Correlation Coefficient Revisited

Elementary approaches to prove basic properties of the correlation coefficient are of pedagogical interest. Besides posing another proof, this article gives variations of the proofs already existing in the statistical literature     

Correlation,Regression Lines,and Moments of Inertia

The R language, a freely available environment for statistical computing and graphics is widely used in many fields. This “expert-friendly” system has a powerful command language and programming environment, combined with an active user community. We discuss how R is ideal as a platform to support experimentation in mathematical statistics, both at the undergraduate and graduate levels. Using a series of case studies and activities, we describe how R can be used in a mathematical statistics course as a toolbox for experimentation. Examples include the calculation of a running average, maximization of a nonlinear function, resampling of a statistic, simple Bayesian modeling, sampling from multivariate normal, and estimation of power. These activities, often requiring only a few dozen lines of code, offer students the opportunity to explore statistical concepts and experiment. In addition, they provide an introduction to the framework and idioms available in this rich environment.     

An Improved Boxplot for Univariate Data

The boxplot is an effective data-visualization tool useful in diverse applications and disciplines. Although more sophisticated graphical methods exist, the boxplot remains relevant due to its simplicity, interpretability, and usefulness, even in the age of big data. This article highlights the origins and developments of the boxplot that is now widely viewed as an industry standard as well as its inherent limitations when dealing with data from skewed distributions, particularly when detecting outliers. The proposed Ratio-Skewed boxplot is shown to be practical and suitable for outlier labeling across several parametric distributions.     

Pmc-optimal nonparametric quantile estimator

According to Pitman's Measure of Closeness, if T₁and T₂are two estimators of a real parameter $[d], then T₁is better than T₂if Po[d]{T₁-o[d] < T₂-0[d]} > 1/2 for all 0[d]. It may however happen that while T₁is better than T₂and T₂is better than T₃, T₃is better than T₁. Given q ? (0,1) and a sample X₁, X₂, ..., X_nfrom an unknown F ? F, an estimator T* = T*(X₁,X₂...X_n)of the q-th quantile of the distribution F is constructed such that P_F{F(T*)-q <[d] F(T)-q} >[d] 1/2 for all F?F and for all T€T, where F is a nonparametric family of distributions and T is a class of estimators. It is shown that T* =X_j:n'for a suitably chosen jth order statistic.     

Data-dependent probability matching priors of the second order

In this paper, we consider the preliminary test approach for the estimation of the regression parameter in a multiple regression model under a multicollinearity situation. The preliminary test two-parameter estimators based on the Wald (W), likelihood ratio, and Lagrangian multiplier tests are given, when it is suspected that the regression parameter may be restricted to a subspace and the regression error is distributed with multivariate Student's t distribution. The bias and mean square error of the proposed estimators are derived and compared. The conditions of superiority of the proposed estimators are obtained. Finally, we conclude that the optimum choice of the level of significance becomes the traditional choice by using the Wald test.     

Robust methods of estimation of regression coefficients 1

Recently, cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon's entropy (see Rao et al. [Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory. 50(6) (2004), pp. 1220–1228] and Asadi and Zohrevand [On the dynamic cumulative residual entropy, J. Stat. Plann. Inference 137 (2007), pp. 1931–1941]). Motivated by this finding, in this paper, we introduce a generalized measure of it, namely cumulative residual Renyi's entropy, and study its properties. We also examine it in relation to some applied problems such as weighted and equilibrium models. Finally, we extend this measure into the bivariate set-up and prove certain characterizing relationships to identify different bivariate lifetime models.     

Importance of the prior mass for agreement between frequentist and Bayesian approaches in the two-sided test

We study distributional properties of generalized order statistics (gos) related by a random shift or scaling scheme in the continuous and discrete case, respectively. In the continuous case, we obtain new characterizations of distributions relating non-neighbouring gos extending some results given in the literature for the neighbouring cases. On the other hand, in the discrete case, we investigate the existence and uniqueness of a discrete parent distribution supported on the integers whose gos are related by a random translation.     

Optimal orthogonal block designs for four-mixture components in two blocks based on F-squares for Becker's models and K-model

Orthogonal block designs in mixture experiments have been extensively studied by various authors. Aggarwal et al. [M.L. Aggarwal, P. Singh, V. Sarin, and B. Husain, Mixture designs in orthogonal blocks using F-squares, METRON – Int. J. Statist. LXVII(2) (2009), pp. 105–128] considered the case of components assuming the same volume fractions and obtained mixture designs in orthogonal blocks using F-squares. In this paper, we have used the class of designs presented by Aggarwal et al. and have obtained D-, A- and E-optimal orthogonal block designs for four components in two blocks for Becker's mixture models and K-model, respectively. Orthogonality conditions for the considered models are also given.     

Estimation of the Memory Parameters of the Fractionally Integrated Separable Spatial Autoregressive (FISSAR(1, 1)) Model: A Simulation Study

In this article, we implement the Regression Method for estimating (d ₁, d ₂) of the FISSAR(1, 1) model. It is also possible to estimate d ₁ and d ₂ by Whittle's method. We also compute the estimated bias, standard error, and root mean square error by a simulation study. A comparison was made between the Regression Method of estimating d ₁ and d ₂ to that of the Whittle's method. It was found in this simulation study that the Regression Method of estimation was better when compare with the Whittle's estimator, in the sense that it had smaller root mean square errors (RMSE) values.