Expressions For Mean Square Error Research Articles

Variance estimation using memory type estimators based on EWMA statistic for time scaled surveys in stratified sampling

In this article, we have proposed memory-type exponential and non-exponential estimators for population variance based on exponentially weighted moving average (EWMA) statistic in stratified sampling. We drive mathematical expressions for mean square errors of the proposed estimators using Taylor and exponential expansions. Our analysis demonstrates that the proposed memory-type estimators outperform the conventional estimators under stratification, particularly, when the information of previous sample is utilized. The performances of the proposed estimators are evaluated mathematically by deriving the conditions in which the memory-type estimators would perform better than their corresponding conventional estimators under stratification. Through an extensive simulation, we evaluated the performance of the proposed estimators across various population parameters, revealing their enhanced efficiency in time-scaled survey. Additionally, a real data application is also used to support the mathematical findings, confirming the practical utility of the proposed estimators. The results of numerical study underscore the importance of the use of previous sampled information which significantly improves the accuracy and reliability of the proposed estimator for variance estimation for time-scaled surveys.

Underwater maneuvering target motion analysis using delayed azimuth angles

Accurate position information of an underwater target is the premise of successfully executing underwater missions, and this paper focuses on underwater target motion analysis (TMA) using delayed azimuth angles. Considering that the underwater sound speed (around 1500 m/s) is much smaller than the speed of light (3 × 108 m/s), the underwater TMA needs to adopt the non-stop model considering the effect of the target movement during signal propagation on state estimation, instead of the traditional go-stop model ignoring the target movement during signal propagation. The existing non-stop model is based on a constant-velocity target, and the model mismatch will make the estimated results biased when it is applied to a maneuvering target. In this paper, we focus on the constant-acceleration (CA) target and propose two methods to solve this problem. One is to extend the non-stop model to the case of a constant-acceleration target. We build the corresponding least squares (LS) estimator and derive the expression of the root mean square error (RMSE) of the LS solution. The other is to directly offset the estimation bias of the CA go-stop model and a bias-compensated algorithm is proposed to improve the performance. Simulations prove that the proposed methods can result in a smaller RMSE than the traditional CA go-stop model. Their RMSEs decrease with the number of measurements and observers and increase with random angle errors. When the number of measurements is small, the RMSE of the established CA non-stop model is smaller than that of the bias-compensated algorithm. As the number of measurements increases, their estimation precisions become gradually equal. The running time of the proposed methods increases with the number of measurements and observers. When the number of observers is small, the bias-compensated algorithm has the advantage of needing less running time. In addition, the running time of the bias-compensated algorithm does not approximately change with the random angle errors, while that of the CA non-stop model increases with the random angle errors.

Enhancing Precision in Population Variance Vector Estimation: A Two-Phase Sampling Approach with Multi-Auxiliary Information

To enhance precision in estimating unknown population parameters, an auxiliary variable is often used. However, in scenarios where required information on an auxiliary variable is partially or fully unavailable, two-phase sampling is commonly employed. The challenge of estimating the variance vector using multi-auxiliary variables is a less explored area in current literature. This paper addresses the estimation of vector of unknown population variances for multiple study variables by using an estimated vector of variances derived from multi-auxiliary information. This approach is particularly relevant when population variances for the multi-auxiliary variables are not known prior to the survey. The paper introduces a generalized variance and a vector of biases for the proposed multivariate estimator. Special cases of the proposed multivariate variance estimator are provided, accompanied by expressions for mean square errors. Theoretical mathematical conditions are discussed to guide the preference for the proposed estimator. Through the analysis of real-world application-based data, the applicability and efficiency of the proposed multivariate variance estimator are demonstrated, outperforming modified versions of multivariate variance estimators. Additionally, a simulation study validates the superior performance of the proposed estimator compared to its modified estimators.

Enhanced direct and synthetic estimators for domain mean with simulation and applications

This article considers the issue of domain mean estimation utilizing bivariate auxiliary information based enhanced direct and synthetic logarithmic type estimators under simple random sampling (SRS). The expressions of mean square error (MSE) of the proposed estimators are provided to the 1st order approximation. The efficiency criteria are derived to exhibit the dominance of the proposed estimators. To exemplify the theoretical results, a simulation study is conducted on a hypothetically drawn trivariate normal population from R programming language. Some applications of the suggested methods are also provided by analyzing the actual data from the municipalities of Sweden and acreage of paddy crop in the Mohanlal Ganj tehsil of the Indian state of Uttar Pradesh. The findings of the simulation and real data application exhibit that the proposed direct and synthetic logarithmic estimators dominate the conventional direct and synthetic mean, ratio, and logarithmic estimators in terms of least MSE and highest percent relative efficiency.

