We focus on the problem of estimating the average vector $\text{\boldmath$\text{\boldmath$\theta$}$}= ( \theta_{1} ,\ldots,\theta_{d} )$ of a random vector $\mathbf{X} \in \mathbb{R}^{d }$ which follows a spherically symmetric distribution. We consider modified balanced loss functions of the form:\\$ \textbf{(i)}\ \ L_{\omega,\text{\boldmath$\text{\boldmath$\delta$}$}_0,\rho}(\text{\boldmath$\text{\boldmath$\delta$}$},\text{\boldmath$\text{\boldmath$\theta$}$})=\omega \rho(\|\text{\boldmath$\text{\boldmath$\delta$}$} - \text{\boldmath$\text{\boldmath$\delta$}$}_ {0 }\|^{2}) +(1-\omega)\rho(\|\text{\boldmath$\text{\boldmath$\delta$}$} - \text{\boldmath$\text{\boldmath$\theta$}$}\|^{2})\ \hbox{ and } \ \textbf{(ii)}\ \ \ell(\omega \ \|\text{\boldmath$\text{\boldmath$\delta$}$} - \text{\boldmath$\text{\boldmath$\delta$}$}_{0}\|^{2} +(1-\omega)\|\text{\boldmath$\text{\boldmath$\delta$}$}- \text{\boldmath$\text{\boldmath$\theta$}$}\|^{2}). $Here, $\text{\boldmath$\text{\boldmath$\delta$}$}_{0}$ represents a target estimator of $\text{\boldmath$\text{\boldmath$\theta$}$}$, $\omega \in [0,1]$ and $\rho$ and $\ell$ are increasing and concave functions. If $d \geq 4$ and the target estimator is $\text{\boldmath$\text{\boldmath$\delta$}$}_{0}(\mathbf{X})=\mathbf{X}$, we provide conditions on the parameter $a$ for Baranchik type estimators $\text{\boldmath$\text{\boldmath$\delta$}$}_{a,S} (\mathbf{X}) =\left(1-a(1-\omega) S(\| \mathbf{X}\|^2)/\|\mathbf{\mathbf{X}}\|^{2} \right)\mathbf{X}$ and reach the minimaxity. These conditions are derived using the radial properties of spherically symmetric distributions, which do not require the existence of a probability density for the random vector $\mathbf{X}$. Furthermore, we extend the obtained results to the case of robust shrinkage estimators of the form $\text{\boldmath$\delta$}_{\omega,g}(\mathbf{X})=\mathbf{X} + a(1-\omega)g(\mathbf{X})$, where $g(\cdot)$ is a weakly differentiable and satisfying some conditions. Additionally, we conduct a simulation study in order to show the effectiveness and the usefulness of the obtained results.

This paper considers the problem of estimating the population mean ybar of the study variate y using information on different parameters such as population mean (X), coefficient of variation (Cx), kurtosis (b2(x)), standard deviation (Sx) of the auxiliary variate x and on the correlation coefficient, r, between the study variate y and the auxiliary variate x through transformation. A class of estimators on the lines of Kadilar and Cingi (2003) has been defined and its properties are studied to the first degree of approximation. It has been shown that the proposed class of estimators is better than usual unbiased estimator y, ratio estimator y R, ratio-type estimator tR and Kadilar and Cingi (2003) estimator y C under some realistic conditions. Numerical illustration is given in support of the present study.

Introduction In this study, we addressed parameter estimation in the linear regression model in the presence of multicollinearity when there exists some prior information about predictor variables that appears as a linear restriction on the model parameters. We estimated the parameters based on Liu-type linear shrinkage, preliminary test, Stein, and positive Stein strategies. The performance of the proposed estimators is compared to the Liu-type estimator in terms of their relative efficiency via a Monte Carlo simulation study and an actual data set. Material and Methods In the linear regression model, the ordinary least squares (OLS) estimator is the best linear unbiased estimator for model parameters when the predictor variables are independent. The multicollinearity problem arises when there exists near linear dependence among the predictor variables. This problem leads to variance inflation of the OLS estimator. Thus the interpretations based on it are not true. The ridge and Liu-type estimators are two methods to combat multicollinearity. The Liu-type estimator is more efficient than the ridge estimator when there is a strong correlation between the predictor variables. We suppose that there is some prior information about parameter vector ,under a linear restriction as R,= r where R is a p2 ,p matrix and r is a p2 ,1 vector. The restricted estimator of ,is obtained by maximizing the log-likelihood function of the linear regression model under the linear restriction. The Liu-type restricted estimator can be defined in the presence of multicollinearity under the linear restriction. We propose the Liu-type shrinkage estimators using the Liu-type and Liu-type restricted estimators to improve the estimation of parameters. We compare the performance of the Liu-type shrinkage estimators and the Liu-type estimator in terms of their relative efficiency using a Monte Carlo simulation study. The simulation is conducted under different sample sizes, n = 30,50, the correlation level between the predictor variables ,= 0: 80,0: 90,0: 95, p1 = 5, and p2 = 3,5,7. To investigate the behavior of the proposed estimators, we define Δ,= ∥, ,􀀀, , 0∥, 2, where ∥, : ∥,is the Euclidean norm, ,is the parameters vector in the simulated model and , 0 is the true parameters vector in the candidate sub-model. We also apply the proposed estimation methods to a real data set. Results and Discussion The simulation results show that all estimators’,performances become better when p2 and ,increase for fixed n. For all combinations of p2, , , and n, the Liu-type restricted estimator has the best performance at Δ,= 0. As Δ,moves away from zero, all estimators’,simulated relative efficiencies (SREs) decrease. As ,approaches one, the performance of the Liu-type linear shrinkage estimator increases. Conclusion This paper suggested the Liu-type shrinkage estimators in the linear regression model in the presence of multicollinearity under the subspace information. A Monte Carlo simulation was conducted to compare the proposed estimators’,performance with the Liu-type estimator. The simulation results confirm that the proposed estimators perform better than the Liu-type estimator when Δ,= 0 and near it for all p2, , , and n.

