We also propose a new generalized exponential type estimator for population variance which performs better than the existing estimators. Four data sets are used for numerical comparison of efficiencies.</p>

The Decision Delay in Finite-Length MMSE–DFE Systems

This paper investigates the performance of the minimum mean square error (MMSE) decision feedback equalization system and optimizes its key parameter, decision delay, to minimize the system MSE. For the first typical scenario where the feedback filter length $$N_b$$Nb is larger than or equal to the channel order $$v$$v, we present an analytical expression of the system MSE in terms of decision delay. A few results in previous literature can be regarded as special cases of this expression. For the second typical scenario where feedback filter length $$N_b$$Nb is less than the channel order $$v$$v, it is difficult to obtain the explicit solutions. We prove that at a given decision delay, the MSE for $$N_b< v$$Nb

Robust Multiple-Antenna Cooperative Spectrum Sharing Design With Random Vector Quantization

In this paper, we propose cognitive overlay transceiver designs, where a primary transceiver pair and a secondary transceiver pair coexist in a network, and the primary user (PU) allows the secondary user (SU) to concurrently transmit its signals at the price of reducing the power of the PU's signal relayed by cooperative amplify-and-forward (AF). Since the considered transceiver design is mainly to devise the precoders both for the PU and the SU at the secondary transmitter (ST), the channel state information (CSI) has to be known at the ST. We therefore consider the limited feedback scheme with random vector quantization (RVQ), where the ST can only know the quantized channel direction information (CDI). Considering the statistics of the CSI quantization error and the linear minimum mean square error (LMMSE) receiver, we derive the closed-form MSE expressions corresponding to the PU and the SU. With the derived MSEs, we propose two robust design criteria. One criterion is to minimize the ST's power consumption under the constraint that the PU's and SU's quality-of-service (QoS; i.e., MSE) can be met. The other criterion is to minimize the SU's MSE when the PU's QoS can be controlled under a certain value and the ST satisfies the limitation of its transmission power consumption. Both the optimization problems of the proposed design criteria are not convex, and the corresponding solutions cannot be directly obtained. We then propose transfering the original optimization problems into two subproblems, where each of them is eventually formulated as a convex optimization problem, and the solutions are iteratively obtained, which is effective. Thus, the results can be obtained with the interior-point method. Simulations certify the robustness of our designs.

Nonparametric Estimation of Volatility Function with Variable Bandwidth Parameter

In this article, we introduce a new method for the volatility function estimation of continuous-time diffusion process dX t = μ(X t )dt + σ(X t )dW t , which is based on combining the idea of local linear smoother and variable bandwidth. We give the expressions for the conditional MSE and MISE of the estimator and obtain the optimal variable bandwidth. An explicit formula for the optimal variable bandwidth is presented by minimizing the MISE, which extends the related results in Fan and Gijbels (1992), etc. Finally, some simulations show that the performance of the proposed estimator with optimal variable bandwidth is often much better than that of the local linear estimator with invariable bandwidth.

Simple regression in view of elliptical models

For the simple linear model Y=θ1+βx+ϵ where the error vector follows the elliptically contoured distribution, we consider the unrestricted, restricted, preliminary test and shrinkage estimators for the intercept parameter, θ when it is suspected that the slope parameter β may be βo. The exact bias and MSE expressions are derived and the mean-square relative efficiency is taken to determine the relative dominance properties of the proposed estimators in comparison. In the continuation, the optimal level of significance of the preliminary test estimator is tabulated and some graphical result are also displayed.