Precision enhancement in variance estimation for complex environmental populations using adaptive cluster sampling

Estimating dispersion in populations that are extremely rare, hidden, geographically clustered, and hard to access is a well-known challenge. Conventional sampling approaches tend to overestimate the variance, even though it should be genuinely reduced. In this environment, adaptive cluster sampling is considered to be the most efficient sampling technique as it provides generally a lower variance than the other conventional probability sampling designs for the assessment of rare and geographically gathered population parameters like mean, total, variance, etc. The use of auxiliary data is very common to obtain the precise estimates of the estimators by taking advantage of the correlation between the survey variable and the auxiliary data. In this article, we introduced a generalized estimator for estimating the variance of populations that are rare, hidden, geographically clustered and hard-to-reached. The proposed estimator leverages both actual and transformed auxiliary data through adaptive cluster sampling. The expressions of approximate bias and mean square error of the proposed estimator are derived up to the first-order approximation using Taylor expansion. Some special cases are also obtained using the known parameters associated with the auxiliary variable. The proposed class of estimators is compared with available estimators using simulation and real data applications.

Refined batch of Estimators for Estimating Population Mean with the help of known Auxiliary Parameters

In the presented work, we have instituted a batch of ratio type estimators for estimating the population mean under simple random sampling with the help of known auxiliary variables or function of auxiliary variables, for more refined result. The expression of bias and mean square error of the instituted class are induced up to the first order of approximation. The strength of the proposed batch of estimators is compared with some existing estimators in terms of MSEs, first theoretically and then a numerical study is also carried out by using real data set to support the findings. It is concluded that the proposed class outperforms the various prevailing estimators in terms of minimum MSE and higher percentage relative efficiency.. KEYWORDS :Estimator, Mean, Bias, Mean square error, Percentage relative efficiency, Auxiliary information.

Efficient Classes of Robust Ratio Type Estimators of Mean and Variance in Adaptive Cluster Sampling

This paper proposes two classes of robust ratio type estimators of finite population mean and two classes of robust ratio type estimators of finite population variance using a single auxiliary variable under the adaptive cluster sampling design. Seven robust ratio type estimators have been developed from each class. The generalized expressions of bias and mean square error for each class have been derived up to the first order of approximation. The exact expressions of bias and MSE for all the developed estimators have been presented. Using simulation studies conducted in R programming, the high efficiency of all the developed estimators over similar existing estimators in adaptive cluster sampling has been demonstrated. . KEYWORDS :Adaptive cluster sampling, Classes of robust ratio type estimators, Hidden clustered data, Estimation of mean, Estimation of variance.

Efficient Class of Variance Estimators for Population using Supplementary Information in Stratified Random Sampling

This paper addresses an efficient class of variance estimators for population using stratified random sampling. The suggested class of estimators using supplementary information has been studied in different circumstances. The expressions of bias and mean square error (MSE) of the proposed estimators are derived up to the first degree of approximation. The theoretical comparison of the proposed and considered estimators is also discussed, which shows that the proposed estimators are more efficient than the existing estimators. Theoretical findings are validated by three different types of real data sets and simulation studies. The numerical results of the proposed and existing estimators are compared in terms of mean square error, percentage relative efficiency and diagrams. It is observed that all the proposed estimators outperform the existing estimators. For instance, the traditional unbiased estimator Ozel et.al [6] and other existing estimators. Lastly, appropriate recommendations have been provided for researchers to use these suggested estimators to solve real-world issues.

Almost Unbiased Optimum Ratio-type Estimator for Estimating Population Mean in Stratified Sampling in Presence of Non-response

In this paper we have proposed classes of ratio-type estimators for finite population mean in presence of non-response in stratified sampling. The properties of the estimators have been discussed. We have also derived optimum choices of scalar constant to reduce the bias in the estimators which make suggested classes of estimators almost unbiased. The expressions of mean square error (MSE) have been derived up to the first order of approximation. We have compared our proposed classes of estimators with natural ratio estimators and with estimator under complete response. Results are also supported by empirical study.

Estimation of heterogeneous population variance using memory-type estimators based on EWMA statistic in the presence of measurement error for time-scaled surveys

In this present article, we have suggested memory-type ratio, exponential ratio, product and exponential product estimators based on exponentially weighted moving average statistic for the estimation of heterogeneous population variance using stratified sampling design in presence of measurement error for time-scaled surveys. Mathematical expressions of approximate mean square error are derived using Taylor and exponential expansions for the proposed memory-type estimators. We have also discussed the situations in which the memory-type estimators would perform efficiently than the conventional estimators. The results of extensive simulation study revealed that the proposed memory-type estimators based exponentially weighted moving average statistic would perform better than the conventional estimators in the presence measurement error for time-scaled surveys under certain condition.

