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To extend existing distance metrics in the $L^{p}$ space, we define a novel distance, named relative Euclidean distance (RED), and prove that it is positive definite, symmetric, and satisfies the triangle inequality. This distance has a value range of [0, 1]. We prove a uniqueness of it in the $L^{p}$ space. The technique for order preference by similarity to ideal solution (TOPSIS) is widely used for multiple-attribute decision problems. The commonly used relative closeness measure in the TOPSIS is not a mathematical distance and thus does not have all the nice properties of such a distance. To improve the TOPSIS, particularly its relative closeness measure, we propose two measures, named optimistic distance (OD) and pessimistic distance (PD), based on our RED, and use them to measure the relative closeness in the TOPSIS. Both measures are positive definite, symmetric, and satisfy the triangle inequality. The best choice in OD (or PD) is on the Pareto frontier. Three examples of performance ranking are given to illustrate the usefulness of the TOPSIS with our proposed measures. [ABSTRACT FROM AUTHOR]

Given multiple statistical estimators, how to rank their performance is a worthwhile problem. We propose an approach to estimation performance ranking based on pairwise comparison. Because ranking is about two or more estimators and is relative, pairwise comparison (e.g., Pitman’s measure of closeness) suits the needs. However, pairwise comparison cannot guarantee transitivity, which is needed for ranking. To get around this problem, we propose a ranking vector (RV) approach based on pairwise comparison. Here, an RV is obtained by using pairwise comparison results without using pairwise ranks directly. An RV provides ordinal information determining the rank and also supplementary cardinal information exhibiting how much one estimator is better than another. Ordinal information is more important and thus is guaranteed by using an order-preserving mapping in obtaining an RV. Our RV approach based on pairwise comparison is also applied to multiple-attribute decision problems. The approach is easily applicable and it does not need data normalization. [ABSTRACT FROM AUTHOR]

If a data set has a large range (e.g., the large elements are several orders of magnitude greater than the small elements), then the median is usually applied to measure its central tendency. However, it has two drawbacks. A novel measure of central tendency called iterative mid-range (IMR) is proposed. It has several attractive properties and can overcome the drawbacks of the median. Estimation performance is often evaluated in a statistical sense by the Monte Carlo method. Given a set of estimation errors, estimation performance is evaluated by measures of central tendency of error, that is, by finding a typical value (e.g., root-mean-square error) to represent the errors. The proposed IMR is applied to estimation performance evaluation, and it is named IMR error (IMRE). This letter advocates replacing the median by our proposed IMR in many cases. [ABSTRACT FROM PUBLISHER]