In survey sampling, most of the research work based on the fact that utilizing the information of auxiliary variable(s) boosts the efficiency of estimators. Keeping this fact in mind we used the information of two auxiliary variables to propose a family of Hartley-Ross type unbiased estimators for estimating population mean under simple random sampling without replacement. Minimum variance of the new family is derived up to first order of approximation. Three real data sets are used to verify that the new family acts efficiently than the usual unbiased, Hartley and Ross (1954), Grover and Kaur (2014), Singh et al. (2014), Cekim and Kadilar (2016), Muneer et al. (2017) and Shabbir and Gupta (2017) estimators.

Auxiliary information plays a vital role in parameter selection and estimation to achieve efficient estimates of unknown population parameters. Dual use of auxiliary information, i. e., \original" and \ranked" auxiliary variables, helps increase the e, ciency of estimators. In this paper, the performance of di, erence-type-exponential estimators was proposed and evaluated based on dual auxiliary information for population mean under simple random sampling. Mathematical expressions for the bias and the mean squared error of the proposed estimators were obtained. Three real-life data sets and Monte Carlo simulation studies were carried out for illustration. The results of empirical and simulation studies indicate that the proposed estimators outperformed their counterparts in terms of mean square errors and percentage relative efficiency.

Auxiliary information plays a vital role at the selection and/or estimation stage to achieve the efficient estimates of the unknown population parameters. Dual use of auxiliary information, one the original and second the ranks of the auxiliary variable help to increase the efficiency of the estimators. In this article, we proposed and evaluated the performance of difference-type-exponential estimators based on dual auxiliary information for population mean under simple random sampling. Mathematical expressions for the bias and the mean squared error of the proposed estimators are obtained. Three real-life data sets and Monte Carlo simulation studies are carried out for illustration. The results of the empirical and the simulation studies, in terms of mean square errors and percentage relative efficiencies indicate that the proposed estimators perform better as compared to their counterparts.

In this paper, an improved ratio-type class of estimators in the two-phase Adaptive Cluster Sampling (ACS) design under the transformed population approach has been proposed. The generalized expressions of Bias and Mean Squared Error (MSE) have been obtained up to the first order of approximation. New member estimators are developed from the proposed class, and their performance against competing existing estimators is evaluated using various empirical studies. The novelty of this design and the new estimators developed therein is further demonstrated using a real data study where the newly developed estimators are used to estimate the average number of thorny plants in plateaus of Western Ghats of Sahyadri from Goa to Varandha Ghat (Bhor, Maharashtra, India).

Several auxiliary information-based estimators of population variance are available in the existing literature on survey sampling. Mostly, these estimators are based on conventional dispersion measures of the auxiliary variable. In this study, a generalized class of ratio-product type exponential estimators of the population variance is proposed by integrating the nonconventional auxiliary information under Simple Random Sampling (SRS). The performance of the proposed estimators was compared, theoretically and numerically, with several existing estimators of the population variance. It was established that the proposed class of estimators outperformed the existing estimators in terms of Mean Squared Error (MSE) and Relative Root Mean Square Error (RRMSE). Moreover, Percentage Relative Efficiency (PRE) of the proposed estimators was much higher than that of their counterparts.

Traditional Ordinary Least Square (OLS) regression is commonly utilized to develop regression-ratio-type estimators through traditional measurement of location. Abid et al. [Abid, M., Abbas, N., Zafar Nazir, H., et al. \Enhancing the mean ratio estimators for estimating population mean using non-conventional location parameters", Revista Colombiana de Estadistica, 39(1), pp. 63{79 (2016b)], extended this idea and developed regression-ratio-type estimators based on traditional and non-traditional measures of location. In this article, the quantile regression with traditional and non-traditional measures of location is utilized and a class of ratio type mean estimators is proposed. The theoretical Mean Square Error (MSE) expressions are also derived. The work is also extended to two-phase sampling (partial information). The relationship between the proposed and existing groups of estimators is shown by considering real data collections originating from different sources. The discoveries are empowering and prevalent execution of the proposed group of the estimators is witnessed and documented throughout the article.

A control chart is an important tool in statistical process control that plays a signi cant role in monitoring and identifying variations in production processes. The Shewhart, the cumulative sum (CUSUM), and the ExponentiallyWeighted Moving Average (EWMA) control charts are commonly used for detecting process shifts. The CUSUM and the EWMA control charts are more sensitive in detecting smaller shifts, whereas the typical Shewhart chart is sensitive to large process shifts. The present study incorporates ratiotype estimators of the population mean based on auxiliary information in the CUSUM charting structure for monitoring the location of the normal processes. These estimators are more e cient than simple mean estimator in the presence of a high correlation between the study and the auxiliary variables. The Average Run Length (ARL), the standard deviation of run length, and the extra quadratic loss are used to measure the performance of the proposed charts. The proposed charts are compared with the existing CUSUM, CUSUM-FIR, and some other auxiliary information-based control charts on the basis of out-of-control ARLs. The comparison reveals the superiority of the suggested charts over the other existing charts. An illustrative example is also provided for the performance evaluation of the proposed charts.