Calibration estimator for a sensitive variable using dual auxiliary information under measurement errors

In survey research, it is commonly assumed that all the observations are measured accurately. However, in practice, this assumption is not achieved due to many reasons, so it causes measurement errors to be inevitably present in the sample estimation. This article addresses the problem of estimation of finite population mean under stratified random sampling in the presence of measurement errors. A calibration estimator is proposed for the sensitive variable by utilizing suitable calibration techniques, which can be applied to obtain the optimal strata weight. The calibration estimator not only incorporates the auxiliary information but also the ranks of the auxiliary variable. Expressions for bias and mean square error are derived up to first order of approximation. Simulation studies and real-life datasets are used to assess the performances of the proposed estimator by comparing them with the contemporary estimators in the presence and absence of measurement errors. The theoretical and empirical studies demonstrate that the proposed calibration estimator consistently outperforms classical mean, usual ratio, conventional difference, Khail randomized response estimators.

Improved exponential type mean estimators for non-response case using two concomitant variables in simple random sampling

This paper addresses new exponential estimators for population mean in case of non-response on both the study and the concomitant variables using simple random sampling. The expressions for theoretical bias and mean square error of new estimators are derived up to first-order approximation and comparisons are made with the existing estimators. The proposed estimators are observed more efficient as compared to the considered estimators in the literature. For instance, the classical [4] unbiased estimator, the estimator of [9], and other existing estimators under the explained conditions. The theoretical results are supported numerically by using real-life data sets, under the criteria of bias, mean square error, percent relative efficiency and mathematical conditions. It is also clear from the numerical results that the suggested exponential estimators performed better than the estimators in the literature.

Ln-type estimators for the estimation of the population mean of a sensitive study variable using auxiliary information

In this article, we offered two ln-type estimators for the population mean estimation of a sensitive study variable by using the auxiliary information under the design of basic probability sampling. The Taylor and log series were used to derive the expressions of mean square error and bias up to the first order. Improved classes of proposed estimators are obtained by using conventional parameters associated with the supplementary variable to obtained precise estimates. Mathematical comparisons of the estimators have been made with the usual mean and ratio estimators using theoretical equations of mean square error. A simulation study is conducted for the evaluation of proposed estimator's implementation using four artificial populations generated through R-software with different choices of mean vectors and variance-covariance matrices. The demonstration of proposed ln-type estimators was implemented through the real data application.

Improved Generalized Synthetic Estimator for Domain Mean

In this paper, we have proposed an estimator for domain mean using auxiliary character when domain mean of the auxiliary character is unknown. The some members of the proposed estimator for domain mean using auxiliary character with unknown domain mean are studied. The expressions for bias and mean square error of the proposed estimator have been studied. The value of the minimum mean square error of the proposed estimator has been explained with the relevant estimators. The proposed estimator is found more efficient than the relevant estimators. The simulation study is also obtained in the terms of an absolute relative bias and simulated relative standard error using the data (Sarndal et al. (1992, Appendix B)). The simulation study shows that the proposed estimator is more efficient than the relevant estimators for those domains for which the synthetic assumption of the proposed estimator meet closely.

Enhancing mean estimators in median ranked set sampling with dual auxiliary information

When measuring the research variable is complicated, expensive, or problematic, median ranked set sampling (MRSS) is often utilized since it is straightforward to rank the components using a low-cost sorting criterion. Using this sampling scheme, many authors considered the problem of population mean estimation with a single auxiliary variable in order to obtain more precised estimators than the traditional ratio type regression estimators. In this article, we extend their ideas based on regression approach using two auxiliary variables and introduce a new regression-type estimator along with its theoretical expression of minimum mean square error (MSE). The suggested estimator's applicability is demonstrated using both simulated and real-world data sets.

Smoothing of Estimators of Population mean using Calibration Technique with Sample Errors

The challenges bedeviling the performance of estimators of population parameters in survey samples as a result of measurement and nonresponse errors are of great concern to researchers and users of statistics. This study suggests new estimators and adopts the calibration approach in smoothing the existing and proposed estimators for optimal performance. We have proposed improved estimators for estimating the finite population mean under stratified random sampling in three different situations: first, the properties of the estimators are considered under nonresponse, then the study of the estimators for measurement errors and in the last case, the estimators are examined in the presence of both measurement and nonresponse errors simultaneously. Expressions for mean square errors are obtained for the suggested estimators. Empirical study has been carried out with two real datasets to validate the theoretical underpinnings of this study.

Chain ratio and product estimators for population mode using two-phase sampling scheme

Background: For skewed datasets, the mode is utilized as a more appropriate measure of location. We formed chain ratio and product estimators for the population mode using two types of auxiliary information under the two-phase sampling scheme. Methods: Expressions for biases and mean square errors for the formed estimators up to the first order of approximation are obtained. The confidence intervals of the estimators are obtained and the aspects related to fixed cost and fixed variance are also studied. A simulation study is performed to support the theoretical outcomes. A real dataset is also provided which was collected from the department of agriculture, United States. Results: The simulation results for the fixed first-phase sample size and the second-phase sample size show that the mean square error is 0.035, the bias is 0.012, the confidence interval is 2.67-3.15, and the cost of the survey under the fixed variance is ₹ 194.67 of the proposed estimator , which is lower than 0.039, 0.055, 2.70-3.20, and ₹ 247.37 of the ratio estimator and 0.078, 0.041, 2.57-3.31, and ₹ 268.73 of the naive estimator . Conclusions: The simulation results show that the proposed estimator performs better than the ratio estimator and the naive estimator .

Efficient Classes of Estimators of Population Mean under Various Allocation Schemes in Stratified Random Sampling

The present paper is an extension of the work published in Kumar and Vishwakarma (Proceedings of theNational Academy of Sciences, India, Section A: Physical Sciences, 90(5): 933-939, 2020). In this paper,various sample allocation schemes are utilized to derive the mathematical expressions for mean square errors(MSEs) of several well-known estimators of population mean in stratified random sampling. Moreover, theeffects of various allocation schemes on the estimation of mean, are demonstrated theoretically as well asempirically. The findings of the study reveal that the Neyman allocation provides a smaller variance (or MSE,as the case may be) as compared to that of Equal and Proportional allocation schemes for the concernedestimators. Moreover, the proposed classes of estimators are dominant over the pre-existing estimators underthe various allocation schemes considered in the study.

Some Improved Combined Estimators of Population Mean in Stratified Ranked Set Sampling

This paper presents improved population mean estimators using auxiliary variables in Stratified Ranked Set Sampling. We have derived the expressions for bias and mean square errors up to the first order of approximation and shown that the proposed estimators under optimum conditions are more efficient than other estimators taken in this paper. In an attempt to verify the efficiencies of proposed estimators, theoretical results are supported by numerical illustrations and simulation study for which we have considered two populations.

Performance of the modified clock skew estimator and its upper bound for the IEEE 1588v2 (PTP) case under packet loss and fractional Gaussian noise environment

Precision Time Protocol (PTP) is a time protocol based on the Master and Slave exchanging messages with time stamps. In practical PTP systems, we have packet loss, a phenomenon where some of the PTP messages get lost in the Network. Packet loss may reduce the performance of the clock skew estimator from the mean square error (MSE) perspective. Recently, the same authors presented simulation results that show the clock skew performance of the three clock skew estimators (the two-way delay (TWD) clock skew estimator and the one-way delay (OWD) clock skew estimator for the Forward and Reverse paths) under the packet loss case in the fractional Gaussian noise (fGn) environment with Hurst exponent parameter (H) in the range of 0.5 ≤ H < 1, where indeed the clock skew performance was degraded compared to the non-packet loss case. Please note that for 0.5 < H < 1, the corresponding fGn is of long-range dependency (LRD). This paper proposes an algorithm that estimates the missing timestamps in the packet loss and fGn (0.5 ≤ H < 1) case. We use those estimates to generate three modified clock skew estimators (the two-way delay (TWD) modified clock skew estimator and the one-way delay (OWD) modified clock skew estimator for the Forward and Reverse paths) applicable to the packet loss, non-packet loss, and fGn (0.5 ≤ H < 1) case based on the same authors’ previously developed clock skew estimators. Those modified clock skew estimators led, based on simulation results, to an improved clock skew performance in the packet loss and fGn (0.5 ≤ H < 1) case compared with the authors’ previously developed clock skew estimators and those known from the literature (the ML-like (MLLE) and Kalman clock skew estimators). With the MSE expression, the system designer can know how many Sync periods are needed for the clock skew synchronization task to reach the system’s requirements from the MSE perspective. But no MSE expression exists in the literature for the packet loss case. In this paper, we derive closed-form approximated expressions for the MSE upper bounds related to the modified TWD and OWD clock skew estimators valid for the packet loss and fGn (0.5 ≤ H < 1) cases.